MOAvnb 1 . CI4CTEMATU3AU|9 TA ySArAIbHEHH9..,' onrng I METoAIB nnlxltvtsrpli

$ 1 .1 . Ilpo norirry no6ygoey nnaxiruerpii

1.1.8, f.

1.2,3 rprox roqox Ha npsMift ogna i rinrxa oAna Jrerr{r{rb mix 4eoua inruzuu (axc. II,).flpra qrolry Moxrnsi Tpt4 Br?rnagxrz BBaeMHoro poslrirqeuus AaHsx roqox Ha rlpg-uifi.1) TouraA rerrtr,rrb milrc B i C, roAi aa arc. III, BC =AB +-l AC, are 1,38 4rvr + 2,72 aw + 1,34 gM. Touy rare poe-uiuqenna HeMorrurrrBe.

2) Toura B .nesczrr uim A i C. AC = AB t BC, ane I,34 ptr +* 2,72 gM, + 1,38 ,qlr. Taxe posrrirqeHHs rorrox rarcorr{ He-MOXInBe.

3) Togi eAnne Mo*cJrr,rBe B3aeMHe poauirqenna rpbox roqorA, B i C raxe: Toqra C .nerrtr.rrr irdnc A i B. fleperiplrmo: AB == AC + BC, 2,72 sNr = 1,38 ,qM -l 1,34 4ttt; 2,72 +nt = 2,72 arr.BiOnoailu: ro.{xa C JreJrcr{rb nri:x roqxauu A i B.

1.3.A, A.

1.4.8, r.

1.S.Ocri;rsxu ea yuonoro M - nnyrpiurnn rovrca ryra.AOB,ro nporrnirr, OM npoxogurt mirx croponaula xyra AOB.Hexafi ZBOM = ,t,.ro+i IAOM = 2x.3a aftc. III2 ZAOB = ZAOM + ZBOM,'l5O = 2x +.r, 150 = 3t, * = 50..BOM = 50", .AOM - 50' . 2 = 100'.Biinoaido: E) 100'.

1.6.3a axc. TII,AB = AM + MB. AM = x cm, BM = (r + 3) cru.

12 = x 1- (r + 3); 72 -- 2x + 3; 2x = 9; x = 4,5.AM = 4,5 ctr; MB = 4,5 + 3 = 7,5 (cu).BiOnoai1o: B) 4,5 cu n 7,5 cu.

I)) AM = x cm, BM = (r + 6) cnn.

x'f (x ): 6) = 42; 2x -l 6 = 42; 2x = 36; r = 18.AM = la cw, BM = 18 + 6 = 24 (cur).ll) AM = x cttr, BM = 2x cr,tt.

x * 2x = 42i 3x = 42; x = 42 : 3; x = 14.AM =14clu, BM =2.L4 = 28 (cu).f) AM = 3x ctttt, BM = 4x c:lr.3x-t 4x = 42;7x = 42; x ='6.AM = 6. 3 = 18 cltr, BM = 4 - 6 = 24 (cwt).

A) AM = x cm, BM = 2x ctl.x t 2x = 42; 3x = 42; x = 42 : 3; t = 14.AM = 14 cw, BM = 2 . 74= 28 (cm).

1.9.3a axciouonIII,: IPOM = IPOA + ZAOM.A\ IAOM = xi ZPOA= x * 4O;rc + xc + 40 = 160; 2t + 40 = 16O; 2x = 120; x = 6O.IAOM = 6O'; IPOA. = 60' + 40' = 100'.E)ZAOM=rciLPOA=x-60:r't x - 6O = 16O; 2x - 60 = 160; 2x -- 22O; r = 110.IAOM = ll0"; IPOA = 110' - 60'= 50'.B) ZAOD = rci ZAOM = O,6tcix * 0,6x= 160; 1,6* = 16O; x= 160 : 1,6; * = 100./AOP = LOO'; ZAOM = 0,6 . 100" = 60'.f\ IAOM = tci ZPOA = 3xlx I 3r = 160; 4x = 160; r = 40.IAOM = 40'; ZPOA= 3 . 40" = 120".I) ZAOM = 3x; ZPOA= 5x;

. 3r 't 5x = 160; 8r = t$Q; a = 20.lAoM = 9..20" = 6o"; ZPOA = 5.20" = 100",

1.1O. a)V e l; P e l; S e P8; W e Pqi B e l; <rouxa S.nexrarr nainr toqrcama P i 8>.6) R e m; C e mi B e MR; M e -41?; <rouxa X se.nemr,rrr uin< ro.rxanrra/. i M>.

l.11.OA=OB + AB; AB =OA- OB = 7,3 - 5,8 = 1,5 (cu). _----.--Ocxinsnrz ro.rxr,r A i B rexarr xa ognift ninnparriri o no- 6----;---T

X

rrarr(ciBoro ror{Horo o, ro sonra He poSAiJr,frorr,ca roqKoroO, ro6ro, ro.rxa O He Jrefirr,rrb rvrirx ro.*tarvru A i B. flrcu1o 6 ro.rxa A .nexa.na uintrorrr(aMLr B i O, ro BvxorryBaJracb 6ra pinnicrr OB = OA -l AB. A.te qe lreMorlcJrtlBo,6o sa ymonoro OA > OB. Orxe, rovra B .nexrarr Minc ro.rxaMr,r,4 i O.BiOnoeila: L,5 cm,

1.12, Iloscnesna gzr. N! 1.2.

l) AB = AM + BM; 8 = 3 + 5 - flpaBr.rJrbno. To.rxa M JreJrct4rb rvrinr ro.rxami,rA i B.2) AB=AM + BM;L9 =7 + 12 - npanu.nrno. To.rxaM relrcrarr rrrix roqrcaMl,IA i B.3) AM = AB + M B ; 27 = 22 + 5 - [paBrJrBrro. To-rxa M,nexrr,rrr rrrix< ro.rxandla A i B,4) BM = AB + AM ; 2L = I + 72 - npaBr.rrrbHo. To.rxa A aerxarr uinc ro.tcanrz B i M,5\ BM = AM + AB; 37 = 2l + 16 - npaBrrJrbno. To.rrca -d. .nencrzrr rvrinr ro.rxarvrra B i M.6) AM = AB + MBi 33 = 18 +'15 - rpaBr,rrrbHo. Toqxa B .lrelxr,rrr rvrix to.rxairrr,r-4 i M.

iiiiiiiil;{il*i IEOMETPI9 Einqxina O. 9. ra iH. Ara,4eMi,{nraE piaenu !{ti 625

-""" -

-"-*t--r--1

\2\/_

OB

T -fi-h

Bi4cranefi uinc cycigniuu roqxaMrr Ha o,q[ry Meurue, rlilr{ caMr.rx rorror.

1.7"3 cu.(21 - 1) = 60 ctlc.

BiOnoaiOa: E) 60 cu.

1.8. .--y4A

A) AM = x cltt, BM = (x - 2\ crt.x * (x - 2) = 42:' 2x - 2 = 42; 2x = 44t x = 22.AM = 22 cv, BM = 22 - 2 = 20 (cn).

624 fd* reouerptg sinsxina o. 9. ra ix. Axa,qeuirHri piBeHb liilt{ifli;il

A 3 4

E 1 6

B 4 3

f 5 2

.4 3 4

A 3 6

B 1 4

B b 2

f 1 4

A 5 2

l:Iexafi ro.rrcr.r -4 , B, M nexanr na oAnift npsrvrift, ro4i rimxr oAHa s Hr,rx Jrerrclrrbruix gnorta iurtrt4Ml4, i sa a&cioMoro nrzuiprona4na rigpiertin BrrrcoHyerbcs oAHarr pirnocrefi:

flpr.r posn'aeanni 4iei aa4aui Ba Kon{Horo B yMoB A-A cxJraAaerbc.ff piBHsHHf, oc-noBoro ,snoro e pinnicrr - nacriAox s aftcioMr.r III,: AB = AM + MB.

1.13. AO = 18 cM' OB = 14 ctn.

ro ,-l do = 9 cu; ona ' lon = 7 *r.Z--- 2

KM = KO + OM = 9 + 7 = 16 (civt)'

Bilnoeidu: 16 cur,

1.14. AO : OB = 3 : 5; AB = 48 cu.3a anciomoro eumiproeanna ni4piaxia AB = AO + OB'

IIosHaquMo xoe$iqient nponopqiituocri l'To4iAO = 3x; OB = 5x; 3r + 5x = 48; 8x = 48; x = 6'

AO -- 3 '6 = 48 (cnt); OB = 5 '6 = 30 (cu)'

Bi}noai1u: 18 cu i 30 cm.

1.15. OCrti.xrx[ TorIRa B uoXe sHaxOAI'ITI4c,fl no o6ugna 6onz ni4 rovxu C' To poBUItl

HeMO ABa Br{na{fiu.

L) Mc=f,ec=rz,z=oMB=MC-BC=6-5=

x--T---i'

llpz reperrHi gBox lpsMux a i b yreoprororrcx napu cyuix-rrux (1 i 2, 2 i 3, 3 i 4, I i 4) ra neprzrcarbHux (7 i 3, 2i 4) xyrin. iix s,'racrzsocri:L Cyma cymixuzx Hyrin gopinuroe 180".2. Beptr.rxarrri xyrr piaui.

t.2'l .72+108'=180'.Bi0noeiOt: E.

llpz neperuni.qBox trp.rMilx a i b ci.rnoro c vrBoprororlc.E napz nyrin:nuyrpiuHi oAnocloponHi - 3 r 6; 4 i 5;uuy:rpiurni pianocroponni - 3 i 5; 4 i 6;nignonigni 2 i 6; 1 i 5;3i7;4i8;rrosHimsi picaocropouHi - 2 i 8; I i 7.,flxrqo npu [eperr4Hi ABox trpfiMfix a i b ciqsoro c eiryrpiuui(sonuinui) pieHocropoHni Hyrrz pieni, a6o cyua nHyrpiuuixo4nocroponnix xyr:in cranonnrr 180", a6o eignonigni Hyrzpisui, ro npaui a i b raparemHi (i naenaxu).

1.22. ZB = Z1; l5 = 17, srt< neprrzxa.nrni.To4i, ocni.nrxz Zl = /7, ro ZB = 15, ane qe eHyrpiurni pianocroponni nyrr nputrpsMlrx a i b i ciqsiii c. 3a ossaxorc naparearHocri upeuux o ll b.Bidnoei}u: a I b.

1.23. ZCAB = /2 + ll = 72" + 72" = 144".Ocniasxu Ba yMoBoro CD ll AB, ro Z3 = ICABaH nuyrpiuHi piaHocroponHi, roai l3 = L44".Bi}noei1u:144".

r.--.....

1

2l MC = |eC = e cu. To'{ra C .nexzrs uim ro'rnarvrr M i

MB=MC+CB=6+5=11 (cu)'Bi}noai}u: 1 cu a6o 11 cnn.

1 .1 6. Hexafi IMOC = x, roai IKOM = 2r' Ocxi'rr'xra npomiur'

OM npoxo4urr, rvrixc ctoponarrrz nyra KOC; ro 3a axcioryIoro

rurvriprosanur xytin lKOC = IKOM + IMOC'Maeuo pinHanHfi:, + 2x = I53; 3x = L53; x = 5L'

IMOC = 5L"; IKOM = 2' 5t' = 7O2"'

Bi}noai)t:51'; 102".

1.17. MK = x cv; AM = (r + 5) crl.; KB = 2(x + 5) cxt'

AB = AM + MK + KB (anc. III').xt(xi5)+2(x+5)-75' A

x + x + 5 + 2x + lO = 75 4x + !5 = 75; 4x = 6O; x = 15'

(cu); MC = MB + BC (axc. III')'

I (cu).

--L zcoB:

,"+-;tlt:

MK = 15 cv AM = 15 + 5 = 20 (cm)'

KB=2 2O= 4O (cn);-4J( =AM + MK =2O + 15 = 35 (cm);

MB = MK + KB =15 + 40 = 55 (cu)'Bilnoei7a: 15 cw; 2O cnn; 40 cl.,r; 35 cn; 55 crt'

1.18. Hexafi IAOB - 4aHzii rcYr'

IlpouiHr OC npoxo4zrl uint fioro cropoHaMr:lAOB=/AOC+ICOB'

/KOC =I Zloc, zcotw2

Sosnilreifi rcyr rpfinyrHuxa gopiruroe cyui nuyrpiuuix xyrie, ue cyui*cuux3 HUM.

1.24. IKMN = x = lMKZ + ZKZM = 72" + 63' = 135'.Bidnoaidu: 135".

1 .25. floanaqnuo xoe@iqieirr nponopqiitirocri r,Togi ZB = 4x; lA- ZC - 3x (Hyrrz npra ocHoni pirno-6e4penoro rptr xyrnuHa pinni).4x i 3x + 3* = 180; 10r = 180; r = 18.

zB ="4 .ta" = 72". ZABD =1ru =I.72" =36".22Bi1noeilu: E) 36".

1.26. B; I; E.

2r- ztoc +L zcon =qt

=f,t^o" + tcoB) =

1 .1 9, Hexaft npaui o, b i c neperunarorl'c;r B r' O' PoarlsneMo

npaui b, c i d'.Bo:r.:rl 3a yMoBoo rar(oxc [eperrluarorbqfl B oA-

nifi roqqi. Ilpunycrzrvro, ilIo q.tr rotrna ni4uinna ni4 rounu O'

ToAi npsui b i c marorr gei cninsHi Toqr{r4' rrlo cynepeqr4Tb

rrac:ri4ry a axcioura Ha.{e}r{Hocri. Otxe, npaui b, c i d raxotx

treperr4Harorbcfl e rouqi O, ro6ro sci 4 npami fleperr4narort'cfl

e oAHifi ro'{qi.

IKOM = tKOC + ICOM =

! ztto".2

g 1.2' Onopxi Sarrr Kypcy nnaHiMerpia

626 {:it rEoMETptg SincHixa 0. 9. ra ix. AxaAervriqHrr piBeHb :{ii;$ii:iil i:ii:iil:ili;,:t FEOMETPIC Einsxina O. 9. ra in. Axa4ervri.rHvri piBeHb iL::l 627

1,27. ZAOB = 180'src pogropgyrrrft;,/AOC + ZCOB= 180'src cynixni; ICOB=18O" - /AOC = 180" - 4a" =132".ICOB - piBno6eApeHufi (OC = OB nK pa,qiycn).tC = ZB = (180" - tCoB\ :2 = (180" - 132') :2 = 24".Bilnoai1o: B\ 24"; 132; 24".

1.36. r, = Q,$1.r * 0,5r = 18; 1,5r = \8; r = 12;0,5'L2 = 6 (cu).BiOnoeidu: B) 1-2 cu i 6 cm.

1.37. S = nPi nf = 9n; r = 3 (r > 0).l=2nr=2n.3=6n(cn).Bi0noeida:,4) 6n cu.

L39, 3a reopeMoro flitparopa g aBDC:BCz = BDz )_ DC2; BDz = BCz * DCz.

BD = \@ 4 = J6-4X5-4) = Je = B (cu).AC=AD+DC=2+4=6(cu).

1.4O, I euna1orc. tOKC = a ODC sa rinore-uyeorc i rocrpaM nyroru (OC - cni.nrxarinorenyaa, OC - 6icercrpzca Kyra C)."Ioai DC = KC = 5 cm; AC = 2DC = 10 crq.AB=BC=6+5=11 (cu).P,*"= 2 11 + 10 = 32 (clt).

1.38.a,=2ri r=?, ,=9=t." =nf =n.3,=9n(cu,).2' 2Bi1noai1u: A) 9n cu2.

s^,""=lec.BD=+.6.8=9 (crvrr). A22Bi|noei|u: B) 9 cn2.

1.28. R=o=!6

'1.29. P = (6 + 7 + 8) . 2 = 42 (ctt).Bidnoaila: I) 42 crvr.

1,3O. Biinoai1a.' A) 1 i 2.

I I auna0orc. Ana-norivno,DC = KC = 6 cu; AC = 2DC = 12 cu.AB=BC=11cu.P^*"2 . 11 + 1"2 = 34 (cu).Bi0noai1a: B'1 2 a6o 4.

1 .31 , d AOB - rp.EMoxyrrrnirt AO L AB sx pagiyc, npoBe4errlrfi y rorrry 4otraxy. Togi

Oe = 1OB= 6 cu grc Rarer, Iqo rreficr,rrb TrporI,I Kyra 30" d = 2r = 2OA= 12 clt.2

Biilnoeila: E) 12 cu.

1.82. a=2ort2 co"; a= 2r; r =t='!P= 10JZ (cu).

Bidnoeilo: B) 10rE crvr.1,41. AC = 2R g'r<,qiaMerp RoJra,

IABC = 90o src nnrcairlrft xyr, r{o crurpaerbc.s ira 4iauerp.oo lo

BC = AC . sin LA = 2rR sin 60' = "i" = nJa.

Bidnoeida: f) -Rv6.

Saaerxnicrr ir.rirx croponoro d i,qiaroHa,rrro d rnagpara: a=a"li

1.43. a = 2r = 2 . 5= 10 (cu). it = L}r[i cv.Bi|noaidu: f) 10rE cru.

il:iliiiil:li FEOMETPI9 Eincuixa O. f . ra in. Ara4ervrirxraii piaexs :i:iii; 629

Cepe4ra ninis rpanerlii gopinnroe nincyui ii ocHoB.

'l ,33. Ilosga.ruMo AoBlnlrgy uenruoi ocnoBrr Ba .r cM,

,r+(.r+8) 2r+8PeAH.SJIIHIS: ,

= 2

=I+4. NlaeMo:

Bi1noei1a: B) 6 cu.

1.g4. s =!0,o,, s=1 ro.86=r8o (cru,).2'" 2Bidnoaila: B) 180 cm2. n

6i.lrrtroi - (r + 8) clt. ToAi ce"

xL4=10;.r=6.

1.35. Y dABC: /A = 25" (ei,qnosi,4ni),lC = 4O' (aepruxa;rriri),tB = L8O" - (25' + 40") = 115'.Bi4noeidu: A) 115'.

Sa.nenrnicrr rrirx pagiycauu -R i r ta cropoHoro rl [paBnJrbHoro n-rcyrHr,rna Br,rpa-

)fi aerbc,fi TarctrMr?I tpopuy.nauz:

n a n r

3 NG zrJB 2rRt

4 RJ' 2r ,Ji RJ'2

6 R2rJg 2rnl3

3

ftJB2

5cu

628 fiti: TEOMETPIS rincxiHa O. L ra in. Ara.qewrivHhr piBeHb ilitltiiiil:

,{ra rpuryrnuru no4i6ui, arcrqo:1) gea xyrra oAHoro rpr.rxyrunrca ni4uori4uo piBHi ABoM nyrarvr iHlroro;2) 4ni cropoHu oAuoro rpuxyrxura nponopqifiHi 4aou cropouaM Apyroro' a Kyrrr'yreopeai urrMil cropoHaMra. Pieui;3) niAnosi,4Hi croporrr4 rpzryrnizxis nponopqiiini.flpama, flKa neperr,ruae 4ni croponrz rpl{rcyrauna i napaxe.nsua rpetiii croponi,ei4runae ni4 nroro rpr'IKyrnr{rt, troAi6Hufi AaHoMy.

1.44. Ocximnu AB I DM 3a lMoBoro; ro IBAC = IADM aH rgyrpiurui pisgocropon-

ni (civna AD). IACB = IDCM flK BeprrrarlbHi. Toai yABC * aDMC 3a ABoMll

IIYTAMII.

49 - g, Ac-=1, Ac-3'1'5 -4,b (cM).cD MD 1,5 1 |

AD = AC + CD = 4,5 + 1,5 = 6 (cru).

Bidnoei}o: B) 6 cm.

l

1.45. 180(z - o) : 160n; 180n - 360 = 160n; 1802 - l60n = 360;

2On=36O;n-18.Bi0noai}t: l) 18.

,(iaroHa.ni pou6a nepnen4ury.rapui. ,(iaroHa;ri pom6a neperr'firarorbcs I roqKoro

neperr.ruy 4i,rarrca nannil.

1.5O. Ilponegeuo ni4piaox AC.Cyma rcyrir a,AJ{C 180", ro4iZKAC + IKCA= 180' - 96" = 84".ZKAB t ZKCD = L22" + 142" = 264".ZBAC + ZDCA= 2640 - 84a = r8Do.A"ne qi rcyru niryrpimHi o,4uocropoHxi K

npv npauux -AB i CD i ciqsifi AC. Ortte, AB ll CO.Bi}noaida: AB ll CO.

1.51. Hexafi AM = x cm. ro4i BM =(x - 3) cu aa yMoBorc.BK = BM, AM = AZ, CK = CZ ar ei4pianz Aornqrrux, npoBeAeH[x s o4niei rouxuAo riorra. Tottty AZ = r cM, roAi CZ = (x 4) cu.P"*"=AB t BC +AC=2x-2(x - 31+2(x 4\=2x+2x 6'2x-8=6x 10.

3a yuoeoro 6x -.7O = 28, asiAnr 6r - 38; - = 6:.

AM = Az= o] t"*t. BM = BK = u}- r = af t"*t.KC=cz=u*-n=zj- tcnr).

Croponr.r rpuRyrHHKa:

AB 6r-+B]=92 (cnr. BC=s!*2133333Bidnoeida:9-2 "*, s? .*, 8? "*.333

1.52. [un..ra6.n. ao Nb 1.28.

fi=sJ5 clu; a=ar6=sJ5.16=g @wt); a=2r.1!;d s 9./s 3./3

f=- 2,lS 2J3 23 2

Bionoeioa: tG "*.2

x Zx-4(crvr). AC = u*-r ri = sl rc*1.

x-4

a

I

S;1:;11:;;;: 1.46. Hexafi BD - 18 cv, AC = 24 cw, ro4i

l$*liiii: Bo -18: 2=e(cu); oc = 24 : 2=72(cm)-:l:i:li:i:i:fii 3 o BoC (to = 9o') Ba reopeMoro fli(paropa:iiili::::::::i: BC2 = BO'z t OC2:

[iii:$ji:i Bc = Js' *r2' = Jsr.M4 = Jr% = 15 (irvr),

:rj:+::x::::!:: P^.-- , L5. 4 - 60 (cM).'.{i:i::.:+..ri d-iioaiao: E) 6o mr.

1,47. IlosHaqnMo MeHrrry cropony napaJreJrorpaMa t cM' a cymixcny 3 Hero - (x + 8) civl'

Ilporrznexcni cropourr napaJreJrorpaMa pinni, romy p = (x + (r + 8))' 2 = 4x + 16'

Ba yuoeoro 4x + 16= 48, aeiAxu 4x = 48 16 4x = 32; x = 32 i 4i x =8' MesuacropoHa flapaJreJrorpaMa - 8 cu.Bidnoai|a: A) 8 cn. B

1.48. aABC-pieHo6e4peHufi, AB=BC,1BCA=IBAC nslr Ryrrr npz ocroni. IMBC = IKBA, 6o xorxnrzii / / \ \is Hrx e cyuoro ryra ABC i o,4noro s pinunx ryr:in ,/ / \ \ABM|CBK. ,/ / \ \aMBC = dKBAsa cropoHoro i 4oova nprrrerra-

'./ 5 4 \O

Mu Kyrauu. A C

1.49. IABC - npnMoxyrull frr (AC LCB sa yrvroeoro) lffiq.3a reopeuon fli$aropa AB2 ' AC2 - BC2; / \ l: \Au=tpiz"=,lisnuq=t5oe=13(crvr). i \ l.:)Bi7noeida: 13 cvt. \ \ l-/\-Y

630 iiii: TEOMETPIS Eins*ixa O. 9. ra ix. Axa4ervrivHrf piBeHb :ii:i:i:i:ii::#:

1,53. Hexafi BE = 7 cM - Bficora pieuo6eApenoi rpa-ne4ii ABCD. 3 aABE (lE = 9O") AB = 2BE = 74 cut.(ICarer, rqo JreJ(rrrr, rrporn ftyra 30o, gopinnroe no.no-enui rinorenvau.)Cepe4Ha .niHis rpaneqir 4opinuroe nincymi ocHon: ABC+AD

2

= 2AB = 2 . 14 = 28 (cu). Orrxe, cepeguR linia rpaneqii 4opienroe

D

, A-ne ocni,rrrr n rpaneqiro Mo'fiHa Brrrrcarr4 Ho:ro, ro BC + AD - AB + CD =2A

i =ta (clc)'

Bi}noaiOu: 14 cm.

1.54,ABCD - AaHa rpaneqia, AB = CD; IABC = L5O".

Cepe4Ha.ninia 4opirnroe 16JE cnr. BE LAD - Bzcora

rpaneqii. AD + BC = 2.16.yE= S2aE (cm).Ane AD + BC = AB + aD (gvs. safla.ry 1.58).

ASl;iscu AB - AD !- BC - 16rF cv.'' 2

i

1

I V vorupunvrHrzrc MolrcHa Bnr{carr4 Koro, sxqo cyma fioro rporrz.nemurx cropin1

prBHl.L

ii:iilli;i::::.:i rEOMETPI9 Einsxina O. L ra ix. Axa4enivnnfi pieexu:.{il 631

1.55. IABC - piBuo6eApeHuit (AB = BC'1.

BD -- 15 cM - Blrcora i Inegiana rABC. AC = 16 cu - ocuoBa.

3 aABD: eO = ! nC = 8 crvl; AB2 = AD'z 't BD2;2

Au = 'le' *tb" = J64+2% = tE89 = 17 (cu).

Bi?noeidd: 17 cw,

1.56. Ilprrvrirra. V uigpytnury Aorryqeno noMrrrrrcy'

ts'yuori rtae 6Yrra AC..= 26 cu.

aABC, AM L BC, AB = 101E cw, lB = 45'.

3 aABM: AM = AB ' sin 45" = |O\E 4 = r0 (cvI).J2

3 rACM Ba reopeMoro fli(paropa: AC2 = AM2 + CM2;

cM =,[Ac, - AM" i c

cM = Jze -to'= !tr6 -10Xr6 +10) = "86'36 = 4'6 = 24(ctl);Bi1noai1r': 24 cv,.

1.57. ABCD - poIvr6; AD = lO cwr, AC = 12 cla.

Pagiyc nnr,rcauoro n pou6 KoJra * qe Br4cora aAOD - OK.

3 aAOD(IAOD=sO',AC IBD\: OD=LeC=a "*;ADz = AOz .t oDzt oD = ^[ADr- Ao,; A

oD = $o, 4 = fioo - 86 =.uGZ = a (cNr).

s^ ^^^ = L ao. oD =!. 6. 8 = 24 (ew2).22A.ne S*o, =Lln'nrtBi1noetla: 4,8 cut.

1.58. Ilpone,qevo OC L AB, togi OC - ri4cranr siA I{eHr-pa Korra O 4o xopguAB. aAOB - pirno6e4pernfi.AO = BO = 15 crvr - paAiyot xo.na. Bvcora OC

e rvreAianoro: AC -- BC. B rAOC: AO2 = AC2 1- OC2;

Ac = ^[Ao' -oc' =.Eb' -tr' =

= ftb -1acb +1r, = &.n = e (cu).

Bilnoai1a: t8 crtt.

1.5g. ABCD - parruir rapaJrerrorpaM. ZA = 6O" 'BE - 6icexrPuca lB. AE = 6 cm,ED = lO crq, AD = 6 + 10 = 16 (cu).IAEB = ICBE ai nnyrpiruni pissocroposginpuAD ll BC i ci'rnift BE. To4i AIABE - pisHo6eApelruii, AB = AE = 6 cut.

S = AB. AD.sir,./A= 6.16.sin60' = 96 f = 48JB (cu'?)'2

Bi\noei\a: 48JB cvl'.

692 l#j reouerplf sincHiHa O. 9. ra ir. AraAeniqHri piBeHb il{til{lill.l

1

150' = 30'. BE =;AB = 8Jd cu (ar xarer, uto rrernurb

oK=r--#=T=4,8 (cu).

,fkcrqo xopgn /3 i CD ro;ra neperlrnarorbcfi e rouqi S, ro AS . BS = CS . DS.

1.60.,43 i CD - aagairi xop4ra.S - cepe4ura xop4u AB, CS = 5 cu, DS = 20 cru.Hexaft,4S=BS=xcu.AS.BS=CS DC;r-x=5.2O;x2=lOO;x=1O(x>0). '.AS=BS=10crulAB=2Oc.lqCD=5-20-25(cu).Bi?noai1a: 20 crvr i 25 cv,

,flxrqo e ro.rrrcra rroaa KoJroM rrpoBeAeno civuy i 4orra.rny, To lcBa{par AoB}nr,tnl4 Ao-rnvnoi 4opinnrce Ao6yrrcy ci.rnoi sa ii sosHiurHro qacrnHy.

1.61. MC = r c]Mj AM = (r _ b) cM;AB = (x + 5) cna. MC2 = MA. MB.x"--(x-5).(x-5+x+5);x2 = (x - 5)' 2x; x2 = 2xz - lOxix2-l0x=O;x(x-10)=0.x = O - He Ba,qoBoJrEI{ge yMoByaa,qavi (r > 0); .r - 10 = O; r = 10.BiOnoaida: lO cm.

1.62. Hexaft MC = x cu, ro4i 3114 = (15 - x) ctu, AMMC2 = MA . MB; x2 = (x - Z). (lb - r);. x2=l\x-x2-30*2r;' xz I x2 - l5r + 3O - 2x = Oi 2x2 - 17r + 30 = 0;D = b2 - 4ac:= 172 - 4. 2. 30 = 289 - 24O = 49;

L7 !7 77 -7- = 2.52 x^ =17 *7

= 6.xr.r=--{-t xr= 4 " 47) MC = 6 cut; BM = 15 - 6 = 9 (cna);

2) MC = 2,5 ctn; BM = 15 - 2,5 = 12,5 (cm).Bilnoeila:6 cnn i 9 cu a6o-2,5 cn i 12,5 cn.

1 .63. 3a s,rac'runicrrc Aorr,rrrHr{x, npoBeAeHr,rxa odniei roqrr Ao xotra: AD = AE = 6 cut;' BD = BF = g ctn; CE = CF = x cttr.AB=6+ 9=15(cru); AC=(x +6)cru;BC=(x+9)cm.AB2 = AC2 t BC2; 152 = (r + 6), f (r + 9)r;225 = x2 + l2tc + 36 + x2 + 18r * 81;2x2 + 30x* 108 = Oi x2 + !5r * 54 = O;

Jcr = 3; tcz = -18 - rre Baa4oBoJrbr{s€ yMoByaa,qa.ri (r > 0).CE = CF = 3 cnt. AC = 3 + 6 = 9 (cu); BC = 3 + I = 12 (cru).

s-,. = | AC . BC = L= .g .tz = b4 (cur).22

Bi1noeidv: 54 cwr2.

1,64.ABCD - rpaneqir (BC ll AD), BC =8 cw. AB =6JI "r - MeHrtra cropoua,sr cni.nrnuft fiepnerrAnrdyJrsp ABox napaJreJrbHnx fiprrMt{x . lC = 12O", Ilpone4euo

CK L AD, rogi CK = AB = 6& cw, AK = BC = 8 cM sK rporr4J,ren*ri cropoura npa-moxyrlauna ABCK.

il}:ii:iii:i:l FEOMETPI9 EincHiHa O. 9. ra iH- Aragervri.rxr,rfi pieexu ii:i: 033

V aABE /A = ta0" -nporr,r rcyra 30o).

Bilnoai}a:8Jg cM.

Eaco

E

C

B

D6 cru E 10 cnr

=(r-2)cu.

60'

IKCD = IBCD - 90" = 120" - 90" = 30'.

3 :CKD: KD = cK ' tg 30"; KD = 6J3'* = O t"*1.J3

AD = AI< + KD = 8 + 6 = 14 (mt).

s.^-^ = BC' M .cK =8

+,14 6'6 = G6'6 (cm').22

Bi|noei\t: 6616 cn'.

1,65. Hexafi ABCD - sa"qarra rpaneqis+, BC I AD;BC = 14 cttr, AD = 4O crr.. CE LAD,CE = 39 cM - Br{cora rpaneqii, AC - \i 4iaroua.nr.Ocri:rr,nu EaBKoJro rpaneqii-4'BCD orrucano r{oJro' qe

ognaqae, uo ABCD - piaHo6ivna rpaneqia.

Toni ED-AD-BC -40 14 =t3 (cM);-22AE = 4O - 13 = 27 (cu).

3 IACE: AC2 -- AE2 + CE2; AC = JSr, + n' =,ttsz:]+ zzs = JrrSo = rr&0 (cu).

B ^.CED:CDz=CE'+ ED2; CD =J3y -1# =rGr1-16' =JG90 =13f0 (cm).

s..-^ = f a-n.cE. 1.40.89 = 280 (cv,).22ocni.ntrz on[caHe HaBr{oJro rpaneqii KoJro oAnoqacHo e onrrcaHuM HaBKOIO aACD'

ro fioro paAiyc li no:nna gnaiittr a gopuy;rn: R - AC 92'AD '4S

^ 15Jto .13,/10 40t1 = -25 (cy).

4 .780Bi1nosi}a: 25 cv.

1.66. a) ABCD - rpaneqia (BC ll AD). AC = 15 cm; BD = 2O cttt.

CE L AD, CF L AD - Br.rcorr4 rpaneqii; BE = CF = 12 cttt.

3 aBED: BD2 = BEz I EDz;

ED = JED' - BE' = Jf -tr" =.,@o -M4 = Ji56 = 16 (cu). "

3 IACF: AC2 = CF2 l AFz;

AF = $e -cr" =,llb" -rr' = ^ln{-r44 = Jal = g (cm).

Bpaxoryrovz, qo BC = EF, maemo:

AF + ED =AE + EF + FD + EF -AE + EF + FD + BC =AD + BC'Ssitcu: AD + BC = I + 16 = 25 (cu).

s-- - -- AD ! BC . BE =ry.t2 = l5o (cm,).-rpan. 2 2

Bi0nosi0a:150 cu2. A6) Ana.noriuno.AC = 30 cru, BD = 26 cv, BE = CF = 24 cvr.

B d BED: ED =,tED' - BE" =.1T6' - 24' - $?6 -5?6= Ji00 = 10 (cu).

B rACF: AF =.[Ae 4F" - $o' - 14'= ./600 - 5?6 = JEr4 = 18 (cu].

AD + BC = AI + ED= 18 + 10 = 28 (c*t).

s^-^- - 2,8 .24 336 (cn'z).-rp^n 2

Bidnoeido:336 cu2.

634 i:::::: FEOMETPIS EincxiHa O. 9. ra in. Axag,ervrivHrf piBeHb :iiii:iiiitiii

1.67. ABCD - pon6. OK L AD - Bucola ^AOD,oK = 6 cv = r. AC = 24 cv: Ao =l oc =r, .*.'2

B aAoK 1,zK = 9o"\: sin ,zoAD = ?! = + = i,oA722ro6ro ZOAD = 3O'.

3 .AOD: oD = Ao.ts30-= t, f - ar6 tmt), BD = 2.oD = sJ5 (cu).

tl1-s*,, = -d,d, = ;o" "o = ;.24.8J3 = 96J3 (crra').

Bi|noei|a: 96rE cm'z.

l.68.3uait,qeMo nrorqy IABC ea (popuy.rroro fepoHa:AB+BC+AC 77+25+28n=

' =-z =35'

S*" = J p@ - o)(p - b)(p - c) = JBb L0'?. 18 = 210 (cM);

Are S*" = S*o r Sn"n = l= AB. Ko - + BC .oF,22

ae KO = FO = R - pa4iyc ganoro xona. A

Macmo: ln zs*\n.tz =zto,22

A

szipnu R(12,5 + 8,5) = 210; R = 10 (cm).S..-_.__ = nR2 = n. IO2 = 100n (cu,).Bi0noeidu:100n cm2.

1.69. ABCD'- trapar'rerrorpaM, y snoro AB = 4 cm, BC = 6 cttr.uirN 4iarona,raMrr relr![Tb MeHrrra cropoHa: ZBOA = 60",3a reopemoro xocunycin:AB'= BO' + AO' 2BO .AO. cos 60'.IBOC=180"-60'=120".BC2 = BO2 + Oi, - 2OB OQ. cos 120. =

BO2 t AO2 t 2BO . Ao.tos 60".Toai -4-8'? - BC2 = 4AO . BO. cos 60o; A

1,7O. Bi}noai1a.' f) {r + 5\ . 2 = 2A.

1.71.

1,72. Bi1noeida: R.

i*liiiitiiii TEOMETPIC Sinsnixa O. 9. ra ix. Axa4ervrirur,ri pieexr ll:j:

-\,

oY

flporu rocrporo xyra

p 6cuC

D

AB2 - BC' = -4. AO. nO !; 2AO BO = 6? 42; 2AO . BO = 36 : 16; 2AO ..

BO = 20.

s*," =L.2Ao.2Bosin6o" =2Ao "o + = ro f,*= r0r,E (cu,).

Bidnoaida: 10.,6 cm'.

91,3. 3a.qa.ri i ueroguii po3B'e3yBaHHe

60"

635

1.73. /AOC = )ci x + 17r = 360; 18r = 360; * = 20'

ZAOC =2O". Ocxilrxr.r rpagycna rripa nllucaHoro xyraABC Aopinnroe no.nonr'rui nig-

uoei,4uoro qeHrparrbuoro xvra AoC, to IABC =)z'+oc =f; 'o"

= 10''

Bidnoaida: B) 70'.

1,74.ltarer , npra.nernraft,4o r{yra e 60" e [porr4Jre)rcHuM Ao Hyra 30o i 4opirnroe no"

.nosrzui riuoreuyer: 10 : 2 = 5 (cu).Bilnoaidv: B) 5 cm.

1.75. rABC - pisno6eApegzft. Ba ynonoro AB = BC = 5x;

AC = 8x; 5x * 5x + 8* = 18; l8x -- 1'8; x = I.AB = BC= 5. 1 = 5 (6r); AC = 8' 1 = 8 (clt).

,AA =!dC = 4 cw. P"*, = AB 4 AH + BHi2"*

La2. ABCD - Aanuft napaJreJrorpaM. flpone4ervroBE L.AD, BF I CD * Br{corra raparreJrorpaMa.BE = 8 crvr; BF = 12 cv.. ZEBF = 6O. ICBE = 90'.3a axciouoro nrauiprcnannx ryrinICBE - ZCBF + IFBE; ZCBF = 90' - 60' = 30".

BaBCF: BC=*-!+*=-+= =!Z=?=4 rc*1. A

cosZCBF . cos3Oo Ja JS

s*"o = BE BC = 8' + =L#= 64JE (cu').Js

Bi|noai|u:64Jd cu'z.

1.83, ^ABC - up,nMoxyrrrr{ft. (lC = 9O").

BC = 6 cm. AD .- ue4iana, AD = 5 cwr.

D - cepe4rana.CB, roruy CO =|AC = 6:2 = 3 (cm). C

- ''z = .,6t- =,lrs-s =.,,[o = a 1cm;.3 aACDI AC=]AD, -CD3 aABC: AB' =,t@ + BC' =.[1" - g = fi6+ g6 = Jaz = zrTsBidnoaila: 2Ji3 cu.

I]errp roaa, onucalroro HaBKoJro rrpfiMorcyrnoro rpr,rxyrnar{a, Jreficr,rrr, tra cepe-4uni rinoreuyar.

1.84, Cepeguna rinorenyera O 4anoro npsMor{yruoro rpr.rryrura-. Ka ....- qe qeHTp onrrcauoro KoJra, rouy OA = OB = R = 17 cru.AB=17.2=34(clr).3 yABC: AB2 = AC2 * BCz;

Ac =,IAts" - Bc, = $nz - to, = "fu4

- g0XB4 - s0) =

s*" =!nc "" =*.16.80 = 240 (cm2).

Bidnoai1o: 24O ott2.

,14.G4=2.8=16 (crrn).

,{,'ra poar'asanna qiei sagaui cftopucraeMocs MeroAoM nJrorq: nJronly rpr4ryrrr[xaanaft4ervro aa ipopivry.roro lepoxa, a noriu, osaroqu nnouqy i cropony, s (bopMyra

1g^ = =ah anaftAeuo AoBlrcr,rHy sncoru h.'2

1.85. flano: IABC; AB = 29 cu, BC = 25 crvr, AC = 6 cm.BD - nucora, [poBe,qeHa 4o menuroi croporlr AC.Ba Sopuy;roro lepona:

AB+ BC+ AC 29+25+6 60n= , = 2 =t=30 (cu).

BH3AC8

8cu

N(cm).

$11tr1:i.J:li{$:r.:*.:

72 = 5 + 4 + BH; BH = 12* 9 = 3 (cm).

Bt}noailo:1.8

1.76. Bilnoai}a; B) 2, 5 i 6.

a 24 24Jg1.77. R= F= F = ._=8J5 (cm).

Js J3 lJ

Bidnoeila: A) 8G cu.

1 .78. s = s.;Jr. sin 45" = 45J, 'l = 45 lcwr'\-,t2

Bilnoai}a: B) 45 crvr'?.

1.79. Bi\noeida: l).-1

1.8O,HexaftAC= 8 cn; BD= 6 cu. ToAi AO = ;AC=4 cmi

OK L AD, OK -- r - pagiyc nnzcanoro r(orla.' 3 IAOD (ZO = 9O'): AD2 = AO2 * OD2;

AD='[4\g" =Jro-9 ='l%=5 (cu).11

s*oo = iAo. oD = ;' n' t = 6 (cm'z).

r =

2s*o, _r_.p =y= 2,4 (cM).S*oo = -rAD OKi OK AD b b

Bi}noai}a:. B) 2,4 cu.

oo =.f,ao =z cM.

=&s.L %=J6 b.b'o4 =6.5.2=60(crur).s*" =!rc.8D; BD = #=f = zo t.*1.Bi0noei1u: 2O cw.

Bicercrpnca Kyra rp[Kyrur]rrca 4irurr rrporr{Jrextuy cropoHy Ha rigpienn, trpotrop-qifi ni npu,'rer;rraM cropoHaM;

1.81. aABC a rDBE, r"*u #;= #,nB. AC 6 10

= tb (.*).ssinxu AB = on = +

BiAnoBiAb: 15 c:[{.

636 i:1: feOUefRlg SincHiHa O. 9. ra in' AraAeilri'{Heril piBeHb :i:iii:i:i:i::ii: l:*l{iiiliitrji TEOMETPI9 Eincxixa O. 9. ra ix. Axasenirxraft pieenu {{";il 69z

1 .86. Hexafi -4BC - Aauufr rpr{rcyrHr'Ix' lC = 9O".

-AJ) -.6icexrpuca xyra A. CD = 24 cu, BD = 51 cu.

AC 24lVlaevo:':: = - :. Hexafi x - Koe$iuieHrAB 51

nponoprliituocri. Togi AC = 24x cv, AB = 5Lx cv' cBpaxonyrouu, qo BC = CD + BD = 24 + 5L = 75 (cu)'cr(raAeMo piBHsHHfi. Sa reopemoro fli@aropa,4-82 =AC'z+ BCzi

(51x)2 = (24x)2 + 752i 26OLx2 = 576x2 + 5625t 2O25x' = 5625;

, = Fro*, , =f,. dc = rn

?=40 (*r). s*" =i"" "" =; 4o 75 =1500 (cu'?)

Bidnoaidu: 1500 cv2.

1.89. 3a Hac.nl4nou g trpaBuJra rrapaJTeJrorpaMa:t_t_t

A4 = :UB + AC\, BB, = i@d+ BC), CC, -':(CA'2'22',{o4aruo novrTeHHo ui pieuocri:

1_A4 + BB, * gg, = ::(AB + AC. BA + BC - CA t CB) ='2'

1_t=' taB+ BA t Aa, CA r BC r cB)=' .0=o.

'l-'t.1

tDopmyrn KoopAilHar cepe.[ilHr4 eiApisxa: * ='' ;x', o = atJ-lz

1,9O, Posl,ricruuo.rpane4iro ABCD e cucremi RoopAruar rax, rqo6 ii BeprrrzHr MaruHoop4ruaru: .4(0; 0), B(a; b), C(a + c; b), D(2a + c; 0).Hexafi MNPK {oruptrnyrHux, BeputrHx rKoro c cepeailHaM}, cropiu rpanerliiABCD. Snafigeuo ix xoop4unarlr :a $opmynaMr4 noopArnar cepeAuHu ri4pisxa.

Toqxa M: * = o*o =9, ,= o-b . b, u(o tb).2 2 " 2 2 \2 2l

ToqxaN: x-al c t a =2o

t ",,=b-b =b, N(?!*",a].22"2\21

Toqxap: r_a+c+2a+c _3a+2c. u=b*O =L. ,( Ao*2".!-).22"22\22)

To.rxaK: ., -o+2a+ c ='o! " r, = o; x(2o! ",o\.22\2)

+ CB).

1.87. Hexafi ABC - lanzfi tpranyruizn,BD = 14 cu - itoro me4iana, AC = 24 cut,

flossaqzuo AB = x c1a', roti BC = (r + 8) cm.

IloaHaqznro 4as apyuuocri IABD = ct'

rcfi IBDC = 180" - cr (ax cynixuzft z IADB).3a reopeuoro rocnnycin s tABD:AB2 = AD2 + BDz - 2AD ' BD cos ct, A

1te AD=:AC=72 cv.'' 2

x2 =722 + 142 - 2'!2.L4 cosa (1)3 rBDC: BC2 = BD2 + DC2 - 2BD'DC cos(180'- cr).

(r + 8)'z = 142 a 122 + 2 72'14 cos a. (2)

flo4amo pirHocri.(1) i (2):x2 + (x + 8\2 = 2. (12'z + 14'z); x2 + x" + 76x + 64 = 2' (L44 + 196);

2x2 * l6x + 64=288i 392;2x2 +'l'6r- 616 = 0; x2 + 8x - 308 =,0;D -_ b, - 4ac = 64 + 1232 = 1,296 = 36';

R-36 -8 36*,= 2 -l4i xz-- <U-HeSa,zIoBoJIt'HfieyMoBy'

AB = 14 cm; BC = 14 + 8 = 22 (cv). P ^ o," = 74 + 22 + 24 = 60 (clt)'

Bir)noeida: 6O cv.

1.88. Asanoriqno Ao 1.87. 3 aABD 3a reopeMoro (ocrHycrB:

AB2 = AD2 + BDz - 2AD . BD cos o..

3 IBDC: BC2 = BD2 r DC2 2BD ' DC cos(180o c).fligcraniruo BHaqenHfi :

142 = 142 * x2 2 L4 x .cos cx;

222 = 14'? + 12 + 2. 14r ' cos o.

,{o4arro noqreHHo 4ei ocraHHi pinHocri: A2x2.1'!42 = 222i 2x2 + 196 = 484:. 2x'z= 288;x2 = 144; x = L2 (x > O). AC = 2 72 = 24 (cxt). P

^*" = 14 + 22 + 24 = 60 (cm)'

Bi}noaidt 6O cm.

,{oregemo, rqo MNPK - traparerorpaM.KqopAzsaru cepeynHv giarouani MP:

a 3a+2c b b,l c 2a,c ,t, hx=' - i u '22"22

ICoopgznaru cepe,qrrurl,4iarona.ni NK:2a+c 2a+c

+,,2

Koop4rznanr a6irarorrca, orrxe, 4iaroHali MP i NK treperuHaTorbcfi i tovrcoro nepe-rrany 4i.namcs naeni.n, A 4e oanavae, ryo MNPK - naparerorpaM.flopinxanuu a6c4ucu roqox N i I{ (eoxz pieni), po6uuo BrrcuoBon, u1o 4iaroua.nr

NI{ raparemua oci op4auar. Aua.norivno, opAunarrz ro,{orc M i P .qopieHrororr 1,2

rony MP OX. Bucuoeon:4.iaroHari napaJerorpaMa trepfleH4hTcyJrspni, a qe oaua-qae, qo MNPK -

pon6,Morxna raxo;x o6quc,rzrtr AoBr+(ilHu geox cycignix cropin ea SopMyrom:a = ,[*" - *,1' * 1y; u;' .

24 cu

!,*.-.......-.r.'

$$liiliiii

2a+c b+O b: u-2"22

. .2

uIo ql\. 2)

A(0;0)

2a+c=a\? r. b\' lto+"\' tbt2z l*lD-tl =!t z)-lz) '

MN = MK.

Bucnonox: y fiapaJ'reJrorpatta MNPK nci croponu pinHi, ro4i MNPK - poru6.

iiii;:ii!{till TEOMETPI9 EincxiHa 0. 9. ra ix. Ara4euivrrafi pieexu lif 639

MN=

MK=

Cyua eenropin:B/\

-,/ \bv\aGc

atb

- trpaBI4JrO TpI4ItyTHr4I{a:

,qn+nC=dC:

- npaB[Jro napaJ]eJorpaMa.

dn+dn-AC.1_

Hac,riaom: AO l(AB + AD)'' 2

638 *:::: FEOMETPIS EinqxiHa O' 9. ra ix. AraAervri'fHnf piBeHb ::iiiiiiiii:ii:ij

1.91. Poslficr]rMo cr,rcreMy KoopAr{Ear rardvM

vnnou, rqo6 fioqarorc KoopAnHar a6irascs3 qeHTpoM BrrflcaHoro B KBa'zlpar rcoJra' a cro-pour4 rcBa,[para 6ylu napale;rlni ocsu no-

opAHHar.Bisrueuo sa rconi goeilrry rcqxy A(x; A),

orrycrlrMo a nei nepnen4rar(yJr*prr AB, AC,

AD i AE rra croposr rcBaApara' ae B(-R; A),

C(x; R), a(R; u), E(x; -&\'Bupaa Ars cynrra xnagparis eiAcrarefi siA

ro.*cu -4 go croPin r(BaApara:AC2 + AD2 + AE2 r AB2 =

= (x - x\2 + (A - R)" + (x - R)2 + (A - U)' =

= (x - x.)2 + (A + R\'+ (.r + .R)' + (y - v\2 -+ (x

= fu - R\' + (x - R)2 + (Y + R)' + (x + R)2 =- v2' 2Ra + R2 ! x2 - zRx + R2 * y2'l 2Ry +

= z(xz + y2) + 4R2 -- 2R2 + 4R2 = 6fr.

Tecr flnc caMoKoHTPorlloqacruHa 1

3. E.

19, /ACB - 90" - enncaauir, lqo cnzpaerbc.s na xop4y-48.Tonry.43 - 4iauerp xona, AB = 2R.3a reopenoro lliQaropa AB2 = ACz L BC2,

AB = .lTb\ B" = Jrzb - 64 = .,489 = 1z (cu).

n =!an = rz :2=8,b (cm).2

fpagycna mipa nnr.rcanoro rtyra AopiBuroe no.noszni oi,qnoniguoro rIeHTpaJrBHororiyTa.

20. zABc =! ^oq zAOC = Bo" . 2 = 60..

2tAOC - pinno6e4peHrfr/A= ZC = (180" - 60") : 2 = 60..Tomy aAOC - pienocroponnifi.AC=AO=R=5cu.

21 . p,axo KoJ'ro B qeHrpoM O. AB = 8 cnn - xop4a.OC I AB - ri4craHr aig 4enrpa Ko,'ra Ao xopAr,r.OC = 4 c*r.

3 rACO: dC = | dn = 4 c.,l.Li AOz = OC2 * AC2i2

Ao =lE+ 4" = do - 16 = 4Ji &yr\.

t = 2n. Ao = zn. ali = anJ' k$.

1, r. 2. L.

4.8. d=aJz; 2R=ali; a=4, s=o" =+=2Rz (cw2)'Jzz

5. A. 6. B.

7. I' Cyrrixcni xyru rpaneqii r cyui cr:anoa.n'srr 180")c + 5x =180; 6* = 180; * = 30' 30'' 5 = 150" - rynr'rft r<vr'

8. E. 9. B. 10. f; 11. r. 12, I.13. B. 14. r. 15. A. 16. E.

Hacruna 2

IIpu neperrni grox npartux yrBoprororbcff napra cyuirxnr'rxtu-

""p**.o""r.rx xyrir. Beprraxa;mRi rcyrn pirni' Cyua

cyui*uiax xyrie 180'.

17. flpz nepenani np'fiMrx a i b yrsopunucn 4 xyrr a6o 4ni napz cyvrirxuzx: Zl i l2ra l4 i 24.l|+12+13+14=180.+180.=360..Yerrepruftryr4opinrrroe360.-240.=120".Cyrvrixnraft 3 rrrlM Kyr: 180' - 120'= 60"'

II{o6 anafirr,r ai4cornoae ai4notreunn attrcra q' Ao trrrcJla b, tpe6a o no4i'nlrrta na b

i oAepxane qficro B].rpasr4rra y rigcotxax (y 4ecar:xonomy .qpo6i KoMy nepe$ecrr4

rra ABa Bnarcfi nPanoPlnr).

60:120=0,5=50%'

18, r, 3x =18O; 4x= 180; r = 45.Cyua rocrpr,rx xYtin: 45o + 45' = 90'.Cyua rynrax ryris: (45' '3)' 2 = 27O".

Bilnoei1a:90" i 270'.

+R)2+(A-A)'=

R2+1c2+2Rx*R2=

2rJ322. au= r, o" r- pa4iyc BnrrcaHoro KoJra, a06 * cropoHa

u*o"o. lu = 6.6 crnr. z'f

= a"la; '= u{t;, t

= 9 (cn).3 2J3

23. Ocxiar,ru r:rpaneqiro nnnbaso rco.no, roAB + CD = BC + AD.

nrecrranyrnuxa. 3a

Hexafi I - roe(pirlienr nponopqifinocri,ro,qi /ll = 7x c;la, CD = 9x cw.CepeAus ainia rpane4ii o6qracnroerr,cs Ba OopMyroroBC+AD ^ BCIAD

Z 3a yMoBoro 2 = 32.

Tooi'B*cD =82, 7**9* =82; 16* =B2t'222

AB =7 4=28 (cut); CD-I.4 = 36 (cnn).

8x=32;x=4.

Tpaneqiro Molr{Ha Blr4carrr B rco,ro, sxrr{o BoHa pirro6iuHa.

24, HexafiABCD * pirno6iuna rpaneqia, Brrrzcar{a B Ae-sx-e norro a pa4iycon 4 cxn, BC ll AD, An = Cp.ICAD = ZBAC sa yMoBoro, IBCA = 30o ea yuoeoro.

- ZCAD = ZBCA = 30'sr rnyrpilrni pienocropoauirpr trapareJrbHrrx fipffMr4x BC i AD i civnifi AC. Togi A

ZBAD = ICDA= 30' 2 = 60o. Ocrcinrxu cyma xyrinBCD iADC gopirnroe 180o, maeuo:IACD = 180' - (IBCA + tADC) = 180' - (30' + 60') = 90".

,:iltillliilii TEOMETPI9 sincHiHa O. 9. ra ix. AxageraivHnil piBeHb ilil Oar

2l v"i rA.ri lt) s.. q. I

640 il{: TEOMETPIfl EincxiHa O.9' ra ix. Ara4eMi'{Hrft piBeHb:i::.::fi:lliti:*

Tarcrau uranor,n, aACD - npamoxyrnuii, a qe oBHaqae' Ep AD - AiaMerp KoJIa'

rl

'loaiAD=2 4-8(cm)'

an =L ln = 4 (crvr), sK Karer' ulo rrer(rrrb rportr r(yra 30"' flpone4eruo CE LAD'2

B ocDE: cE - cD.sin6o" = !!: =2.,6 (cvr).2

Bilnoaila: 2.tE cl..

25. 3a yrvronoro BD = 2O crrn, CD = 14 cu'-- r"*"= * + BO + AO = AB + BO + OD ='AB + BD'

IABO = r DCO aa trepiloro ogr{ardolo

(BO = CO, AO = OD 3a YMoBoro'

ZAOB = IDOC stt aePrlrxalrni)'3 pieuocti tPuxYtnuxie:eA = co = 14cm. P^oou= 14 + 20 = 34 (cu)'

26. Hexair ABC - Aas:zft rpqnyrHrll('

zA= 45", lB = 60" BC = 3J6 cm.

Ilponegewro CD LAB.- .ti saE

B t BCD\ DC = BC sin 60' = 3J6 ' V -- , t,*l'

29, O6,twcxzvto neai4ouy cropoxy-4C. BD LAC - Bucora pinuo6egpeuoro rprxyrurr-

Ya ABC, roAi 8D - ue4iaua, An = \ AC.'23 rABD (lD = 9O"): AB2 = AD2 't BD2i

AD = '[AB'-BD'=,t86'-Lo" = Jee -10X26+101 =.[to.so =4.6=24 (cvt.AC=24.2=48(cm).

AC = 1O.24 = 240 (cm').

2S 2.240

S*" =

sp

Lao2

4804. U {cM).

lo0AB+BC+AC 26+26+48

27. flozsa,*rttto aa r xoe@iqient nponopqiiiuocri'rolrt BD = 5x, AC = l2x'

L.s*.tzt =l2o; 3ox2 = l2o'; x'= 4;2

0).

To.ri BD =5.2= I0(cvr): nO=f,en=5 cttn:

AC = 12 ' 2 = 24 (crvr); OC = 12 cw'

B aBOC: BC2=BO2 + OC2; BC=J|";Lr' =tE|+L44 ={OS =fa "t"'P*"r= !3 ' 4 = 52 (cv)'

3a yuonoro

x=2(x>

45 4.240]t=_ =33.8 (cu).AB.BC.AC 26.26.48

30, V aABC ZC =9O", CD - 6icexrpuca /C,AD : BD = 5 : 12. CM - weqiana, CM = 26 cur.Ba snacrr4eicrro 6i€exrpzcz Hyra rpnnyruunanona 4iaurr cropoHy Ha sigpisru, nponopqifini

.ACAD1nprlJlerrtl M cTopoHaM: Atsc BD 12

flosuaqueuru xoe(piqieur nporopqifinocri l, uaemo: AC = 5x, BC = 72x, V nparro-xyrnouy rpnryrnrlxy ABC qenrp orucaroro KoJra - qe cepeAlua rinoreuyeu -4-8,rotty AM = CM = BM = R = 26 cm, AB = 26 . 2 = 52 (cm).Y rABC Ba reopeMoro flitparopa: AB2 - AC2 * BC'; 522 = (5x)2 + {l2x)';25f + L44xz = 27O4; x2 = 16; x = 4 {x > O).

AC = 5 .4 = 20 (cu). BC = 72. 4 - 48 (cu).P^*"= AC + BC + AB = 2O'+ 48 + 52 = 120 (cu).

31. Hexafi -4-BC - panufr, rpraftyrHlrr, AC = 5 cw, BC = 3 cu.CO - horo me4iana, CO = 3,5 em.

,{o6y4yelro a AC B lo raparerorpaMa, npoAoB*cr{Brxt4CO ga roqrty O na ni4pisox OD = CO = 3,5.Poar.nsnemo rCBD, y ,sxoro CD = 2CO = 7 cm,BD = 5 cvt Ba BJracrr{BocrfiMr{ rrapaJreJrorpaMa.3a reopenoro rocrauycieCD2 = CB2 + BD2 - 2CB . BD . cos ICBD,

cB2+BD2-cD2 9+25-49 -15 1cos Z('Bl)

32. V pirHo6ivHifi rpaneqii -ABCD (BC I AD) AB = BC == CD; BD - 4iarona,rr,, CF - nucora, O - roqra ixneperrzHy. CO = 15 cm, OF = L2 cv. a BOC cq a DO.Foa ABoMa.r{yramu (ZCOB = lFOD an neprr,rxa"rrui,ICBO = IFDO st* eHyrpiuui pieuocropogHi npr tra-paJreJrbrnx npsMr.rx BC i AD i ciqnift BD. A

. ICBD = 12O".2CB.BD 2-3-5 30 2

Ocxi.nrru cyci4Hi xyru napaJreJrorpaMa n cymi cranoe,rsrr 180', r:oIACB = IaO" 120" = 60'.

28. Hexaii eucota BD 4i;rurl cropony 4C na ei'4pisru

AD i CD, AB = 2$ cNr, BC = 5 cr$' 1A = 6O" '

B rABD.. BD = AB.sin6o" =&f= 3 (cu)'

3 r BDC: BCz = BDz * DC2 (teopeiua lli(paropa)'

Dc = JEc, - BD, = ^ts,- =,lri-= n4o = a (c}a).

{acrrxa 3

Sna.rens,s r i -R (pa4iycin Bnucauoro fi onucasoro ria auafigemo e tpopuyn 4,rao6.rucreus,fl n.noql rpzryrHurca.

abcS=DrTa S=-'4R

DM

642 lti; reouerptg EincHiHa o. 9. ra in. AxaservrivHur piBenb :i{iiii:i;r* ifiiii;iiii;i TEOMETPIC EincHixa O. 9. ra in. AxaAeuiqnrafr pieexu l!;il 643

Tl

ffis*fr$ffi

3 nogi6irocri rpuxyr:unrcin Br'ItrJIr,IBae nponopqifinicrr' rignonigrrax croplH:

coBo155oF= oD= n=1'

Ocxi,rrxv 3a yMoBoto BC = CD, to a BCD - piuHo6e4pen uit i ICBD = ICDB, toil

Do - 6icexrpuca ZCDF. Sa sracrneicro 6icexrpucrz #=#=iflognaqunro noeQirlienr nponopqiiiuocti r, roAi CD = 5x' DF = 4x.3 aCDF Ba reopeMoro fliiparopa CD' = CF' * FDz; 25x2 = l6x2 + 729; 9x'z = 729;

x2=8t:x=9(x>O).CD = 5. 9 = 45 (cu)r DF = 4 9 = 36 (crvr).

3a yuonoro BC = CD = 45 ctn. Ocsint'wuABCD * pinno6ivHa rpaneqia, roAD = 2. FD + KF = 2FD + BC = 2' 36+ 45 = 117 (cu).

s - BC + AD gp = a':lll .27 = 2187 (crsrl.

22

MOAvAb 2. BCryn [o crEPEoMETPli

S 2,1 . Ocxoeni noxqrrn crepeoMerpii. Axciorvlu crepeovrerpii

2.3. B) 1, 3'i 5.

2.9.frrqo n.norqi,rurz c i B neperutarorbcrr tro npfi-nrifi a, n.rrorqtnr 0 i y - no npmrifi b, nprauouynpaui a i b neperr.trarcrscn n rovqi C, ro qnTorrxa EaJrex{r{rr ycim.rpr,ou nnorqr,rnau cr, pi y, a ana.rrzrr,, aa arciouoro II, rourca C laarre-rrnrs i npauifi c neperaHy l,rorqlrn o i y.

E 2.2. Hacnlarr s arcioM crepeouerpii

Teopeua l. {epes nparry i Toqrty, qo He Harer$rrrt, ifi, uoxna trpoBecrrr nroq}r-ny i 4o roro nc riarxia o4ny.Teopeua 2. ,flHrTIo 4ni rovxa npauoi na;rexarr, u.norqraHi, ro i Bcs np,fiMa HaJre-xrrr qifi nnoulnni.Teopeua 3, 9epea rpfi Torrnrr, rqo Ee Ha,'retfiarr uprrvriii, Mo)+cHa npoBecrrr nJrorqu-ny i go roro :rc ri.nrxz o4ny.Ilpra poar'aeaxni crepeouer:pfiqnrzx sa.qaq fleprn oa nce neo6xigno cnpo6yoarznu4i.nuru trJrorqrrHy, rraM caMrrM eniuur,r sa4avy 4o n.nanitrerpuvnoi.Il.norqr,ruy oAnoaHarruo Br{aHat{arorb:1),qni npsui, rqo fieperr{rrarorbcfi;2)nparra i rovra, rrlo He HaJrextratr qifi rparaifi;3)rpra roura, sri ne nexa:rr na ogHift nparuifi;4)4si napa;re;rr,Hi upaui.

2.1O. B, A, E, }rC.

2.11. B) 2 i 3.

2. 1 2. I) Hecrcin.renuy rinrrcicrr.

2.13. I\ MD.

2.14.t) era.2.15. D 1, 3. 6.

2.16. A) (ABC) i.B) (MBC).

2.17. B) fl'arr.2.18.f)1i4.2.19. B, r. '

2.1 ,A) o.qua.

2,4.A) o4na.

2.2. n\ 6egni,a.

2.5. A) o,qria.

2.6,Hexaii a t b * 4ani npami, qo Marorr' cni;trny rouxy A, to,4i aa axciouoro II, ve-

pes Hux MotnHa trpoaecru nJrorurrHy, ao roro r+c ri.nrrcr'r o4uy. Ilpurryctrmo, ulo ic-

Hye Apyra cni.nrua roqxa npflMr,rx a i b - ro'rxa B' To,4i icuye Apyra trJrorqftEa' Iqo

npoxoAr{rb vepee npanni a i b, ane qe cyflepeqzrr, axciorri IIr. Orxce, Harne npfiuy-rrleun.tr HerrpaBumne. Ilpaui a i B neper:r'rnarorbcs lTr'rfire s o,4nifi ro'{qi.

2.7,flpunywulno, rqo rouxr A , B i C ne Jrerftarb ua ogniit npsuift. To4i uepea ,4ni npa-

vi AB i AC Morrcua rrpoBecrrr nJrotr{rlHy, i 4o roro x ri.nsxu o4uy (arcc. IIr).Are qe cynepevurs yltoni, 3a flKoro rovxra A, B i C lerxatr y ftorr(Hifi s 4eox pianrzx

nJrorqIaH. Or*ce, naure npunyu{ennfi HenpaBrrJrbue.

Toqrur A, B i C ;rexarr na o4nifi npauifi.

2,8,IJexafr fiJrorqfiHr{ a i p neperunarotics no npsmift a,

. b c ct.3a anciorroro II, nparaa o uicturr' yci cni.nruirovnu n;rorqun cr i B.Hexaft npaua b neperlruae n,'roqtruy g s ro'rqi ATo.ui A ..- cninrna rorrra nJrorqlrrru u i B, A e a'

2,2O, Hexafi nparrri a, b i c npoxo4nrb qepea rovxy O i nepe-rr4uarorr, npauy d oi4noni4no B rorrxax A, B i C.Ilpnrra d i ro.rxa O (O e d aa ymororc) o4uosHatrno Brrarra-qaloTb nrtolqr{Hy cI.lConrsa a flpsMrrx a, b i c v,ae B nroulrrHoro a 4ri cni,ruri rov-Kr, - roqxy O i roury neperrrHy 3 rpsMoro it (A, B a6o C).To4i aa nac.nigxorvr a axciorvr crepeoinerpii (reopeiraa 2) npami a, b, c i d tencarr r og-nift n.norqr,rni, ro6ro ro.rxr A, B, C i O .nexar:r s o4Hiti nloqnui.

2.21. Hexair ABC - 4anrzft rpnryruntt, BD - fioro *re4ia-lla, M - 'rorrxa neperfiHy rre4ian, rovxz A, C i M rrew'arsy n,rorqr,rni 6.Toqna D HaJrerfi r4rr, r'patljfi AC, a sHairrrrb, i n;roquui 6. To4isa reopeMoro 2 upnrra DM netxwrs y nlorqr.rni 6. B e DM,ro,qi B e 6.

2,22. Hexafi a - Ea:eta nprma, 3a asciorvroro I, icnyerouna A, rlo rre JreECr{Tb na npamifi o. Ba reopemoro1 uepea np,EMy a i rovrcy A MorxHa npoBecrra nJrorrlr4 -

ny or. 3a arciomoro I, icirye rouxa B, r{o rre Jrerfir.rrby n:rorqr,rni crr. Ilpore4euo qepes fipsMy a i ro'rry BnJrorqr,rny ctr. Il.norqlrnra o, i o, piaui, ToMy rrlo roqxaB n.norqrauu cx.2 He J'IelrdffTb y n.nouluui cr.

I.. flxa 6 He 6yna nrorrtrr- | IIr. flxuro aei piani npa- | IIn. flxu.1o 4si piani n,ro-

n]a, icnyroro rorrxu, u{o I mi trarors cniar,Hy tovxy, I rqr'IH}t MarcTb crri.lsny rou-

Axciomz crepeouerpii:

HaJre?+(arb uiii naorqilni. I ro .{epe3 IJ14x }lo)rcHa npo- | Ky' To BoHIz treperr'IHa}orb-

i roqmfi, srti He sarexars I necru nroqllHy i 4o roro I ca no npauiii' u{o npoxo-ifi.Ac d,C e o., l x ri,rtHu o4Hy. l Aursvepea ulorouxy.

644 l::I: TEOMETPI9 EincxiHa O. 9. ra ix. AxageuivHnfi piBeHd !tt{li:{:it* itlililii;:t IEOMETPI9 SincxiHa O. 9. ra in. Axaseuirnrafi pieenr tj:i:f 645

r

2.23. la:ai rpffMa a i ro.rxaA Bu3naqarorb nlonqlnry cx (te-operra 1). flrcrqo npaua b npoxogurr, vepea rovxyA i ne-peruuae npsMy a n roqrli B, ro nparvra b Mae s nllotrIunorocr 4ni piani cniarsi rovnu (A i B), rouy Jler+curr y nro-rqzri cr (reopeua 2).

2,24, fl.xr4o ro.rxu A, B, C i D He rlexcarb a oAHift n.norquxi, ro sci sour,I i l*cogui rpuB Hrrx He Jrelr{arr na o4nift npnuift, Tomy xoxua B qorr,Ipbox MorrtJrr{Bl,rx rpifior ro-uor (A, B, C\, (A, B, D), (A, C, D) i (8, C, D) ozaravae eAEHy rJrorr{uuy (teopeua3). Ifi vorzptr nroqtrHr e pisnrnmlr, 6o arrqo 6 6yAr-sni Asi s sux a6irarucs, rorouxra A, B, C i D Jre)fia;ru 6 n o,qnifi n:rorquni, rqo cynepevurr yuoni.Bidnoaidu: 4 n.nonlunr.

2,25, a)V npocropi icnye 6eariu rovox. ui4miuunx ri4 4anoi ro.Jlcl,r.A, Biasmeuo tovrcyB. gepea roqrcrl A i B npoae4ervro npamy. 3a axciorrtoro I, eisrveuo e npocropi rou-xy C, ara He Harrexcrrrb npamifi .43. 9epea nparry AB i rourcy C uoxcsa [poBecrl4eArrHy nJrorqraHy. Ocri.rirru i ro.*cy B, i roqxy C rvrolrtna sa.nrra 6eariqqro cnoco6in,ro i nlorqun vepea rovry A Mo)fiHa flpoBecrn 6eaJriq.6) 9epes Ani Aani roqrcz -4 i B npone4euo eArrHy rp,ffMy AB (axc. Ir). 3a axciolroroI, icnye 6eeri.{ Toqox y npocropi, gxi ne na.nexarr' nparvrifi -A3. 9epea rcor+*ry rarcyrouxy i nparvry-AB Ba reopeMoro 1 rvrorxsa trpoBecr[ nJrolqr4Hy. Orr+ce, vepea gri r:ov-Kil MorfiHa npoBecrn 6eaaiu n.roullrx.n) ,flrrqo 4ani rpra roqxn He Jrer{arrr na o4xifi np'nuifi, To sa reopeMoro 3 uepea uuxMOlrcHa npoBecTrr eAr{Hy rrJrorqrrHy.flxrqo xc rpr{ Toqrr,r Jre}rcarb ua o4nifi nprrvrifi, qepeo ra}tx Mo}Krra fipoBecrrz 6ea,ri.{nJronltrH.

2,26. Hexair ABC D - gauuir uorupurcyrxr{n, giarona-ni axo-ro AC i BD neperanarorrca n rovqi O. 3a axciouoro II,npnui AC i BD, rqo ueperrnarorbcs, oAnoqacHo BI4BHaga-

rorr, nJ'rorqzHy o, Ocrimxr,t npsui AC i BD .ne:rcarr y.n.lro-quni c, ro i roqxr A, B, C i D ;rerxarr y qifi nnorquni.

2.27.IIponegeuo npsrvri AB i AC.3a arciorvroro II, npore-AeMo rrepea Hrrx fiJrorqilHy o. 3a yMoBoro M e AB, K eAC, ro4i rosnu M i K .nen<arr y n,rou4uni cr, a sHaqr4rb,i npana MK aer'wru y n.rorqnui o (TeopeMa 2). X e MK,ornre, X e (!.

2.28. ABCD - pom6, a ll BD.Ilapa;ren xi npnMi a i BD oA-HOsHaqHO BrrSEaqaroTb trJrOrlUEy C[, TO'rKa neperr.rHy gia-rona.neft pou6a O rraJreJrc[Tb uparuii BD, a or:rce, i nro-u4vui o,. Togi giarona,rr /.C poilr6a rraJrexrrrrr, nnorqr.rui o(reopeua 2). Taxlru vr.rxou, yci BeprxrdHri pou6a i upauazr Jrerfiarb e oaHir? naouruni.,{ore4emo MeroAoM ni4 cynporueHoro, rqo npaui CD i a neperunaror6cs. flpianycrr.r-Mo, rr{o BoHr,r He fleperrruarorrca. To4i CD ll o, i vepes rouHy D, rqo ne Jrextllrr' rranpamifi a, npoxoar,rrb 4ei npaui (BD i CD), napa,re.nsni c, rrro cytrepequrr axciouiV (axcionri napa,rerbnr4x ),Bftcxosox: npaui CD i a reperuHaromca.

$2.3. flepepisr

2.3O. B) (B,BD); I) (/rD1); A) (,438,).

2.31 . B) t, 3, 4.

646 lt{: TEOMETPIq Eincxixa O. 9. ra ix. AraAeuri.{Hufi piBeHb :;:iltii::iiiil

2.36. PQ c (SBC), rosi X e BC. 2,37.To,rxu D i M nextart i e n.norqr.rni(AMD), i n nrorquni (CDB), oruce,DM - fi:r.ia [eperr.rny rlux nJrouI{H.

B

iti:::jitii$l: TEOMETP|9 EincHira O. 9- ra in. Axa4eruirxrai piaexu {J:l: 642

l

t^:x: i

I

I

2,34. Toqxa neperuny X nparvror KI i n.ro-ll1'uau (ADC) - qe roqr(a neperr,rnynpamoi ,I{F npauoi DC (DC c ADC);ocxi.nrxr.r X - cni.nrsa rorrxa firorquH(BCD) i (ADC), ro BoHa rrexurl Ha upfi-ltifi reperzry qux n,rorqzu DC.

2.35. EF c (C\CD), nxotlnuu (C.CD)i (ABC) neperuuarorbc.fl no npsuift CD.Or*ce, uryrana rorlrda - r{e roqra ne-perrany 1{ npsMr4x EF i CD.

l

i

i

2.33.8) 2 i 4.

'flxrqo naorqzna rpani Mrroronpa'Hrr*a i n,rorqzna nepepiay rvrarorr ,rsi crinrsi'roqftr{, To BoHrr fieperuHarorbc.a no npamifi, qo npoxoAr4Tb vepea r1i ro\nu (niHianepcmuHg naou4un).flpaua, no srcifi fiJ'roqr.rHa nepepiay neperHHac nJrour,rHy rpaui, Hasrnaerbcs cri-dom ntouqunu nepepi,ty.flo6y4ona repepiaie Muororpangunir ipynryerrcr ga nac;riaxax s axcioM cre-peorrerpii:1. ,fxrqo gri l,roqzulr uarorr 4ni cniltsi Torrnr{, To trpffMa, npoBeAeua vepes qiTOqKr.r, e npfiMoro nepeTr,rHy AaH[x nJrorqrrH.2. fxrqo 4ei napaae.nrni r,rorqrana uepernyri Tperboro, ro npami neperuHy na_pa,re;rrHi.cyrr merogy cnigir noanrae n no6ygoni 4onoui'Icnoi fipfiMoi, axa e ao6parxennarvr.iriHii nepernxy civHol n.norqunr4 s nJ,rotrIr4noro rpani $irypu.v sara.nrnouy Br4rraAlcy ciuna n,ro'4una leperrruae [Jrou-rr4Hy xoxnoi rpaHi nrno-r_o,rpaHrrr{Ka i ronrry 3 rrp,fiMax, rra fircux Jrertarr, pe6pa mrrororpannxxa,II{o6 no6y4ynarn c.ni4r.r Ha rpansx i pe6pax, oorpi6"o,I ) no6y4ysaru roqru [eperr4Hy caigy

" ,pu"",r";

2) gni rburn nepepisy, sxi rerxarr r ognifi rpaai, e,e4naru.

ry,B

2.38. fl;rorqranu (ABBr)i (BCC,) ltarorr.+si cniru-si roqxu B i 81, roMYBB, - myraaa.ninis ne-peTrrEy qr4x nJrorqr,rH.

B. C,a-7"'77"'',"| 1/IV

2"39, y nipauigi ABCD roqxa M - roqxa reperlrfiyrregian rpani rASC. flnou{nna nepepiay, u{o npoxo-

Arrrr' qepea pe6po DC i ro'rxy M, neperunae flJrolqu-

uy (ABC) no npsuiit CM. DK - roqrca neperr.rny npauoi /NCM i nnoqann (ABD)- CiT - //f \Ha nroulr.rHa Ivrac rsi cni.mni / Illtt \roqxw D i K o rpannro -ABD, d( ll/lhA .) nroai DK - riniq ilepernHv \ lll-lrut r/civnoi n.nonquu u i rpaui ABD. \[l'/'-aDCK - u:Yxanuft nePePio. Y

2. 44, TIo6y ay emo nepepia xy 6 a ABC DA rB rC rD, n no -rr{uHoro tlc, fiKa rrpoxoAarr, qepeg Bepurr.rHy B, i tov-xnMiN,geMeAAr, Ne CCr.Ocri,ruxr ro.rfir4 ,8r i N - cnilsHi .4JTff n,tou{Hrr L{

i (BB.C\), ro nparra BrN e np,fiMoro neperrzny 4uxtrJrou{}ru. 3'e4naeilro rorrHra B, i N.Anarori.{ao BrM - npfiMa rreperrruy nJrotquu txi (AAP\), Bnaft4emo caiA npsnoi BrM na nro-nluni nlrmnroi ocnonr ry6a. Ide cni.nsxa rovHarrrrorqrrH (AAP) i (ABC), roMy Mae rre?r(arrr Hanpruiii .4-B - npnmifi treperr{Hy qux nJrorrlr4n.AB^MBt=X.Ana;rori.ruo anafi4euo c.ni4 Y npsrroi BrN na nlorquri nzlxnroi ocnonu xy6a.To.rrlr X i Y -: cni.nsni rorrxrr nJlorquu u i (ABC), roMy flpsMa XY e npauoro ne-peTh Hy rlrrx n,lou{ff H.

XY nAD = K; XY aCD-- L. M i K - cni,rrni toqrcn DrrJrorr{EH a i (AArDr),N iL - trrrorq.un ai(DDrCr).3'e4naemo sigpiexaMra M i K, K i L, L i M.BILKM ruynauufi nepepia.Morxe rpanzrucb, ntro ror{Hrr M i N posrvlirqeni rarcrur{unoM, rrlo rpnrvra XY rre rreperuHae pe6pa Hnxnr,oiocnorz ry6a. V qronry B[naAKy moxua snaitrz rovxyfieperr,rHy Z npawoiXY a npflMorc CD a6o nolygyearuNK 'l BtM IMK ll B,N).

2.45. no6y.ayenro nepepis niparui-pn ABCD rrJrotqrrHoro, qo rrpo-xoAZrE,qepeo rovnu I{, S i P, .qeK e AD, S e AC, P e (ABD). IJo-BHaqnMo ii d. tx . (ACD) = 175.ToqHzl{i P.irexcarr y (/BD). IIpo-ne4emo npauy /{P , KP a AB = L.' Toqru S i .L ,remarr y ({BC).TTpoeeremo SL. a Kl,S - uy-xaNnit nepepie.

V 4pyrorrny BlrnaAr{y ro'rxa P ua n.norqurri(/3D) posuirqerra raftnM qfiHoM, r4o flpfiMaKP i nigpieox 1B He neperunarorr,cx, To4isuaxoAr4Mo cttil, X - Toqrty neperr,rHy nps-mux KP iAB..KP

^AB = X, KP ^

BD = R.flponogr.rmo npauy SX: SX

^ BC = @. r{orr,r-

pr,rxyrnlrx KRQS * uryr<auzft nepepie.

TecrAnc caMoKoHTponlor{acrrana 1

2.8. 3. A. 4. B. 5. A. ', 6. E.

DA

2 .4O , V rty6i ABC DArB ,C ,D , r - O -- rourca neperruy Ara-

roga-nefi rparri A,B'C'D'. Ciuna rurouluta (ABO)'

1) flaorqrzHa (AbO) neperr'rnae nJ'ron{rruy (APpr)no rpanifi KL, KL lL A3, ocrcixr,su AB 1l (ArBrCt)

i KL c (ArBrCr).2\ (ABO) a (da,n; -- AL, (ABO\ o (BB,C') = BK.3', ALKB - uyrcaHuft nePePie'

c1 . 2.4l.Ilo6y,qyerronepepianpsMoryruoronapaae'nenine-ga ABCDATBTCTDT n.noulrnoro' ffKa rrpoxoAr'rrb qepec

Hiuqi At, C1,- I; pl6"p, ttlo BrrxoAsrr' a eepurunu Dt'Iloagaqurvro cirrHy nrorlr4Hy {ALC p).

- (ArCrD) n (A,B,C,) = A,C,;,' (A,C,D) a (AA,D\ = A,D;

(App)^{ccp)=DCf'aArCrD - utYnauuk nepepis'

2.42.Hepea rouny B i nparay /D Morrcna npoBecrr [irorq[-uy (noana.rrzuo ii a) i ,qo roro x ri.nr,Hrjr oguy' 3nafigeuo,

sK rl,rr rrJroilIl4Ha rleperrrHae rpani i pe6pa ry6a'Ocrinrrcu roqxu A i B aelrtarr y unorqrzni cx, roABca.fl.noulunlr (AAID \) i (BB p J ndpa:re,rrHi, roMy rr'roq?rna ofieperr,IHac ix no napaleasHr'IX [psMr{x: BCr I ADrIllynautii nepepis - ABC.DT

2.43. qepee,qiarona,u -4D, rpani AA.D TD i cepe4zny M pe6-

pa 8.B, uoxua npoBecrrz fiJrorllrduy i 4o roro :x rilsxrao4ny. floaHaurzvo ii q..

Ocrcinrxu ro.rrra A i M e cui;rrur{Mrr AJr'fi rrirolqugcl i (-4BBr), ro AM * rrpfiMa rreperrHy 4I4x rrrorquu'3'eguaervro ro.rxNA i M silpiaxoNr.fl.norqunu (AAp) i (BB1C') napa,te.nsni, roMy nrrorqrrua

ct neperr.rHa€ ix no napa.ne.nbHrax rp.fiMr4x. qepea rovnyM napale.;ruro npa*rifi,4-D, [poBeAeMo nprIlry MN (N -roqfia [eperrrny o pe6porvr BrCr).OcHi.nrxn ro.rrrra N i D, - cni.nrui toqnpr rrJlorqr4H a i (ArBrCr)' to NDr - uprma

nep-er:rany qux nJrotr{r4 Ir.Ana.rorivuo, AM * tpawa fieperr'ruy [JroqlIH o'1(ABB.)'Illyxaurafi nepepia -

rrorrlpur(yrrrvo AMNDT.

64a :#!: rEoMETPlg Einsnixa o. 9. ra ix' AxagervrivHr'ri piBeHb i!:#:itiiil:i

9.A. - 10. B.

1. B. 7. A.

8. A B B I A4 3 5 2 1

]lilfiiiili!.r fEOMETPI9 Einsxixa O. 9. ra ix..Ara4ervri,rnrai pieexu filli: 649

15.

HanpuH.;ra4, MN - cepeAssr ttiuis rABD.1

MN =;DB.Ocxi.nrxn rpan.fiMrr xy6a e pinni HBaAparu, ixni Aia-rorraJri rar(ofir pieni, touy MN = KN = MK.

11. E. 12. t. 13. f, .U. 14. B.

16. B, f.|iacruaa2

17. ,{onegemo qe rBepAxeuHfl Mero,4oM ni4 cynporznnoro. flpznycrumo' qo npflMa 4

neperrrHae nci croponra rpunyrurxayrouraxK(7(e AB)' L(Le BC)iM (M e AC)'

ocrci.nsru TpfiKyTHrrn JreE(r,rTb y ,4eanift n.norquni cx,, MorI(eMo sacTocyBaTu anciomy

BSaeMHOrO poSMlrrIeHHfi TOqOTC Ha nJroil{I',rHr.

flpaua a neperrllrae ni4pieon AB, roiuy roqrcrz A i B ;rexarr yHax signocro A.AraroriuHo, roqxn C i B nentarr y piaulax nlBrrJro-

u{nHax. Ale, bcnilrxu npsMa a nepernuae ni4pi-aox AC, To roqftrr A i C raxox Jrelxar-s y piaurxniennoulunax aigHocno npannoi a. IIe osnaqae'rrlo roqxa C o4novacHo Mae SHaxoAurr'Icfi B ABoxpiauux ni elroulnuax siAuocso npxlroi a' rqo He-

Molr{JrqBo. Haue npznyrqeHH.a He[paBrlJr]'He.Ilparwa a He Mo)r{e neperuHarr't rperro cropouy rprrnyrrruna.

18. Hexafi rpra npami b, c i d tpoxoq.flrb qepea rouxy ? inonignoyroqfiax P,QiR.9epea rpalry a i Torrxy ? moxHa npoBecrr'I fiJrorquuy(nocuavumo ii a) i ao roro lr ri.nsmn orsy.To.rxra P i ? npamoi b Ha,telxarr nJronIXHi oc, ro4i aa

reopeMolo 2 PT c d. Ana,rori'{Ho TQ c ct, TR c a.P € a, Re oc, ro4i PR c ct. Q e PR' rouy @ e g.Bracnoeoft: rourcrl ?, P, Q i R Jrelnarb r ognifi n.nouluHi.

19. Hi.

20, rlepea npami -A-B i ,4C, ulo neperr'rnalorbcfi u tovqiA, uorxH€r nponecrll nJrorqr4ny (.4-BC) i go roro xt ri.nr,-

rcrr oAHy. Toqrru M i N HaJre)fiarr, ni4noni4no flpfiMrrM

AB i AC, a or)fie, rerrtarb y nlouluui cr.

BracnoeoH: rouxu M, N, B i C Jrextarb n oguifi n;ro-qrzni.

21. [si Bepmrnu -A i B rplrxyrnlatta ABC i ro.{ra ne-.perrrrry lregiar M Jrelrcarb y nrouluni a. AMr, BMr,CM, .._ megianu rpr{rcyrHrrxa, AII r BH z, CH

" - ftoro

Brrcorr.r, I{ - roqrca fieperrrHy Brrcor.M e u, M" € cr, roMy MMrc a, Q e MM* C e a.3ri4cu ezrnzsae, qo BC - d", AC c d".

Toqnu A i fI, .ne;xarr y n.norqzni u. H e AE' sri[-cuHea.

22, (xrxrx"), (xrx"xn), (xrxrxn), (xrx"x). 23. lue. !& 2.24.

.c24.

pianux ninnrorqil-Da

2s 4 26

A1 MilsP)@ -w^r:.u''";.

'' 9acruna 3

29. Hexafi npnui a i b neperranarorr,ca n rovqi O, Ilpama c He

npoxoAr,rrb qepes roqfiy O i neperzuae npami a i b riAuonia-nosroqxaxMiN.Ilpxui a i b aa arciouoro II, ognoanavno BusHaqarorb n;rotrI[-ny cr. Touxn M i N npauoi c Harexarb n:roulnui d,, a or,fie,i npnma c Ha-flexctrrr, n.rrorqtt ni cr,

3O. Hexafi Aec.rrrb fipffMrrx fipoxoAflTb qepeo Toqriy O i neperuHarorb trpsMy a. r{epea

np,fiMy a i rovxy O uontHa trpoBecrz nroulr.rHy o, i 4o r:oro rr ri:nnu oAny (reoperyra 1).Koxcsa a Aecsr[ trpsMr.rx Mae B rrJ'rorlrrHoro o 4ri cni;rrni roqrcr.r: O i rouxy neperlHyB npflMoro a. ToAi aa reopeMoro 2 xorxsa B qux trpsMzx rex[Tb y uorquui ct.

31. Yripani4rKLMN P e LK, Re 1{N, S e ML.Ilo6yAyeno nepepis nipauigu uoqzHoro (P/?S).l) P e (LKM), S e (LKM), roruy PS c {LKM),PS=(PRs) a(LKM).2) P e (LKN), R e (LKN), romy P-R c (LKN), I'P1'r=(PfiS)^(IKl/).Bnaii4eno 4ni croponn (pirypn repepiay: eigpiexz PS i P.R.

Toqxa S - cninrHa roqfia ABox nr'roilInu (PRS) i (rNM).

aC

MK

tA

nepeTI4HaIoTL nP,flMY 4 BrA-

650 r:;:::: fEOMETPIS EincHixa O. 9. ra ix. Axa,4euivHhfi piBeHb l!i;i::;:+i;ii: ili::ili*iil: TEOMETPI9 EincHiHa O. 9. ra ix. Axaseuivxrafi pieenu f:;1 651

r

3.18, ABCD - naparre,rorpaM, a MBCN - rpaneqis'rrlo Marorr' cninrny cropoHy BC. BC ll MN sx ocnosurpaneqii, BC ll AD .alc nporr4Jrelr(Hi croponz napaJle-

JrorpaMa. "lop.i MN ll AD sa o3narcorc napaae.nrnoctinpflMnx y npocropi.

3.2O. SABC - rpuldyrua nipami4a. Touru I{, Z' M, N -cepeArzHr.r ni4piaxin SA, AC, BC, SB niAnoni4Ho.

sA = sB = sc = b, AB = BC - AC = a.V ar4SC KZ - cepegua rinia (ea oanavennau).

"Ioti KL = anaroriquo i NM =

M

3,19. CD - cni.rrna ocnoBa rpaneqii4 A-BCD i PZCD'srti He rexcarr e ognifi r.norqusi. @.R - cepegus liniarpaneqir ABCD; QR ll CD. Asa.;rori'mo, MN ll CD-

Toai Qli ll IWt'l sa oaHaxoro napare,rr'nocri npflMnxy npocropi.

zm =!tn=2

3.22. [).3.23. B) CD;t)AB; [) BC;E\AD.3.24.8\ 2 r 4.

fl.norquna o .rirrlrr, npocrip Ha 4na ninnpo-cropz.,flnrqo roqxuA i B;rexars y piaHzxninnpocropax, ro niApisox.A.B fieperlrHae nJro-rr{rrHy cx.. Hannaxz, axrqo rourz C i D.nerxarr,B oaHoMy nisnpocropi si4uocHo cr, ro nigpiaoxCD ne neperunae rlro nJrorqlrHy.

3.25- B) BiApiaoH.Ocxi.nrrcz rrps:o.tiAB, AC tAD ne-

'r{arb Ha o4Hifi n.nor4zHi, To ro.{-

xu Br, C, i D, - cnirrHi roqrrztrJrorrlrH (ABC) i o. Ornre, nonurelrcarr, ira cninrnifi npauifi quxrrJrorqrrH i yrnoprororr niapiaox.

A

V IABC ZM - cepegna nivia,

Lcs -9221cs = 9,22

a,i

I

l,rY:r..-..'....!:ii./-.:;:::.iil+!;it:+:.:.i$ii1:.*':J!ildil*:::::i;Ftil1;!:.:::;i

ilt$*-l:::::.:l

AHanorivHo, KN =

P*"r,

3.21. Toqrcz A, B, C i D ne ,rerxarr, 6 oAniji naorquui.To,txu M, N, K i Z - cepeArrltu ni4piaide AD' BD,BC i AC, CD AB.3a ossase*:r.flw MZ cepeAHs niHis a.A-DC:

MZ t DC. lwt,=-l oc.-2Ana,rorivno, NK - cepe4xa .irini,s aBDC:

Nx =Loc. NK I DC.2

ToAi sa ogHaxoro MZ ll NK i MZ = NK.flonroprenrra nipxynanna, e rprzxyrguxin ABD t A-BC MN I AB t KZ ll AB, sr.ilv.u

1MNllKZ. MN=ZK---AB.2

Are aa yuonoro CD = AB' toti MN = NK = KZ = MZ.gorupuryrnux MNKZ - poiu6 (nporra;rerxHi cropoHu napalellui, rci cropoHupinni).MK I NZ nr giarona;ri pou6a.

3.26.3a yirronoro n.nonluua cr i nJrorq[Ha rpzr{yrnuNaABC treper[-Harorbcs tro rpfluifi BrC,. BC ll B p r BC = t5 cm, AB, : B rB = 2 : 3.yABC * aABrCr 3a ABoMa xliramr.r (ZA - cnin:ruuir., ZABC == /AB:C, ax ei4noni4Hi trpu trapaJreJrsHrx npsMrx BC i BtCr

!ds=9.22bbaa=-+-+-+--a+0,2222

i civuifi -4-B).

"Ioni M - BC , B.c. = AB1 'BC

=ABl B'C' AB

Bi1noeiOa: f) 6 cn.

AB, BCABr + BB1

2 .15= s =tt ("*)'

3.27 , 3a oBHaqeHHsM KM, MN i I{N - nignoniguo ce-pelHi ainii rpuxyruuxin A tOB r B tOC t i A.OC t"toai KM =IUU, rwN =!a,c,, KN =;4c,

o*,, = Lo,", , )",c,.1r,\c, =

- f,,+u, + B,c, + ,+,c1 = |r* ",r.

Bilnoeidu: +* =:'M,B,c,

$ 3.2. BsaeruHe po3Miu4eHHc npeMoT i nnotr1nH14 y npocropi

Moxnuei raxi nzna4nz BBaeMHoro poeuirqenna [pflMoii i nraorrlaur4 y

Teopeua 3. ,fxrqo oAHa B ABoxi 4pyra npaua rar{ox nepertrHa€

napaJreJrr,Hux rrpfiMfix nepeTr.rEae nJrorr{r4rry, Toqro nJrotrI[Hv,

Ilpama rtt Jrexturby n;rorquni cx

nca(Bes,riv cni;nHux rovox)

Ilpaua I reperrruaenJrorqrrHy cr r rouqi A

lr:a=A(O4Ha cni.nrna r:ovxa)

Ilnorqraira cr i npaua anapa"re.nsui

a ll cx,

(Heuae cni.rrnrzx rovorc)

654 ;::# rEOMETPIq Einsxixa O. 9. ra in. AragevrivHrfi piBeHb iii::;i;ii;iiiiii:i ii:;iif;:lltit TEOMETPI9 Eincxixa O. 9. ra ix. Axa.4erri.rxrafi pieenu iil 655

Ilpsna, fifta treperunae gni croponrE Tpr{r(yrnllfia i napa:realna rperifi cropoui'ei4rnnae ni4 nroro rpr{nyrHun, no4t6nuit AaHoMy.

.RNB"r-i--1

FW--T-a--r

!ir;,,. -....{i;i'ri:l{.itr

liiif;::i,;::.li

3.32. a)

ZN/A '-'/

3.28, Pasirue 6y.xo 4one4euo, rqo sci npaui, napa.re;t'ui ognifi a 4rox npnnczx' sxirreperr{Harorr,c,fi, rqo [eperrrEaror], Apyry np,flMy' Jre]fiarb s oAHift naorqzui' Orrxe'

nparui-4Ar, BBliCcrJrelrcarb s o,qnift n;roulrani. Bnauzrl, 'ro'tt+uAr,8,, C'nexatrna nparrifi ArB, neperrzny nJrorr{t4H (ABB1) i o'3a reopemoro'@-a.n eca C, - cepe4una ri4piexaArBr' A qe oenavae, u1o CC, -- cepe4-

n,s ;rini.n rpaneqii -4'ABB, aa reopeuoro rrpo cepeAulo niniro tpane4ii:

ac. =!.(AA +,88,) = I .1^2+tG\= 14 (cM).o' 2

Bi0noaidr': f) 14 cu.

3.29, Maeruo IABC a aADrD, a orlrce, ri4nori4ni cro-

po'r4 uxx rpurcyrunxin noAi6si: #=

4*,t

BC AD, 8. 4,5 8 4,5""1 AB (AD, - D131 .1.5 + l,5

8.4.56

Bidnoaida: A) 6 cm.

3.3O, MN - gatnit' niApisox, av'uir se [eperr{rrae rrJrorlrnxy a, K * fioro cepe4r'rxa'

flpone.nn MM, ll KKt ll NNr, Kfr = 9 cru, NNr = 15 cu'Paninre 6ylo goxe4eno, rqo npaui MMr KKr i Nll, ne-

xarr n o4niii nlolluni' 3a reopemoro (Da,reca K, - ce'peAzna si"qpiaft a M,N r, a KK, - cepeAu,fi ninia rpane4iiMIMNNL1MM,1l NN,):

1yy, = l1MM, + NN,), sP.is$u 2KK,= MMr't NNi' 2'

MM,= 2111, - NNr = 2'9 - 15 = 18 - 15 = 3 (clt)'Bidnoaida: A) 3 clr'

3'31. Hexafi P e u, Z E 0. .IIpoBeIfi PK ll ZM' PK = 6 ctfl, ZM ='9 cu. flapalre.nrui

nparrl PK i LM oguoa]flauHo Bt4sHarraroTb AeJrrdy nJtonII4Hy F, ,srca fiepeTl4Hae nJIO-

rr{r4ny c, no nparrift PZ. Baywonow ni4pisox KM neperunae nJronlr'IHy cr n :ro'rqi O'

roti O e PZ.yOPK * yOZM aa ABoMa liyraMr (ZPOK = IMOZ sxrep'rraxa;rrni, lK = lM sx rnyrpitrni pianoiroponni npu

PK OKMZ I PK i ciqniri MK):

-- =

- .MZ OM

Hexafi OK = r cM, ro,qi OM = MK - OK = (6 - r) cm.

9= ", 6(6-x)=9x; 36 - 6x=9x;irlx=36; x=2,4.9 6-*

OK = 2,4 cv, MO = 6 * 2,4 = 3'6 (cu).

Vci cnimni rorrxr4 rrJron{r{H (ABC\ i o Jrerrtarl sa ix cnimnifi npsrvrift -4-B (axcio-ua IIr). Ocxfumxn X, e (ABC) i X, e u, ro X, e AB.,{;rr no6y4onn ruynanoi rouxr,r X, na nigpiary AC (prc. a) a6o ua iioro npo4on:rcenni -nponeHi AC (pzc. 6) siarMervo rouxy X i npoeegemo nigpisox XX, (X, e AB).

3.33. Ilapaaersiri npaiui BC i AD oAuooaaqrro Blrorraqa-rorb rrrrorrluay (ABC), axa neperlruae nnoqr.rny cL no

pernuy HerrapaJreJrbHrlx rrpfiM{x CD i AB. Ocri.nrrunpama CD q'u,, ro X - roqxa neperprny npauoi CDi n.notqr,rnu o.

3.34. IAXD * a BXC (BC 1l AD aa yuonoro).

^ADAX'' BC BXIjexafi AX = r cM, roAi BX = AX + AB = (x + 2) cu.qe'= = -:-; 2(t + 2\ = 3x; 2x . 4 = 3x1. x = 4.3 x+2Bi}noeidt:AX=4cw.

3.35. Il.norqrana (ABC) np,fiMonyrHr,rxa ABC D nepermtaetrJronlrrHy cr no npauifi AD. Togl aci cniarxi rouxz,n rouy vr.rc.ni fi ro.rxa X, .nencarr, na npauifi.AD.3a y*ronoro DM : MC = 2 : 3'. To,qi 6a y3araJrrtHeHororeopeMoro (Da.neca DX t XA= 2 : 3. Orxe, ro.rxa XMae 3rraxoAr4Tr,rcr na ri4pisxy AD rax,.iqo6 nraxony-Ba;racb qfi pinnicrr.

3.36. Ilo6ynona orucaHa s sa,4a.ri 3.35.3a ymonoro SM : MR -- 2 : 3.Togi, ocni.nrxr MN ll RT, ro SN : NT = 2 : 3Ba yaaraJrbrreuoro reopeMoro <Dareca. ,SZ = 10 cu."SN = 2x, NT = 3x, ST = 5 ti 5x ='l.D; x = 2.sN= 2.2=4 (cm); NT= 3 2=6 (cm).Bi1noei1t:N?=6cu.

3-37.HexaftAB - patazir ni4piaox. C e AB,AC : CB = 2:3.Ocxi.nncu napa.ne.rrni npnui,44r, B B r i CC r rreperr.raarorr,ri4piaor AB, ro sci BoHE re:rcarb n o4nifi n.norquni, araneperrrnae [Jrorq[Hy o uo nparriii ArBr.3a reopervroro @a;reca, ocxi.nrxz AC : CB = 2 : 3,roi-4rC, tCrBr=22$.To4iA.B, : ArCr= (2 + 3) : 2 = 5 : 2.

flpaua i rrJrorqr.rua MoJ{cyrr, 6yru posuiqeui r upocropi rardr4M trr4roM:a) rpsva Jlenurb y naoqrani;6) npama fleperrrHae rrJrorqusy;n) npnua napare,rbHa n.norqrzni.

3.38. Ocri.nrrcu nparriAB i BC neperuxarorr, rrJrorqr4rry c, ro BoHn ifi ue napa-nenrni.AI3 ll DC, BC ll AD sft flport4Jre]fiHi croponlr naparer'rorpaMa. ToAi ra reoperrorc 3(rro4y;m 3), nxqo npaua AB ueperarrae nJrorqrray cx,, ro i nparua DC ii nepernHae.

;ii:illtl:.i;;{'! FEOMETPI9 EincxiHa O. 9. ra ix. Axa4euiqnlrft piaens ii:j 65V656 :i;i: fEOMETPIS Elncxixa O. 9. ra ix. AxagervrisHrfi piBeHb i{tiii*ff

0

A

3.39. Hexaii yABC .lew"yrrs y nlorquHi a, a tABD -e nrouluui F. ln - ix cnilsna cropoua' rqo HaJrelfir4Tb

npnuifi neperuuy fi,rloqrrn cx' i B. Ilpaua CD neperrrHae

r,roqnHr cr i F siAnosiAHo B roqrcax C i D. Eygr-nxaintra npnivra, trapaJreJrbHa CD, neperzuae nJrorrluHr4

cx, i B aa reopeMoro 3 (rrno4Ym 3).

$ 3.3. l-lapanelbHicrb npcMoT i nnotqxHl't

3.4O. I) Besriq.

3.41. B).

3.42. A) oAHy.

g.43. BPt - npflMa reperr4Hy nroqt4H (ABC) i a. BC ll cx.' roMy BC "ll

Bpr''foai IABC N aA-B,C, 3a ,qBoMa lcyraMlr.

AB BC3a yuonorc ABr: BB, = 3 : 1, ABr= 3r, BBr- ia, AB = 4x'

""r- "r"rtAB. B.C. 4x .72BC - '* 1:1 = -^. '" = 16 (cY).

A-8, 3.r

Bidnoaida: B) 6 cm.

9.44. MN - cepeAgfl ninia rpaneqii ABCD, rofi AD ll MN;

MN =AD-* BC: MN = ltlz*gl=13 (cu).

22Bi}noei1u: B) 13 cu.

3.45. 3a reopeMoro @areca N - cepegrna CD, 6o ea yMoBoro AD I a' a suawrt', AD

naparrerrbHa rinii neperzHy (ABC) i Q' P*"o = 2AB + AD + BC = 2AB + 2MN' pe

MN - cepet:ea :rinia rpanerlii. PBID= 2' 8 + 2' 16 = 16 + 32 = 48 (cu)'

Bi)noai1a: f) 48 ctu.

3.46. B) a ll 6.

3.47.8) 3 i 4.

659 t:::i: FEOMETPIS Eincxira O. 9. ra in. Axa.qervrirHufi piBeHb i;ii:1:iiii;ii:i

3.49, Ocxiarxu AC ll BD, ro aAOC N a BOD Ba ABoMa KyraMv.TosiCO:OD=AO:OB=3:5.

CO : 10 =3 : 5; CO=+j=6 (cru). CD = CO + OD=6 + 10= 16 (cu).

Bidnoeidt: [) l6 cm.

3,50. Ocri.nrrlr AAtll CCriAAtll BBr, ro sa osnar{oro napa,re.nruocri npffMr4x BB1 I CC1.3a ymoeorc BBt=CCr Horrpuxyrrr.rx BBPTC iraac 4si npornrerrcHi cropouu, pienii napa.ne,rr,ni: L{e oanauae, r4o BBTCTC : napaJreJrorpau i BtCl = BC.

3.51 . YorzpuH yruux AATC TC , y srcoro nporn,rexni cropoHr.r AtC 1 i AC pieHi i napa-

.ne.nrni, e napaJrer'rorpaMow, rogi CC, ll AAr.AHa.noriuuo, BB1 1l AAr. ToAi aa ocnanoro naparerraocri npnMux BBr ll CCr

3.52. Ilporra:rerxHi cropouz napare- 3.53. Hexaii .43CD pon6, y ;rnoro BD =rorpauis naparelrxi i pinui, romy =AB =AD, MN - tiuis treperuHy a;rorqu-CD = AB - CrDr, CD I AB ) DPr uu pon6a rrour,r4Holo. napa.rre.nrHoro giaro-3a oananoro napa"tensnocri npfiMlrx wati BD. 3a ymororo M i N - cepeAunr4

3.48. Pll ll a, orxe, P" ll SO (S@ - ;rinis rreperrruy rrroqrnn d i (P-RD).

=H; Pn=RS ISP =7 -35.42\cn)tPRa Si?@ o a PfiT (aa 4noua

pr =sQ.pn =3'42 =t8s,R 7BiOnoai1t: B) 18 cu.

.s@KVTaMU t-

PT(ciur).

CD C,D,.Bzcnosor: CCtDp - napaJreJro-rpaM.

ctopin BC i DC, MN = 6 cm.

Toti MN =)"o, BD=AB=6.2=t2(cv).'' 2P*"o = 4AB - 12 . 4 = 48 (cm).Bi1noai)a: 48 cut,

A

3.54. Heiafi P@ - riHia neperzHy trJrorqrEu cr i firoil{r4-an (MNK). Ba yuoaoro MN )1 a, rosi MN I PQ. Touy

r MNK aa QPK, esiAnn y! - !!. flxqo KP= r cm,QP PK

roNK=(r+8)cn.t-t

= **t, 1Br=9r | 72i 4x = 72; x, lB.9x

BiOnoaiOa: L8 ctut.

3.55- Hexafi s -

trror{uHa, npoBeAeHaqepea xiHeqr eigpiaxa OA - rouxy O. B e OA,BBl ll AAr Bi4piaorc OA i npani BBri AA1,'rerfiarr, e oAHifi n.norqnHi, flKa reperrr-Hae uorluHy o ro rpauifi OAr,V IOAA, BB1 I AA' romy a BB'O N rAArO.

Buio*r BB' - oB - 04 'A4 oA 04BB,.OA BB..OA-Ll AA, ,_. , ,.(cv);

OB OB,

osxa.reHss. flpaua naaasaerybctr naparNenaaoa nnotu,urti, JlHrrlo BoHa He Mae ,fioA-

uoi cri.nrnoi Toqrcr4 3 Hero.ossaxa. flxrqo npaua, sr{a He HaJrelfir{rr nnoulnHi, [apa.'IeJlr,Ha nrcifi-ne6yAr nps-uiji ga uifi nnorrllrni, To BoHa lapaJrerr'Ha i cauifi l,noquHi.

Hac:rigorc 3. flxrqo np,s-Ma fiapareJrbHa Hoxnifiia n.rorqzn, ,flHi neperfi-Harorbcff, To BoHa [apa-ae.nr,na i npnmifi ix uePe-T14Hy.

Hasriaon 2. Icnye 6ea,rivtrpsMrrx, RHi napa-ne.nrHio4Hiii i rift cauifi nno-rrrr,rni.

CI

Hac;ri4orc 1. ,flxu1o npsiualapuerbrra rronlzni, roqepea rco)lcuy rovxy qieiIIJTOTqI{HII MO)fiHa npoBec-rH e sifi flpsMy, [apa-ne;nny 4aHifi.

ry,

ififiiiiili rEoMETPlg Einnxixa O. 9. ra ix. Axa4euivHr4i piBeHb ii:ir 6s9

2) 4= !-9 = ro ("*);

Bt AA, = 11j

= so ("*),

a.(a+b\4) A4= o- tct").

3.56, Il.roulzna (BMC) treperr{Eae nrrorqruy (ABC) nonparaii BC. To6ro BC e (BMC)' AD 1l BC sn ocl{oBn

rpaneqii.3a osnaroro uapaae.nrnocri npauoi i trJIoIrIrrH['ocxiarxr.r.4-D ll BC i BC c (BMC), ro,4-D il @Mc).

rqrani, arrfi r{aJreJrcarb i nparri lM i CB (reoperraa 1, uogy.nr 2). A:ie qe HeMornJrI4Bo,6o sa ymonoro npav.a AM fleperrrnae nJrorquny (ABC), axiit r{a;refirurb nparvra BC.B[cnosox: MK i Pq - rrurro6in*ri.

11. DC=!t*=9=B tcu); AB=DC: AD=BC=l-xrw=8=4 (cm).2 2 2 2 -',-"','

Pou"o = (4 + 3) . 2 = 14 (ciu).Bi}noai1u: 74 cw.

12, AB =.5 cvr (erunercrrufi aABC).

A,B, - cepeAHn ainin, ,4r8, = ! as = 2,5 cl'r.2

Bidnoaila: I) 2,5 cna.

19. AtBt,BtCt,CrD,AiDr-cepe,qnininiiei4noni4uorpzrcyrnnxinAPB,BPC,CPD,APD.Cropour.r r{oruprrnyrnr,rfta ArBrCrD, bAsiqi naeumi sa cropo:r':a ABCD.

s,o., =9 l=t.4=12 (cru,).22Bidnoaida: f) 12 cu'.

1 4, 3a reopemoro (Da.neca MO - cepegna :rinist IABC. llO = 1 AC;2

AB = 2 . MB = 2..5 = 10 (cu). 3a reopeMoro lli@aropa,4-B' = BC2 -t ACz;

BC=,[AB'- AC2 = Jt0-6" =fi00-a6 =JA=8 (mr)..ro=;.8=4 (crvr).

Bidnoai}a: B) 4 cu.

15. IABC - aAQD {BC II QD,6o BC ]I a, a QD - Cx i @D C (ABC).AB

= B9, qp =

AQ'ac - (1 F 2)'5

= lb (cnn).AQQDABl

BiAnosiAb: I) 15 cm.

16. AB=lPo"",=4 cm. MN =I^(AB+zK); ZK--2MN -AB=2.7 -4-r0(cu).4 *'," 2'Bidnoeida: A) 10 cu.

3.57. Hexafi SABC - rerpaeAp. flror4rana cx rpoxoAr'rrbqepe3 BeprnfiHy S i nepenuuae pelpaAC i BC ocnosuri4noai4no y ro.ncax M i N' M - cepeguna AC' N -cepeAtrHa BC. ToAi 3a osrraqerrH,fiM MN - cepeAss

nittis IABC i MN ll .A'8. Ocsi;mnu MN c (MNS)'To Ba oauarcoro napa.nearnocri npfiMoi i naolquuuAB ll(MNs).

3.58. Ilapa;re.nrni npami AA, i BBl oArrosrraarHo Br,r3r{aqalorb [JIolqIIHy. BilpisoH /-BHaJrelnrrrE, qiii nnorqlrni, 6o A e AAr, B e BBr.tAOAr* tBoBraa,4BoMa KyraMr{ (zao'd,r-- lBoBl sH Beprr{r{turbgi, lA= lB ax

I$;li,i:,:*X uuvtpiruni c4nocroponni)' 't"A' #,= H= Hi#fiii:::::i'f L\ u = 40 , a4 = ELg =+n =,2 (cv)Fj,:l:::i:::::i:S: BB, OB oB b

t+f:i:ii:iii+ AA Ao ^o b

iiifii:iillil ', "d- a4' oa e'il$?ti:ll:::ll rlosuavnnlo -4,o -- 5x; oB, = 6xi 5x + 6x = 22;

11x = 22: x -- 2.A,O=5 2=lO (cnr), OBr -6'2= 12(cu)'

TecrAnc caMoKoHTPotK)

A,/l/L-A--l /

/;' \o B,/

A' 9ac'ruxa 2

9acruna 1

't. A.

2. B.

s. I A-ftlf----#tEtStlelt I

4. A) AA.; B) BB,; I) CC,.

5. B),4A,; [) BBr

10. A) Mn*ro5irtni, flrc6u tpanv;t MK i PQ 6ynn trapaJreJrburiMr a6o nepetuna'nlrcn'

To Boura refi<arrr 6 n o4uifi uorquui. Togi rouxra K, Q, M i P 'nexcarr s oAHift nro-

660 iii:l reOfuerplg SincxiHa O. 9. ra in' Ara4ennirHuf piBeHb :{::::ililiii

17. Ocxiamrr DDLllAC,to tBDD.a tBAC.

AB.DD. (2+7t.7 7.9AC -- ' 3I.5 (cul.

BD22Bidnoeida: 31,5 em.

18. MN 11 .4-B, romy ^ MCN * oaCB. k =!.5

aABC * npsuonyruuft, lC = 9O', AB = 13 cut, AC = 5 cln. Ocxi.nsxrz xarer,trIo Jrelr{urb trporr{ Kyra 30", 4opinHroe noaorlrni rinorenysu, ro 1A = 30". To4i

IBC=;AB =6,5 cu. MN llAB,romy d MCNa ICAB,

Cix croponn i nepnmerpra nponopqifiui. Ba yuororoCM : MA = 1 : 4, ro.4i CM : CA= P^"**: P^"*= 1,5.

p..,,, =Po,* -5+13+6'5=4,9 cv.-^cntN 2 5

Bidnoai1o:4,9 c*r.

Bp _ p4.AB AC'

6. E) DDTCTC: B'l AtBtCtDt.

7. A) ABBfLi B) DCCpr

B.[ A LE]ls I r I

lclr IL4lrlsllrlzl

9. B) 10 cu.

:,iiilitiiiii TEOMETPI9 SincHiHa O. 9. ra in. Axa.qerui,rnrafi pieexr i;jlj: 6GI

1 9. flparvri,4A 1, BB\, CCli niapisolt ,4.B retr{arb e o4uifi trJrolqrrHr' flfta reperrrrae rrro

rrlrany o no nPauift ArBr.CC, - ""o"o;"

.niuia'rpaneqii A.ABBT Hexait AA, = rc cM' roAi CC1-- Qc + 4) cM'

,", = 4:out: 2CC, = q+ BB,:2'2(x + 4)= r i 10; 2x * I --r * 10; x = 2' AAt= 2 cr't'

20. AK - paa:rrit' siApiaox, Ae a, B e a. B € AI<, BM a'BM = 8 cm. KM n cr = Q. AQ - riuia neperr{Hy trJrotqEH cI

i (AKq. BM I a, touy BM ll A@ (inanure BM nepertuanac;

6n s AQ i nloutunoro cr).

^BKM *

^Af(QiBK _BM. Ao-AK BM,{4+7) 8 --Lt.2=22 (cwr).

AKASBK4Bidnoaila: 22 ct:tt'

21 . PQ ll P,Q, (P181 = (PQR) ^

ct).

PR .Q.n _ Ptq.a PRQ c a P,.RQ,: Tn= en pg

Hexaft PP, = 2x, PR = 3x, ro4i PrR = x'P.R PO x.30p.O.=:t::--:1 =';" = l0 (cm).

PR 3x

22, [us. puc. Ao aaaa'li 21.

PPr= 2t, PrR= 7x, PR= x. 4Qt

*-3 =?r 3(r-3) =2x;3x 9=2x: x=9.r 3'AO = 9 cu; BO = I + 3 = 12 (cu). AB = 9 + 12 = 21 (cv).

26. Puc. i noscuesss .qzs. .4o J$ 25.OAt = x cu, OB, = (r + 9) cu.Ao -40 =?,

* =?:,bx=2(x+ 9); bx= 2x+78;Bx=t8;x=6.OB OB, 5' x+9 5'-'-

OAr= 6 cmi OBr= 6 + 9 = 15 (cM). ArBr= OA, + OBt=6 + 15 = 21 (cu).

27. {epea napare;mui npaui,4A, i BB, npoxo4zrr, naorqrHa i go

roro lrt ri.nrnr'r ogna. Iloeuaunrrno ir B. Poarrsseuo ga.ni lra-JrroHoK y n.norqzni B. QO AA, ll BBr; Q - cepe4usa AB.

Y aBAB,QC cepeass aisis; QC =I"",.

rq risis: o.c = ! da..V aAArB, Q,C - cepear , 211

To,4i QQ, =QC-Qp=;-(BB,- Az+tl--;(11-7t =2 (cm)

28. Iloscsesns asa;r'oriqse 4o Nb 27.

MK =! DD, \.16 = 8 (cm).2'2M,K=\cc,=1.4 2 (cu).'22

' MMr= MK - MtK = 8 - 2 = 6 (cru).

tlacruxa 3

29, Poar.nsueuo rrpecreuH,fi n nlorqnni rpuxyrnuxa-A,BC.KM - tinir. neperrHy rrolqr4urr (ABC) s nJlotqr4uoro,[apaJreJrr,Horo npanrift AC (u).K e AB, M e BC, KM = 4 e\r, AC = I cv, MC = 15 cu.AC c (ABC), AC 11 u, rorry /C I KM (sa o3Haxoo trapa-.nearnocri npslroi i n.noqzuu).r KBM.a \ABC (aa gnoua xyrauz),KM BM tnBM=r.MaeMo: != " ,AC= BC; rro3'aqrBr g r+lb'

4(x + 5) = 9x; 4x + 60 = 9r; 5i = 6O; x = 12.BM = L2 cm: BC = BM + MC = L2 + Lb = 27 (cv\.

30. Hexafi -43C - ganuit rprrr(yrxrn; M, N, K - cepeAunr4 cropin.4B, BC, AC nit-noni4no,'rlepee rouru A, B, C, M, N, K npoBeAeMo napa.ne.nrni npanti AA., BBr,cc\, MMf NNf KK|NN, - cepe4n,s .niHis rpaneqii CCPTB.

NN,=CSEB'i cc,=27tJ1,,1 ,-BB,=2.g ll =?'2KK, - cepegua.riHia rpaneqiiACCA.

AA- + AA.KK, .'--t,---:-:t: M, = 2KKt - CCt- 2. 1O 7

MM, - cepeAns ,rinis rpaneqii MB\r.twttt. = 4 * Bq - 13+ l1

=24 =12 kvt\.'222

= P,lft:Pa =!!,n = 21 (cm).

PR 9xil

$,;r*........uii,'r-$::i$:::.*{/,:/rJ...a\.niJ.i$i.J,rr.-.-.....-.-..iyt4j-r:.:i......!.idt{f.tvtt.:ft a!rj:'/.r..ii:.:.:L.i)ritt:l::tll::.:t:L Jl /rli+-..rr.a.:..'.r:JiI-'.r-r-+...r.r...ir

'idl4JJ:';'.1!a'i\:+if at-rr....{....'.L rl al:++.jJ.!..'riJi4l..-'r::....:.ri

.i'lr:r-.:.-......r...

23. flaorquHa, trapaJreJlbsa cropoui-4-B rpzxynnuxa'A-BC, nepernnae ftgro no npsuifi

A,8,, A, € AC, Bt t BC. B,C = 10 cm.

Ocrci,ront A,B,i AB, ro rA,CB, ^ tACB'AB BC - AB 48,toot 4q 'Bp aoo Ec= AC,'

3a ynonoro AB : BC = 4: 5,

o'*", {f -t; +n,=+- 8(cM)'

Bidnoaida: 8 cw.

24, APr - ;rinig fieperr4uy rrrrorrlrurr rpurcyruuxa ABC

i n.nonlunr a, o lL -43. AP\ | AB.\ABC - tArBrCr:

48, _ \c = B,C . a_8. _ AB A.C = b"

AB AC BC AC o+c24 10 =8 t"*).O6ctrcrnMo: 4Bt =^*lO

25. flapa,renrni npaui AAti BBr oAHoBHarrHo Bu3Ttaqarorb

lJrorqr{Hy, sxa trepertrHae nJrorqrrHy o no npeuifi ArBr,O e ArB'oOAir; aOBBrea ABoMa KyraMlI (ZAOAT-- IBOB, an

neprr.rca.nrHi, 1A = lB.fiIc BHyrpirrrni pienoctopouni nprAAr ll BB, i civniit.4S).4 _ AO _'4,O =?.BB, OB OB, 3

floaHaquMo BO = x cM, ro,qi OA = (.r - 3) cM'

= 13 (cu).

P 30cm A

662 i::i: ;EOMETPIS Sinsnina O. 9- ra ix. ArageuiqHufi piBeHb :i:i:i:i:ii:i:i:i;:: :ilt{t:iit{ TEOMETPI9 Eincxixa O. 9. ra ix. Axageuivxnn pieenr illi; 663

91. ABCD - poM6, a I BD,A e a. flpaua a i npawa BDoAHoBHar{Ho Br,rrtarrarorr, flJrolqnuy. HagseMo ii o. 4 e tt'O e rx (ocHinrr*nO e BD), ro4iAC c cr,6o 4ri ro'rcuAi 0 npnrvoi AC lexa'ru Y nloruuui o.Ceo,,DE0,roMyCDcu.Oexi.nrrrz npnlri a i CD .nexarr, y nlou4Izui cx. To Borlu

ne e mnmo6ixnalru.flprznycrzmo, ryo a i CD napalenrxi. To4i n nlorquHi c uepes rovrcy D trpoxoAurb

ani pieHi np.aui BD i CD, napalelrui np.alrifi a' aile \e HeMoltcJrr{Bo'

Orxe, npaui a i CD nePervuarorr'cfi.

32. Ba yrvlosoro a Ji BD, rouy a i BD nexarr n o4uift n:ro-

uluni. Hasseuo ii cr.

A e o, rovxa neperuny 4iaroua-neft ponda O,rexr{Tr' y nJro-

uluni c, 6o narerxzrt nPauiit BD.

,{ni ro.rxx 4 i O npnuoi AC sa,rerxarl nrorqzxi o' roMy

AC c a (reopenila 2, rvroxytn' 2).\s

Orrxe, pom6 ABCD i flprrMa 4 Jre]+(arr, n o4nifi n.noqrzni, a nparvra b -o3uaqae, rqo npxmi a i b - rtralro6irxni.

s isuriii. IIe

liir!.,........i4?i.....'.:....1.*rjnf,.......,.....,1it'rni,..+.....r."..v.

u J,ir';f..-...4..j...rJrn...r-.4......t!

uJrT{.:'r...4.1.!t.trd^ jlaat-.-i,....'rr,\- Ji[.it^.-.....f tr..L.r6r'11a..+;.f...Li.n^}t ....'...i.'.'t.

Mo.q,vnb 4. B3AeMHE PO3MltrtEHHg nnoll4ylH v nPocToPl

94.1. Bsae&lHe posMiu{eHHc ABox nflou4l4H y npocropi' flapanenuninroull4Hll

4. 1 .8) flapa.nemni.

4.2.A,8, [,.

4. 5.A (oauaxa napa.ne.nsnocti n,roltrun).

Ogsana rapalelruocri rrJron{ilHTr. flrn1o 4ni npaui, ari treperuIralorbcfi, o4niei nloqnnr'r, ni4noei4no napa're'nr'ni

4uolr npanru* 4pyroi nloulrnr, ro qi nJrorqrrur4 lapaJre,'rbur'

4.3.f) 1, 3, 5, 8.

4.4.A, B, L.

4.6.A.

4.7.

4.9.B.

4.10. A.

4,1 1. B) flapa.ne,rrui.

Poara,flHe&ro napr{ rpr{r(yrHnxia OA.B, i AAB, OBpri oBc, ocl\i ocA.Ocxilsrcu nloulr,tun napa,relsHi, ro qi rpaxyrurrcr,rnogi6ui. 3 no4i6Nocri Br{[JrrrBae:

04 =44 =O4. Bp1 _OBt _OCt.OB AB OA, BC OB OC,o_tq

=oAa =4c,.AC OA AC

3 rlHx nponopqifi uacuo, ^o

43 = 49 = ry.'ABACBCI{e oauavae, rrio sa rperr,oro oanaxoro (nponopqifinicrr,ni4uoni4nr,rx cropiir) a-4IlC N aArBrCr.

KZ ll AB, ZM ll BC,nx cepe4ni niriii a-43D i yCBD. (KZM) ll ('4BC) aa reope-

Moro 1 (MoAyfib 4),

664 il:i: ;EOMETPIS Einsxina O. 9. ra ix. AxaAervriqHri piBeHb :i:::i:i:i;i:ili:i:i

4,1 2. A) fleperauarorrca.

4.13. B) S6irarorlcs (cni.nsni npama i roqxa).

4.14. A) (oana).

,{.nn no6y4oeu qiei rounu (4auo urluo6ixr+i npauiaib):1) qepeo 6y1t'rrr+y ro.rxy qpamoi {z rpoBeAeMo npfi-vy b,, b, ) b;2) vepea npami a ibr, axi fleperrHarorbcfi, npoBeAeMoe/4,rrrry flJrou{ilHy (x.

b ll c aa osnaxoro rapaJreJrbHocri npruoi i n.ro-qn{u.

4.15" Hexafi nprui a i b urruo6incsi, BN6epeuo Ha npfi-uifi a (axc, I.r) Aoni,'rrHy ro.rxy A i npoBe,4eMo .repesuer upauy br, napa.ne.nruy b, (rlepea rortcy, slla rreJrexc[TE xa 4airifi npamifi, Morr(Ea trpoBecru e4r4ryrp,fiMy, napa.uelrriy 4aHifi.) Ilpami c i b, aagdrorreAr{Hy nJroquHy B, 3a oairarcoro napaJ'reJrbHocri npa-uoi i nron{znra b li B.Ana.noriqso [poBeAeMo npfiMy dr vepea goeinray rou-xy C: a, 1l c.9epee nparri a, i b npore4ervo rrJrorrlriny o. .(ri nparrni a i br, tqo flepernuarorbcs,urorrlr{Hu B ni4noni4no napalersHi AEoM trpEMuM a, i b, rqo neperr4Harorbcs, rrJro-rrlr{Hr.r o, 3a oaHarcorc napa.ne.nrnocri uJrotrtr4u cr ]] 9.

4,16. Hexafi trJrorqr,rur4 cr i [3 napa,re.mni, i a cc{. IlplrrrycrrMo. rqo fipflMa d treperrz-. Ha€ rrJrorr{r4rry p e 4eaxifi To.{qi M, ro4i rouxa M - cninrsa rovxa n.xorquu o i B,

rqo cyreperrrrrr, ymoni cr ii B.Bucnoeon: rrpflMa a He rreper[Hae u.riorqzny B.

4,17. Itonrsi gni npnrvri, nlo fleperr,rnarorrcx, aa arcio-uoro II. saAarcTb rrJroil{rrny.fonegeno, qo (ABC) ll (A'B'C').Ocni:nrcu O - cepe4r.rua rc6xnoro e ni4piamin AA'BBr CCr ro IAOB = aArOBr Ba fiepmoro oaHaftoro(IAOB = ZAtOBt nn reprunanrni).B pienocri rpanyrurzxia BrrrrJrrrBae: ZOAB = lOArBrro4i ea oanaxorc ilapaJreJrbuocri npaunx aa nnor4uui AB ll ArBr,Arra.noriqno ,4oBoArrMo, ao AC li ArC'3a osHaxoro napa.ne.nrnocri fl,'ronlux (ABC) il (A.B.Cr).

4.18, Hexafi O - 4aua rouxa.

Il.norqranz y npoctoPi Mo]ltyrb:a) neper:zHatraca no npa*tiii, srcfiIo BoHtr Marorb o,4ay cninrny rovrey;

6) a6irarrca, .fircilIo Y Hrrx e:

- 4ui cnirsni npalri, u1o fleperrlHalorbcs:

- cninrna nprua i Torrxa., rr]o iii ne Ha:!exurr,;

- rpu cninr,ni TorIKu, rqo He rblfiarr, ua o4nifi npgrvriii;

e) Syrt napa.ne Jrbluvrtt, To6To He wratn rxo4noi cni.nsHoi ro'rxra'

A B B T T3 2 1 2

4.8.8. r, lI.

iilti;litiiii TEOMETPI9 Einnxira O. 9. ra ix. Axaseuivxui pieenr ll:! 665

4. 1 9. gepes Beprnr{rrr4 trapaJreJrorpaMa r -BCD nponelr'rnpami.4-A, 11 BB, ll Cc1 ll DDr ari neperny.nu n.no-

rrlr,rHy c B Toqrcax Ar, Br, Cr, Di.Hexafi O - roqria neperrHy 4iaroualeii, -BCD' O -cepe 1nlr,a B D. Ilpoeegeuo OO I ll B B I ro4i sa reopervroro

@a.neca O, - cepe4una BrDr. Ana"norivHo MotrcHa Ao-

Becrlr, rlo O, - cepe4uua AlCr A\BrClDt - trapaJre-

JrorpaM. To4i OO, - cepeAHs .niHia rpaneqii AATC.Ci BBlDtD.3a yuoaoro AAr = 4 m, BBt = 3 u' CCr = 1 rr.

AA- +CC. 4+l3 rpaneqii AATC,C: OO' = T =; = 2.5 (m\.

BB tDDt. nn -t.,n , r2 -r.3 BBTDTD: OO, = Ln , --t --t - .

.2o.HexafrB - 4ana i.r*u t o( - AaHa rrJrorrf'Ila.Ilpone4euo qepes roqxy B n.norquuy B, napane.rrnyulorqrani c.Hexai b - ,4oni.nrua npsMa, rqo npoxoAurb rrepe3 Toq-

xy B napa.nearso ct. Bisrueuo n n.norquni cl 4onillnytouryA i npoeeAeMo qepe3 tounyA i npsuy b rrJrolqu-ny y. To4i nJrorrllna Y fieperunae rrrorrl[Hlt a i B no

rrapaJreJrbH[x np,fiMr4x aribr, ale np.alra b, trpoxoAr4Tb

uepes rouny B, a npsMa b raxorx .nerxuru y nnorquui y

i npoxo4ntr vepee ro'rxy B i sa xac.:rigrom 3 TeopeMr'I

4 napa-ne.nrHa ripalrift or-Togi aa arcciomoro napa.nelr,noeri npaui b i b1 Marorb cniBua,4arrz, rouy b c p, qofi rpe6a 6y.ro Aogecru.

4,2i,fIloqnnr.r o i $ ne MoJ+{yrb neperuHarr,rcff. .flrc6r ninorqznu u i B rra;rz cni.nr,iry

Toqxy, To qepe3 qxr roqrcy trpoxoAn rn 6 Asi nJou-u{Hfl (o i 0), napale,trti nlouluuiy. A qe cynelpeuzrr reopeni 2 (uogy.nr 4).

4.22. l\ Ocxi;rrrrl M i N ea yMoBoro e cepeAuHaMrr pe-

6ep AA, i BBr, ro MN I AB. flpz no6ygoni nepepisvny6a n.norrttoro (MON) BpaxyeMo, rqo O - rovxa A1

neperuny giarona.nefi rpani DD ,C ,C . lltrot4unlz (M NO)i (ABC) rapa.nearHi.Il;rorqrzna (BDM) npl,z' nepetuni ry6a yrnoproe ne-pepia - tMDB. Ito:xna s rpauefr AArDrD, AATB.Bi ABCD Mae B rroilIrauon (MDB),4ni cni.nrui rovHrz'

B'e4uanmu axi, orpuryraervro nepepi3.Il.noqranu (MDB) i (Bpp) fleperuHarorr'cs.2) flo6yaycrvro ix ,riHito fleperilHy.MB n ArB, = Xi MD

^ AtDt = Y.

XY - uryxaHa ainir nePerunY.3\ DDPTB - uryxaunti nepepio.AAr ll BBr roMy Ba osHaroro.AA, Ll (BB'D,).

Sr4o,, = BBr' BrDt' BrD, = AtBr' J2 @nuc-

runicrs giarona-neii KBaApara: a = ali ).

Bp,=3^,15.'E=6 (cru).

S"u.,,, = 3,1r.6 = tavE (cv').AB

666 :::# rEOMETPIS EincHixa O. 9. ra ir. Axa.qerurivHrfi piBeHb i:i;:::::J:lia:i

4.23.8.4.24. A, t.4.25. B) AC = 15 cu.4.26. A, B, B, [. I

4,27, A\ fleperuHarcrrca.4.28. A, r.

4.29.

4.30. A, B, .U.

4.31.

4.g2. AC = JAB\ BC'z = Jrs +r44 = t69 = 18 (crvr).

P*,4", = P*" = 5 + L2 + 15 = 30 (cu).

Bi0noei}u: B) 30 cu.

4,33, flapa.ne.irrni n,rorquuu o i p neperznarorbcfi B napa-- reJ'rr,EtrMrr npa:lrutttuAA, i BB, i ni4runarorr nig trr.rx pie-

ni nigpiana: AA, = BBr. Topi AATBTB - raparrerlorpaM(rpouurextni cropoHrr piani i napa.re.nlHi) i -4r8, = AB = a(an nporurelcHi cropouu napa.ne;rorpama).

4,34."|at<. Hanpux;rag, fiHrrlo oArra B napaJreJrbHux rrJroulrrH lricrrrr ocHoBy, a Apyta-rreperuHae cropontr pinuo6e.Apeuoro rpr,rfiyrnlrna.

4,35. (FPQ) ll (DrBrcl 3a o3naxom napa- 4.36. floscreuHg aua"noriqse,re,rssocri uroil{r4rr: FP ll DrB, ax ce- ,qo N! 4.35.pe4ua rinia a DrBrCr, PQ ll BrC art ce-pe4na ,rinia aCCrBr.

4.37. Ocni.nrrcu IMAB + IAQP = 180", a qi nyrz e BHyr-piurHi o4nocroponni nprE rpflMux AB i PQ i ci.ruifi,4@,roAB I PQ.IMKQ = ZMCA, a qi xyrz - ri4uoniAni rpu flpfiMr4xAC i QK i ciqsifi KC, rowty AC I QK.(ABC) il (QPK) sa osHafioro napa.ne.nsnocri rrrorqrrn. A

i:iiiililiitlii TEOMETPI9 EincxiHa O. 9. ra ix. Axa4ervrivxrafr pieeau ;|:!:,i 667

2,5-3=2(vt).

$tnl,'-'............

l:iliti:!::::i::::::.tl\rr... ... .....

M

I

$ 4.2. BnacuBocri napanenbH[x nnou4r4H

1.,flnqo 4ni napanenrni n.norqunra neperrryru rperr,oo, ro npaui Ix neperuny6y4yrs napalelsni .

2. flapa.ne;rrni rrJrorrlt4nrr, neperzHaroqu gni napa-neltni npaui, ri4rlrirarorr ni4Hux pieui ri4pisrcu,3. ,{ni n.noqunn, napa.ne.nr,ni rperifi, rapa.nelr,ni rri*t co6oro.4. flrorqnna, .rrra fleperuHae q.4uy o ABox naparerbHux n.rroulrtH, neperunae i 4pyry.5. Ilpaua, .flHa rreperlrHae oAHy B ABox napaJreJrrHrrx fiJrorqr]H, neperuHae i lpyry.

A 1 2 4

B 3 5

4.38. FS ll CD i Ff ll BD sx cepe4ni .ninil ni,qnoni4xoyADC i aADB.Toai (TS.F) ll (BCD) 3a osganoro napa.nelr"nocti nno-trI11H.

4.39, DPl ll DB sx.nisii nepernuy napurerrr,Hrrx rpasefitrJrorrlr4lron DDtBrB,floaga.rfiMo pe6po ry6a a.To,4i n aAr-LDr i tKBC ArD, = BC = ai

a-L-- KC =!a.'2tArLDr= a KBC 3a ABoMa KareraMrr' 3 pieuocti rpr'r-

xyrnrlrcin LDr= /f3.Ana.norivno D, K = BL' IJ,e osnavae, go LDrKB - ta-paJreJrorpaM (nporr,r.irexni croponrr [onap]ro napareJrb-

4.4O. Bi,4oryro, qo cr ll I, MK I NL. Togi, epaxonvroutr,

rrlo sa BJracruBictro napa.ne:rnrrx nJroulrarr MK = NL,po6rauo Br{cHoaon, n\o MNLK - rapaJreJrorpaM(MK NL. MK = NLl.Hexafr. ML = 14 cw, NK = I ct'r, MK : MN = I : 7'O - rouxa neperuny giarona.nefi napa"re;rorpalra. ToAi

MO=OL=l ML-7 "*,21

NO=OK=1NK =4 cm.2

Iloasaqruo NM = 7x, MK = 9x, INOM - a.3a reopenroro xocr4nyciB:MNz = NO2 + MO2 - 2NO ' MO 'cos s';(7x)2-42 +72 *2.4.7 coso") .l49x2 = 16 + 49 - 56 cos a. (1) tMK2 = MO2 + OK2 - 2MO 'OK' cos(180" - a); rt

4,42, flpami OA2 i OBz, rr{o neperlrHarorbcfi, BrraHarra-rorb n,'rotqlrHy, sxa neperr,Inae rrJrorr{r{Hrl o i B no na-paJreJrbHr,rx flps Milx A\Br i A2B 2.Ocxfursrcu A)Bt ll A282, ro tArOBrN rA"OB"

, 40 _ 4,8,. 16 48,.' 40- A"B,' (r4-n - 50 '

aB. =16_i9= 2o (crur.40

4.43. a)

/ o',L),",

/,,*A",f

ni) i D,K Ll LB.Bucnosor: (Bp$) ll (BDL) 3a o3Hatcon napa.ne.nrnocri rrlorql4H (,qni npsui, rqo

neperrruarorl,cs, oAniei nJroilIrlrru - DrBri DK - napa.tenrui ABoM [psMr4M Apy-

ro1 -DBiBL'1.

a) 3'e4naemo rouny K g ro.rnauu .F' i M, a ix - is roq$orc N, .qe.F, M i N - sia-noei4no cepe,qrrrru BiApisniB AP, CP i DP. To,qi FK ll AB, MK I BC sx cepe4ui niniirpurcyrnuxin APB i BPC.3a oananoro napa,'renrHocri nJrorrltrH (FKM) ) (ABC).6) Hexait M - cepesuna AB, N * cepeAuHa BC. 'foti MK'11 AP, NK il PC sH ce-pe4ui ;rinii aAPB i, tBPC.Korxna a,nap npauux AP i PC ra KM i I{N, rqo neperlruarorbcfi, BrraHarrarcrb Aesr{yn,aorqr,rny. KM ll PA, KN I PC, rony (MI{N) ) (APC\.

l#l&;ri;rfin

XtfriiJ.:::i:iJ::

[5]f$l:j.;:;:;:;;j

ffififfii,*jj't

(9x)2 =72 + 42 + 2 7' 4 cos s. (2)

,{o4auo (1) i (2): 4gtc2 + 8lx2= (16 + 49)' 2; l50x2 = 130; * = 1 (r > 0)'MN=TcwiMK=9cw..P*r"*= (7 + 9). 2=32(cnr).

4.41 . Ilpmri OF i OP, qo fleperr4narorbcfl' Bllsrraqaror],

nJrorqr,ruy, .fiKa neperuaae flJrolqr'rur{ c i B no napa-JleJIbH[x np,rurax FrP, i F rP r.'I opi o F rO P, t t F

"O P r,

OF. F.P OPseiwu oi= ,,r,= *,Hexafi AP, = ,c, ro,4i OPr= 2a ', 4.

9= t :B(x+4\=sx;5 x+4'3.tc+L2=5x;2x=12;x=6.OPr = 6 .*.

4,44.Poss'aaanrss asa,rori.rne 4o Nr 4.17.

4,45. BrarouaeMo AoBeAeHH.rr, uo6y4ynannrr.r rani nlorqnnn.PoarrsneMo rpz nonapuo ur.ruo6ixui pe6pa ry6aABCDAIBLC.D1 AlBr DtD i BQ.flo6ygyel,ro repepiau xy6a n,rorqlrnavu (BrDrC) i (A,BD).I-[i n,nouluux Maror.t 3 ai4noai4nrur.r rpanflMu rcy6a no 4uicni.nrni Toqlifi, a orrfie, fleperr,rHarors qi rpaui no flp.rrMr4x,rqo flpoxoAfiTr, vepee qi To.rxfi. TaxuM trrauoM, ABTD'Ci a BDA, - ruyxani nepepiez,BD ) BrDr CDr I BAr ax 4iarona.ni nporra.nerxnrax rpanefirapa;re.nrnocri nrrorrlrH (Bp1C) ll (BDA).L{i napa.ne;nui fiJrorqlrHlr rigrnnarorr sa mr.rlro6irnnrx npsMux pinni si4pisxu ArB'BC i DDf

4.46. [ni npsui a i b ne e napaJre;r;Ht4Mr.r, 6o r:oAi6 ArA" = BrBr, a ca yMoBolo noirrz piaHi..flrrqo npnrvri a i b neperunarorrc.fl, To rrepec Hr{x Mo}r{EaflpoBecrrl urorqr4uy, sra fieperr{Hae ir.rorqunra cx, B i y norrapaJreJrbHr,rx flpffMrrx ArBr A2B2i ABB,.

4A, = Arn,

.ArA" BrB"

62B,B" 5'

(

ry6a. To4i sa ooHar{ox)

3a reopelroro @a,reca:

- B.B^ 23a vMoBoro:' BrB'^ 5

BrB" = (clt); B,B" = 6 + I"5 = 21 (cu).5.6

- = lt)

2

h

/A. L)",i/t

/ o,l-?:,/1f)u1,""1

668 iii: feOugfplg Sincxixa O. 9. ra ix. AragerurivHhfi piBeHb :i:::li:i:::::li::: :ilil{liltil TEOMETPI9 Einqxixa O. 9. ra ix. Axasewrirnnft pieenr fi!: 669

A'A' =?' 5 =2-; 44 =!-5 = rz.s (cu): Ar4, = 5 - 12,5 = 17'5 (cru)'

A"A. 5' 4L, 5 2

fxrqo lx npami a i b nurro6ixni' ro qepe3 rouxy Ar npoBeAeMo npauy bt ll b' sxa

neperue t"rio.,tr""t B i y o rovrcax C, i Cr.

Arb"= BrBr, crC"= B"B" an riApialtrz [apaJreJlr,Hux fipflMr{x' o6mexeni rapaJreJrb-

HI{M h nIorqIIHaMl4.

,{ari posr'aeanufl sBoAl4Tr'cfl ,4o Blzna'4ny, Hoau npaui o i b ueperr.rnanrr'ce'

Bi}noai}a: 17,5 crvr e 21 cN{.

$4.3. l-lapanenbHe npoeKqiloBaHHq' 3o6paxenxn nrocKlrxi npocropoeux OirYP Ha nnortlt4Hi

4.61. Hi. Eo aa npaenramH napaieJbHoro npoenuirosanri,r npoexqii napa.lelsuux pis-Hr.rx ni4piarie - u.e napaJreJrblri pieHi eigpiaxu.Tanzrvr .rznom, ao6pa:nennavr napaJreJrorpaMa y BaraJrr,HoMy BlnaAHy mae 6yrra na-paJTeJrorpaM.

4,62. l) Ilpsui,43 i,4rB, o4HoexaqHo Brauaqarorr,[JrorqlrHy, fiKa fiepeTlrnae nJrorqr,rHy o no rpxuifi-4r8,. Tomy, axrsqo AB

^ a = C, ro C e ArBr.

2) IIpw napaJreJrbHoMy npoexqiroeauHi a6epira-rortcfl nponopqifi aicrr ei4pianiu o4Hici npauoi,rouy eo6palrleHnsM Toqr{u D 6y4e rovna D, - ce-pe4una ArBr.

^, A,D. AD"' DrB, = DB'

4.63. 1) ArBrCrD,; 2) Dpi 3) AtD; 4) A,B,C,D,;5) AptC\D; 6) Api 7) r BrDrCr; 8) a-4,-B,D,.

4.64. L) flpoenrlis a DB.C na rpanr B,CTCB - ei4piaox BrC.2) flpoexqia r DB.C ua rpans ABCD - ^

BCD.

s^-"^ = lrc . cD =\.6. 6 = 18 (cur,).223) flpoexqia r DB.C ua rpaas ArBrCrD, r BrCrDr.

ttS*,".o =tBtct qDt=t6 6=l8 (cu'z).

4.65, ABC - piauocroponniit rpratyrnzH, rArBrC, - fioro npoexqia. .

Y aABC BD LAC, D - cepe4rzna AC, M - cepegutaAD, KM LAC, KM 1) BD.

4.53.

4.54.1.

4.55. r.

4.58.,{ur. puc. Ao aagavi N! 4.57. flxttlo(App) | (ABC), ro AA, = BB, = CC,,

romy A-ACCr, CICBB. ATABBT'- napa',TeJrorpaMlr (nporra.nencni croponu pienii napa.ne;rni).SeiAcra ArB, -- AB, B1C\ = BC, ASt = AC '

'ABC = tArBtCr Ba rpboMa croPoHaMI4'

Ar

4.47. A, t, L.4.48. A, B, f.4.49. D ni,qpiaox.

4.50. A, E.

4.51. A, B, B, A.

4.52. A, B, f, A, E.

4.56.

4.57.

IIIo6 nznonaut qe saBAaHHs' Tpe6a ctroalarny BctauoBr{rz nigHourenna

Ocxillxu npft [apaJ'reJrr,HoMy fipoex- BuimeaHHi a6epiracrsca nponopuili-

^nicrr, ra napaae.nr,nicrr eilpialcis, /l\

lff,H:Xll""ru nPoe*Ilirc nPnmoi

/ I \l) nponecrr ve4iagy BrD, rA,BrCr: / I | \2)poa4irurnaigpiaoxAr.B,raurir- / h L \

rouxaMr; ' A M D C

3) nponecru M,K, ll 8,D..

ACBC

Tarun xe 6yAe i siAHome n " fl.B,C,

Ha rparrrzqi epyvno sHafirz, "*y tuct""y crasoszrr AC niqAB (AtC, ni4 '4tBt)' 4.66. y potn6i ABCD AF - uryHana eucora (1A = 60"). A.F ll BK, p,e BK - nncora

i rve4iaua pirHocropouHroro tBCD.fIo6y4oea:l) AtBplDr - rapa,'rerrorpaM -eo6paxenna pov6aABCD.2) K, - cepe4rana ei4piaxaCrDr.3) BrI{1 - eo6paxteuHa ezcoruBK. }

4) Af 1ll BrKr- uryxarre eo6pa-lxellrrs sucorz -Af.

4.59.1. Ilpoexrliero npaivroi e npsMa' a npoercqiero eiApiaxa - eiApiaor'

2. Ilpoenuii napaJreJlbHhx rrpflMilx naparelsni'3. Bignotreuun AoB)rczH ni4piexia rapaJreJrbuu x rrpfl Mr4x a6epirarorrcs'

4.6O.,(.nn no6y4oera 3o6palxerHfl rerpaeApa aocrarHbo no6y4ynaru 4osinsnrafi tpu-

KyrHr{K, rqo ao6pantae ocHoBy rerpaeApa. Iloriu tpe6a nu6parra arcy-ne6y4r rouxy'

rxa6y4eao6paxaruBepIiII4HyTeTpaeApa'is'egnarr.riiauepurr,rrralrl{MIIoIoRyTHu-xa. Orpurraxi ni4pisxn - ao6pa:rcenHa 6ivnrzx pe6ep'

670 i;::::: rEOMETPIS EincxiHa O. 9. ra in. Axa4ervrivHnf piBeHb :*1i:iiiiiiiii;i

4,67,Hexaft ABC - 4aHzfi npauorcyrHufi rpzrcyruux, lB = 6O", BD - 6icexrpzcarcyra B.BiAomo, rqo 6icercrpzca Hyra rpr,rT(yrHr,rxa 4i,rrzrr nporrrJrex(Hy cropoHy ira ej4pic-

rcu, npouopqifini npr.r.reuzm cropoHaM rpr*yrHtrKa: + = *4! drc cn =! nnCDBC2flri rarer, qo Jrercr4rb fiporr4 Hyra 30'. Togi + :'+ = ?"CDBC|

liiiittitl{i TEOMETPI9 EincHixa O. 9. ra ix. Axasenivxrfi pieexu f:id 671

BJracrr.rsocti eo6pamenua npocroponux (pirypi. flponto.rrlriiisi ei,4pisnu 6i"ypz so6pamarorbcs Ha nnoqnni pficynxa eiApis-

KAMI{.2. flapare.nrHi eiApisxz @irypt eo6paxarorr'cfl rra nlorquni pr4cyHxa napaJleJrb-

arairnu ei4pieramz.3, Bianon]eung ,4oBlnr4u ni4piaxin ogniei npamoi a6o napale'nrnux npaur'rx a6epi-

ramTbcg.

AE I 2 4 5

B

A B B I A4 1 2 5 3

Orxe, rqo6 no6ygyrarra so6parxenua 6icexrpucu BD, norpi6uo rla 3o6pax(eHHi ftarera

AC nossaqururovrcy D, rarc, ilIo Crnr=f,4o' i o'egiraru ii s Br.

A\Kl'\ \\^l \"l---,-

.L \" "U;P"

Tecr nflq caMoKoHTponK)

9acrl*ra 2

17. fl.norqrira, ,fi$a fipoxoAr4rb .repes uparri KP i CH,qo neperr4Harorbcfi , neperrrHae napa"re;nxi nJrorqr,rnra

o i B no napaJreJrbnr4x np,tMrrx KH i CP.yKOH * aPOC sa ABoMa ByraMt4 (IKOH = IPOCsK Beprqr(aJrbHi, ZKHO = IOCP ax nnyrpilrni pie-nocroponni npu KH ll CP i ciunift C,FI).

KO _ RH . op _ KO 'CP _ 9,6 . 4,5 = 3.2 (cv).

OP CP . KH 13,5

i

I

I

i

iil

ri

ii

c

9acruxa 1

9. E.

/ ",D:'/

T,L)U,'

18. Ilprui A4,2i Bp2(A4r^ BrB"=C) reperrrHalorbriapa.ne.nriri rrJrorr{r{Ht4 o i B no upaMrrx ArBl i AzB2,ArBr ll ArB2. To4i a,4,CB, N a ArCBr.

94_94ca2- cBz'

. Ocrci.nrxr CA, t Ay', = 1 : 3, ro CAr: CAr= | : 4.

"Bt -!, cB- = 3 (cru).124

!i;il,\'r.,-.'.'..-..'...:

$fi$$*i*iIfifrdiitriiiiii*fr'.iltirri.ij.,iiitr r;;\:i'*.;.:.;....1i;/iNll.il-..+r:...r....

16.

1. B.

2. B.

3. B, A.

4. I.5. A.

6, .lleperunarcrr'cs a6o napa.ne.nr'Hi.

7. t.8. A.

10. B.

11. r.

12. E.

13. f.

14, A) iE(o,qnor. Inarnre ri.norqr.run cx i B

rra.tu 6 cninrni Toqxrr, a qe cynepeqzrbyrroni (o I F).

15. B).

19. fl.norqr,rna (/3S) neperunae n.rrorqaHrr o i B no npa-*rux ArB, i A282. ArBt ) A2B2ll AB.Ba yuonoro A, - cepe4una AS, A1 - cepe4r.rra ArS.ToAi sa reopeMoro (Daneca B, - cepe4rana BS, .B1 -cepeguna BrS,ArB" - cepeAuff .ninis a-A-BS:

4r-,=+on=1= t8=9 (cu).22ArB, - ""p"g"*

niiria aArSBr:

+n,=\+n,=1 s=4,b (cu).2" 2

,BiOnoaiOt: 4,5 cttt.

2O. OcNi.nrxu (AlDpt) 1 @DCr, ro ArDrll AB, Dptl DC,Bpt ll BC, Apr I AD.D, - cepe4una DC, rowy ArDr, DlCr BtQl iArB, - ce-peAni ninii niguorigno tADS, DCS, BCS, ABS,

n O, =l dO= 4 em: B,C, = ArD,= 4 cwi2'c,o, =!co =lan =44 = s (cru).22P+a",o, = $ + 3)' 2 = 14 (cm).

i::ilti{llil rEo M ErPl s

22 vcirnp.lo u.. q. 1

ooK-L - roMH: oK =

oL = KLOM OH HM

A) K! -oK . 6 =o!- o* - 6'10 -I2 (cM);'HM OM 5 10 5

KM = OK + OM = L2 + LO = 22 (cvr);

B) oK =aL : oK =oJ:gy=t-? =B Gvt; KM=B + e- lz(cM)i'oM oH oH 12

B) Kt =9K, ox=[!:9M =3 9= 6 Gwl;KM=6+8=14(cu):.HM OM HM 4

y1 9L =9L:6 =J_ ; ou =9-q=12 (cM); KM=i2+8=20(cM):'oH oM 9 0M' 6

n\ KL =oK, oM-oK:-gM =!!5=6 (cu);KM=6a4=10(cM).--. HM OM. KL 10

672 il:l rrouerplg Einsxixa o. 9. ra in. AxaAeMi,{nni pieeus iii:i:iiii:iii

21. Ocxi;rrxrz a ll 0, ro.4-B ll APr AC ll APr BC ll BprToAi aASB- aArSBr, aBSC- r B1SC1, ^,4SC-^AlSC1.3 no,qi6Hocri tpnrcyrnlrxin Brarrr'rr.rBae:

s4=A,4 =54. Bp1 =54=SC'.sB2- A2B2- SA2' B2C2- SB2- SC2'

sq =

s.4, = 4c, .sc, s4 4c,

3 qax nponopqift maemo, *o € = ++ = t+.

48, 4C, B,C,

Einnxina O. 9l ra in. Aragerui,rnnfi pieenu i{!: 673

D

r

A aHauurr,, Ba rperboro oguartom no4i6nocti r:puxyrHrxin (ea rpr,olta croponanrr'I)aArBrC, a tArBrCr.

.sA_ 23avmonoro f= . Toni"sa23

Maemo: P*,",.,

=?. p.. ^^Pno^^^"^

Bilnoei}u: 2O cw.

22. floscseHsa i puc. gue, go N! 21.flic:rs 4onegeuH,s noAi6nocri rArBrCri tA"BrCrpoBB'flBaHHff Mo)fie 6yrrr raxrrM:

3a ymonoro s,a! : s-a! = nb ,2, 'osi ++ = ++ = Y = +48, 4C, B"C, 2

3ai4cu,4,c, =#=!!i h =6 (cm).

s.,"- = A,c:h =q'tr erE (*r,)."M,4', 4 4

Bidnoai0a: 9J3 cm'.

23. 1) AC a BD = K, rouy WC) ^ a - AB,(ABC)o9=CD,an CD.2) lAKB - ZCKD rrt eeprurcarsui,IABK ICDK ax suyrpiuui pisnocropoHni.\ABK * tCDK sa ABoMa KyraMr{.

$ AB - 84, nx =92'BK 4'5 =ro {cv).CDDKAB2Bidnoeidu: 1O cm.

24. ALB. n A',8" = Q, (ApAr\ o u - AAz,(APA) n B = -B,-B,, AA2 ll Bp2.

"A{rq a a BrBrQ Ba ,qBoMa RyraMu.

P =% - ++ =1. roar eA,- x*r; @8, = 3r cu.aA" aA, ArA, 1

ArB, = 6 cut, ArB, = QA1 + 8BrroMy , + 3x = 6; 4x = 6; x = 1,5. QA1= 1,5 crvr.

BiOnoeiOu: 1,5 cw,

25. Ocxinrxz o B i sa yrrnoeoro ArA, I

nlous,ua A,A, B,Br= CrCr.B182ll CtC2, ro aa s,Tacr[eicrro napateJr]'Hilx

Togi aa oaHaxoro lapareJrorpaMa, ocxi,rsxu trporu:reffi-ui cropoHn pirni i napa.ne;rnni. ro ArCrCy',z, CrB\82C2,A1B1BA, napalaeJrorpaMu. Ix nporra.nerxuicropouupinHi: ,4rC, = ArCr, B,C, = BrCr, ArB, = ArBr.Touy r-4rBrC, - rArBrCr Ba rpboMa cropouaMrr.

1lSo'u.., = S*,u..

)BtCt 4Bt=t 3 4- 6 (cv').

Bi4noai1u: 6 ctrr2.

26. Ocxi;lrxu npil napaJT eJrr,Hoivry npoemqioeaHHi a6epiraeT tcs nponoprlifinictr, niapic-xis oAsiei npnmoi, ro ArK, i KlBt = ltf( : KB = 2 : 3.

.- r''B,.2 75 2ToMy,41Kr -' .' = *=ti (cu).

oaBi4noaidu:6 cyt.

674 ::ii: TEOMETPIF EincxiHa O. 9. ra ix. ArasevivHnfi piBeHb i:iiiiiiij;iiiiii

48, = 4C, _ BrC, _?.48, 4C, B"C, 3

= Pon

"=""'2 _ (8 + 10_+ 12). 2 _ 20 (cnr).DO

ffij{fi$i$$

27. Hexafi AtBrCtDt - npoexqia rpaneqil ABCD.Ocni.nr,rcr npfi napaJreJrhHovy npoenqirosasuia6epiracrscs naparersuicrs npflMux ra nponop-qiituicrr niApisRie o4Hiei a6o trapaJreJrbnrx trp.rr-

mux, ro ArBrCrD, - rpaneqia, BrC, ) ArDr.MrN, - cepeAH.fi nillia, MrOr: OrN, = $ ' $.Togi M,O, = 3.r cm, O1N1= 5)c cM. 3t + 5x = 24;8x=24;x=3.MtOt= 3. 3 = 9 (cm); O,N, = 5 . 3 = 15 (cm).Bidnoeidt 9 cv i l5 cu.

28. Ocxi:rrnu (ABC\ ll a i AAr ll BB, ll CCr I DDr ro ArADDr, D\DCC. C\CBBri AABBT traparrerrorpaMr. Togi AD = ArDr, DC = D1C\, BC = BrCr AB = ArBr,to6ro ArBrCrDl - r{BaApar, pinuufi gaHouy (ax napalerorpaM, y ffrdoro nci cropo-su i aiaroHa,ri pisHi).

I'

I

S-,u,qq =S*"o = 4S*ou=4

Bi}noai}u:128 cu'z.

\ eo. no = a 1.s.8 = 128 (cu,).22

r{acrnxa 3

29.

1) 3o6par:renusu pirno6ivnoi rpaneqii..43CD (BC 1l AD, BC < AD) e goni.nrna rpa-ne\ia ArBrCrDl, y flKoi BlCr l) AtDr BtCl < ArDr ocxi.nrnn npz napaJrerbnoMynpoexuilosaHHi :6epiraorsca naparersuicrs nprMhx ra nponoprliiluicrs si4piarciso4Hiei a6o trapareJrbur.rx np.fi Mr4x,2) floenauzmo M, e BrC, rarc, rqo BrMr: MrC, = I : 3,

,{ra ao6paxeHHfl traparerbhoi npoeHqii trepneHAlxyrspa, npoBeAeHoro B ToqKuM 4o ocnoou AD, cxopzcr:aeruocr ni4omum B Hypcy n,rauiuerpii @anrom: npRua,flrca [poxoAurb qepea cepeAZH[ ocHon rpaneqii, nepneHAu KyrspHa Ao H].rx.

3) floaHavnuo P, - cepeguHy BtCr K1- cepe4nuy-4rD,. flo6y4yerrno npsuy PrKr.flponegemo MrN, PrKr MtNr iryxanrafi nepneHAr.nyJ'r,flp,

q B:+ =!; n,*r, = 18 : 4 = 4,5 (cu). M,c,= 4,5, 3 = 13,5 (mr).M,C. 3 '

30. Hexaii AB 4anzil ni4piaorc, Ar i 81 npoexqii roqox A i B Ha n.norquHy c,l) AAr ll BBr. Ilpoee4euo uepes npaui -4-4, i BB, nrorquHy. BoHa reperHe nroqzHy

Hexait O - cepeArrHa BBr, ropi AO o a = C, C e AB.2) tAA,C ," aOB.C (OBt ll AA).

oz-=!nn.=4:2=2 cv.'2AA. A.C

Hexail B,C . x cm, ro.qi Oi,= i,C,8x-2x+18;6;r=18;r=3.BrC = 3 cm; ArC = ArB, * BrC = I *BiOnoeiOa: 12 cm.

illiii{iiiill TEOMETPI9 EinqHixa O. f,. ra in. Axasewrivxnfr pieens fri:ir 675

22+

i

l

ii

i

I

8 r+92r

3 = 12 (cu).

31. Ilo6ygyeMo nepepis xy6a nrorqunorc (L4rCr) - uo-rr,rprrxyrulrrd AATCTC; uepea gri upnui -AA, i ArCr, rqorrepeTr,rHaroTbcfi , npoBeAeMo nJrorqr,rHy, BoHa nepeTr,rHae

(ABC) no npauifi.4C, AC ll Apr fl.norqrzna (MNK) ne-perr{Hae rpaus AATDTD no npanrirl KM (4ai cnilrsirouxz). Anaroriqqo (MNK) . (ABC) = MN. Il;toqu-uy (ArBrCr) trJlorquna nepepiay [eperr,IHae no npalrifiKF II MN.Orxe, KM ll AAr, MN ll AC (cepe.que:rini* arADC). BaosHarcoro napa.ne,rrnocri [Jrorr{r{H (MNK) ll (AA.C).

32. Ocri.nrrcu ro.mu A, B, C i D He Jregcarr, na oAnifi nlo-rqvrni, ro npaui -48 i CD - unlro6ixni, a anauurr, .ne-

,rcaTb Ha Aesrcrrx [apaJreJrBHr,rx fiJrorqr,rHax 0 i B.Cuoci6 no6yAoBr4 Tariux rrJrolqrru onncanraft y sa,qaqi4.15.Tper* nnorquna Y napaJreJrbHa xoxnift o npaivrr,rx .43i CD, a sna.rrarr, i n;rou{unam o i 0.r{epee upaui AC i AD ra BC i BD, qo reperl4rrarorbcs,npoBeAeMo nroquHrt.(ACD);y = x4s, MK ll cD; (BCD) n y = NP, NP ll CD,rori MK ll NP.Arra,roriuno, MN I KP. SeiAcr po6uMo Br{cHoBoK: MNPK - uapaJleJlorpaM.

MOAvnb 5. nEPnEHA]4Kv.IlgPHlCTb nPCMl4X I nroult4H y nPOcToPl

$ 5,1 . IlepnengrKynepHicrb npeMt4x y npocropi

,[ri npanri y npoctopi Haar{Baror}'cs nepneru0urcynapH'tt'Mu, ar"ruo Borru fieper?rna-rorrca ni4 [psMraM KyroM.Bnacrunocri neplerrArrrcyJrspnux rrpfiMr{x y rrpocropi1. {epee 4oni.nrny rovxy npsuoi y npocr:opi MorfiHa upoBecrr'r trepnerrAr4rdyJr.Epuygo irei npamy,2. ffrcrqo Ani npgrui, cNi nepetr,rnarorrcs, si{noniAuo napa.nelrni ,4aoM nepneHAtr-nyJrfipnl,rM [p.EMHM, TO BOHI,T rerft flepneuAr{Ky.napni.3. rlepea 6ygr-arcy rorrry flpocropy, rqo He HaJre]lcllrr nprMii, Mo]rcHa fipoBecrrilnp.fiMy, nepneH,qHXyJIspHy Ao Aanoi.4. flrcllo npsnra [ep[eHArr]cyJrspna Ao oAniei B ABox napaJreJrbHilx trpffMrrx i,rexursB Hr{Mr. B o,qnift uouluni, to soua rreprrenArrxyJrfipna i Ao 4pyroi npauoi'

5.1. A. s.2. A. 5.3. A. 5.4. B. 5.5. B. 5.6. f.

Croporra tABrC - 4iarona.nr rpani, rqo uae tpopnry rcBa?qpara i nrzpaxaerrca ue'

pea cropouy aa (popivrynoro a = aJ2.

5.7. AB1 = BrC = nC = 4x!2; Pona" = 3. alz = lzxli @vr).

Bi}noei}a: l.

S.g.And.noriqso ,qo N! 5.?. C'ropona LDTAC = 8J2 c*.r-t2-

. -(a'E)"'6-o+'" t;

-M,A( 4 i:! = 32Jg (cM'?)'

Bi1noei}a: E.

676 i:i{: reouerplg Einnnina o. 9. ra in. Ara4ervrirHnft piBeHb iii:iiliiiii:j

5.9.allON = aNOM = tKOM ca ABoMa lcareraMr4 (OM = ON = OK za ymonoro). Togi' KN = NM = KM, To6ro tKNM * npanrz.nrnr,rfi.

3 .NOM: NMz = NOz + OM2; U.tW = nl"' + "" = "JZ.

P*xr=vaJi; So*r*=US=+42

5.10. 1) ,{zo. Ni 5.s. PMc = Sa"l, = 5.5J, = 15rE (cu).

2) B aAoB: AB - ^[Ao\ oB' =,ls' - g = ^lz5 +s6= J61 (civr).

B tAoc: Ac="[Ao\oc, =E.s'=Jrb-% ="60 =s.E (cnn).

B aBodz BC = JoC\ oB" = ^ts" * = ,lrs +86= ,,6T lcnn).

P*" = 16T + rJz + "67

= 5"t, + 2.,6T (cu).

3) B aAoB: Au =,[Ao\ oB' = 17 + t'z = J9+ 16 = 5 (crvr).

B dAoC: AC = {Ao' +oe = J"z - rz = Jg -%= J34 (eru).

B rBoC: BC = .tje *oB, = ,lt, * = J% + 16 = GT rcm).

P*"=5+J3++Jar (cu).

4) 3 r,AoB; Au = ,[Ao\ oB' =.8 + 4a' = al6 (c*).

B aAoc: AC = ^[Ao\ oct = ^ta\ ra' = o.u4o (c*r).

B a BOC: BC = Joc\ oB' = ,16c\ 4a' = a",[g (c*r).

e* = o(JB +.u[o + JG) "..

5,11. Hexafi O iOt- qenrpra nporurerxnlrx rpanefi,43CDi ArBtCrDr xy6a - rorrrlr [eperr,rHy giaroua.neft nna4-parin rpauerl.AO = ArOr AO ll Apr rowy AATOTO - rapaJrerrorpaM(nporrz,rexni cropourr pinni i napa.ne.nrrii).Ocxiarxi,r 6iuai pe6pa rcy6a nepneir4nryngpni Aoftoro ocnon, ro AATOTO - npsMoryrrux, OO, I AOi ool L ArotAna-nori.rno AoBoAr{Mo, rqo OO, L BO i OOr L BOr

5,1 2. Iparri Aarroro [psMorcyrnoro napaJrerenine4a - fipaMoryrHr,rxra. Croponu nepe-pioy ..._ a BCrD awaxo4uwo aa reopenoro fli$aropa.BC! = Bgz + CCI; BC, = J7* "";

. DC, = Oncc: = '["\u'; BD = 'lEc\ DC' = J*;a.p*or ='[o' n t + nE * u' + Ju' - r'.p*",o = JB"l& + Jg' + +' +'l 4' + 6' = .i6- 86 + Js + 16 + Jro + eo =

= JE + rlis * "lii = BJb + b + 2Ji5 (crvr).

lltiiliii:l:ii: TEOMETPI9 Einsxina o. 9. ra iH. AraAeMi,{nufi piaenu:j!!: 677

5.13. Poss'saaHus asalotiqHe .4o J$ 5.12.

BD = ^,tEC' * CD' = 't"' - b" = J4' - r" =

= Jro+ 4 'fzo

= zG (cul.

c,D -- ,En' , ccl = i;;t "' = Jr'z + 8' '= 1i - ea = ,6a = 2r,47 tcm).

: -tfl",, ," vl

Al

B

DA

5.27. AB L a, CD ll AB flK ocHoBr.r

rpaneqii. Tozi CD -L o ga sa-c;ri4nou B oBHaKlr fleprrenAr4]ry-,rcpnocri np.flMoI i rrJotrll{Hu.

5.24. HB L(ABC),CK I HB,romyCK L(ABC)(na-c.iri.aox a oBHafi r{ nepneH4u rcy,rspnocri npnmoii nlorqlrnn), ToAi npgma CK nepneHAr4r{yJrfpHa6y,qr,-,sniii np,rmifi, rqo re:xurr y n.lrorqfiHi (.43C)i npoxo4urr qepea rorrlcy C. Orxte, CK )- AC.

i

a

BC, = Fc' * cci = ff - "z

=,tU - e' = d6+ 64 = Jao = a.6

p*o", = 2J5 + z^,Ei + +'tB = alr + 2fi7 (cu).

(cwr)

5.24. t.

TerpaeAp HaarrBaerbc,fi npanunbtlutt, srtrrlo aci itoro rpani - pienocropouui rpu-

KyTHI4rtI4.

5,14. Hexaia ABC\ - gart:rir tpanuntuuii rerpae4p. M -cepeguna AD, K - cePeguna BC.l) vAKD - pieuo6e4peuuir, 6o AK = KD stx ne4ianz(a sHavurr, i nucoru) pionocroponHix rpnnyrnraicin '4-BCi. cDB.MK - we4ia:aa r'A.I{D 3a yMoBoro, a snaqzrb. i eracora:

MK LAD.2) AHa,'roriuHo, rCMB - pieHo6eApesuk,6o MC = MB'MK - v'etiana i eucora: MK L BC.

5,29 . ABCD - npane4ia (BC ll AD , 1A = 9O").I)(ABC),MeLOcxiarrn I -L (ABC), ro MB I AB.Ocxiarxu r:panerlia ABC D npavoxyrHur{a3a yMoBoo, ro AB I BC.Tanunn vlrnom, npaua -AB trepnenAznyJrapHaABoM np.flMrrM MB i BC nrlorrlr4Hu (MBC),a rle osaaqae, rqo,4-B -L (MBC) (e a ymonoro).

5,3O. ABCD - npsmoxyrnux, roMyAD I AB. AK L,(ABC), rovyA]< LAD.flpnma AD nepnenAr{nyrfl pna ABoMtpamuvAK iAB nnoqnuu (,4-BK),nni neperzHalorrca n rovqi A. To4iAD L(ABK) Ba oaHarroxl nepreu,qu-rcy;rapnocri npfiMol i n,roq[Hz.

Hffi$i: /iiu;ri'^..r-.r.tr

i1.i$ffi

D

I

5.31 , Vci rpani xy6a - rcBaAparq, rouy AC L BD. HexaiiM cepe4uuaAB, noanaqzrro I{ - cepe4zHy-4D. TogiMK - cepegaa rri:r.itt IABD: MK ll BD. To4i MK LAC.flpoee4euo MN ll BBr MN LAB. Ilaorqnna nepepiay ne-' perrrHae rpanrA,B'C,D, no ni4piany NP, NP )l BprMNPK - uyrcaunfi nepepia, npsMonyrrrrn.Il.norqrana repepiay (MNP) napaae,rrua r.norqr.*ri 4iaro-HaJrr,Horo nepepiey BBtDrD, ocnimnz npoxoAurb qepea

4ai npaui, ani ueper:zuarorbcfl, flapareJrbHi 4eou npa-MuM nroqnHu BBtDtD.BBtDp 1,4C. Biaouo, ruo flKUlo trprMa nepneH.4r{KyJrfipHa o4uiil a aaox napaJrerb-Hfix trJrorTlfiH, ro BoHa neprreH,qr.rxyrflpna i gpyrifi. Orxe, AC I MNPK.

5.15. A. 5.18. B.

5.16. A.

5.17. B.

5.19. B.

5.20. B.

5.21 . B.

5.22.8.

5.23. B.

5.25. 3a yMoBoro rrz n, [pt4 Ltrr'oMy rn I cr. AoseAenro, rqo i n I tx'

Poar:rsneuo gei npaui c i d' tnoulu]f.u cx, nHi neperzualorbctr' Ocxinr'xn n I cr'

ro m L c i m L d'3a oguaxoro. A ocnilsxra n ll m, ro n L c i n .L d' Togi fipflMa z

nepneHAuxyrlrpHa ABoM npfiMuM c i d nnoqu:rru cr, rrci neper:zuarorbcfi' Suavurl'n I cx (aa osnaroro).

5.32. Hexafi ABCD raparerbrpaM, BK f (ABC),FD L (ABC). Tosi. BK ll DF. AB CD nn npor:u-.nerxni croponu fiapanerorpaMa. flni npami AB i BKlJ'rorrlr4Hn (,43K) napa"remni ri4noei4Ho rpauuu DCi DF nnoryuun (DFC). 3a oaHanoro (ABK\ ll (DFC\.

5.33. Posn'saanns ana.noriqse ao Nh 5.32.

AD

iif;tiiitif* rEoMETPlc Einsxina o. 9. ra ix. Araseuivxui pieenu;!:i:i 629

55.2, flepneHguKytepHicrb npeMoTra nroull4Hll y npocropi

flpama, sna treperr4Hae rrJrorqrrHy, rlaar4Baerbcn nepneu1urcynnpnom go qiei n;ro-

rrlrrnr, slcnlo BoHa neptreHArrr{ynffpua Ao 6yAr,-sxoi npauoi, qo 'nbxrzrr' Ha qifinrouluHi i npoxo4lrs qepes TorJKy ix neperuly.Oasana. flHrqo npama trepneHAZnyJrsprra Ao ABox trp.trMrlx' sri 'nexcatr y n'norqlrriTa rreperrrxarorbcs, ro BoHa nepneuAttny,'rJlpua go 4aHoi trrolrluHu'HaoriAxu. 1. .flrqo nJrolquHa neplen.qflIdyJlfipHa 4o o4niei 3 ABox rrapaleJrbuzx

npffMr{x, To BoHa nepneH.Ilu}tyrfipHa i 4o apyroi npnvoi.2. ,[si npsui, nepileHAnxyrspni ,4o o4uiei fiJrotr{r'rnu' napareJrbHl'

5.34. Hexafi/BCD - aattufr 4onfurrnrft vorzpr4r{yrgr4r{.

AC n BD = Qi BO = OD. M e.(ABC); MB = MD.BD = 6 cu; OC = 4 clu; CD = 5 cv. tBMD * pinao-

6eApesuii, BM = MD 3a yMoBoIo' romy ocxi.nlrcr'r

BO = OD, ro MO - sficora i MeAiaHa d BMD,

oo=!nn=9=s ("*r.22aCOD - npauoxytnrzft (erranercrrcuft), CO LBD- Taxuuuunou, BD LCO, BD LMO'..L BD, rouy BD L (MOC\ 3a ooHar(oro flepnerrAnr{yJrspnocri npauoi i n'noulunra'

$ 5.3. IlepneHgltKynep i noxnna. Teopeua npo rpl4 nepneH4llKynepn

5.36. E) OB < OC.

5.37. E, r, A.

5.38. I)1i5.5.3e. A)

5.4O. I) MA L AD i MC'L CD sa TTII.

5,41. Bi1noei)a:2J IHAC;4) IBAH;5\ IHCB.

5.42. ,{iarona.nr KBaiqpara a = "J2 ; L= oJz.

A 3

E 1

B 4

f 5

A 2

A)l=z^tE rE=4"(cm).F) J=4Jt (cM).

B) t = 4^tE. Ji = +.2 = 8 (cu).

D t=lJr.E=o (c*).

A) 1=6J2 (crvr).

5.43. S =AD2; AD = rlS "*.3a reopeuoro fligaropa /Jl1'? = 1lD2 + MDz'

AM =,[S +to" = Js+ roo ("*).

A) m= vEl +100 = 11 (cu).

n) "i = 16O + rOO = 14 (cu).

D m=Jaa+too =12 (cu).

r1 -=./os+roo=13 (cu).

Al m = JffO+fOO = 16 (crvr).

5.45. 1) BD; 2) B,C;3) DCr.

A

5.46, MA L (ABC), AC I BC oa yMoBoro.

MC L BC aa TTII. a MCB - npauo*yruufi.

5.47. y Nb 5.46 6yno gonegeno, u\o a MCB - upauorcyrnui,3 tABC: AB = 5 cm (erranercrxzft rprzxytrrarc),3 aACM:

Lag = ,tqM' + Ac' = llr' + 4' = v/iz4 + 16 =

= fi60 = a.,60 (cm).

3 aABM:pm =,[ettt, + as, = Jb, -tr, = Jrb+u4 = JrOS = ra

5.48. 3 ro.mn A Ao nJrorrlr4Hr,r clc npoBeAego rrepnerrAlr-ttynap AB = 12 cm i noxuly AC. "fozi IABC * npa-nor<yrnrafi. Hexaft BC = x cur, ro1i AC = (r + 8) cm.3a reopemoro fli@aropa AC'z = AB2 * BC'i(x+8)2=122*x2ix2+76x+64=L44+x2;l6x=144-64;l6x=8O;x=5.AC=5+8=13(cu).BiOnoai0u: LB crc.

(cnn),

D

Bi4piaor AB - nepneugtrxyJrflp, nporegenvfi s lo'lxu AAO frJrorqrrHrl C)( .

AC - nox:u.na, BC * upoerqia noxu.rioiAC na nJroulr{Hy o[.

B - ocnona flepneuArrrcyJr,Epa, C - ocnosa noxr'rroi'

BracrNrocti reprres,qrftyJrspa i noxra;roi, npoBe,4errfixs o4riei rovrcu1. Ia r:ovrcr.r, rqo He HaJrexctrrr n.nou4uni, Molrcna rrpoBecrrloaun i ri.nrnra oAilH fiepneuArrxy.nrp i 6esni'r noxfirrzx.2. ,{oaxnna trepneuAfircyJrspa Meurna 3a AoB]rcr.lHy 6yAs-sxoi noxnnoi'3. Pisui noxuai rrnarorr pinni npoenqii i, nanuanu, uoxr.ni, rrlo Malorb pirni npo-

enqii, pisni iuim co6oro.4. Bi.nirua noxtrJra uae 6i.nr'ry npoenqilo, 6i.nsnrift upoerqii niguori,4ae 6i'nruranoxrrJra. A

Teoperua no rpr{ repreuAurcylapu (TTII)flxr4o npama, flpoBeneHa na nlou4rai qepes ocuoBynoxz.noi, nepneHAl4xyJ'r.fipna 4o 'ii npoenu,ii' ro BoHa

lepnerrAlrxyJrapna i Ao noxu.noi. I nannaxu: axrqonpffMa, rrpoBeAena qepes ocHoBy [oxI'{JIoi n-a nnqu4zni'nepnerlAr4rcyJr,fipHa Ao noxr4lrol, To BoHa rrepFenAr4rty-

.nnpna i 4o upoexrlii noxz.noi.

A 3

F 1 o

B 2 4

A .dcM B

680 i:::::: FEOMETPIS Einsxina O. L ra in. Axagervri'lHrft piBeHb ii:ii:'.:*itii! $iC.t:;ii:il:.ii rEOMETPI9 Eincxixa O. 9. ra iH. Axa,qeMi,{nraft pieexu l.{"! 681

W,ttt,.

!ftffii{:fi;:1:;:

5.49. AQ L a, AB i AC - noxz.ni, QB i QC ri4noni4no ix npo-

exqii, 3 trpffMo(yrHl4x rpuHytHuHin ABQ i ACQ:

BQ'1=ARz'AQ?'BA = {Au" - AA' = 'l:.s' 1r' =.lf€p -r44 = fs = e 1".CQ2=AC'-AQ2;ca = ^tls' -rz" = Jr25 - r44= r,Ei = g (cu).BiAnosiAb:5 i I cm.

5,5O. SO ,L cr, SA - noxtrra, AO - li upoexrlia.SO = 15 cu,AO = x cv, SA= x * 3.3 tAOS (ZO = 9O"):

SA'z= AO2 + SO2; (x + 3)2 - x2 + 152'* + 6r't 9 = x2 + 225;6x=225 9;6x =276:':c=36.Bilnoai}t: AO = 36 cm.

5.51. AO L a, AB i AC - noxnn| AB i OC - ix npoenqii.A.B = 26 cv, OB = 1'O cla, AC = 30 cm.

3 IAOB: AB2 = AO2 l OB2;

AO2 = AB2 _ OB2 = 262 _ LO2 = (26 10X26 + 10) =

=16.36=576.3 IAOC: ACz = AO2 'l OC2; OCz = ACz AO";

oC = ,tTo' - sza = Jr00 - 5?6 = .tm = 18 (ma).

5.52. MA L a, MB = 13 cira i MC = 15 cru - roxu.ni,AB i AC si4noni4Ho ix nPoenilii.Ocrci,,rr,xn MC . MB, ro AC - 6i,lr,ura npoexqin.Toruy AB = x cM, AC = (r + 4) cu.3 IABM: AM2 = BM2 AB2; AM2 = 132 - x2 = t6g x?.

3 yCAM: AM2 = CM2 - AC2;

AM2 =I52 (x + 4)2_225 _ x2 _ gx_ 16== x2-8x+2O9.flprzpinuaemo orpramaHi snaqeunfi .

!69-- x' = x2 - 8x + 2O9; 8x = 2O9 - 169; 8r = 4O; x = 5.

AM = $6s - b' = "tT6, - % = ^1144 = 12 (cu).Bidnoaidu: l2 cttr.

5.53. cx B, rovrca K j' D ner.<aru y n;rorquai a, M i N -y nrroquHi B.KM L c\, ror.i KM I P. Ocrixrru DN * KM, roDN/p. Tlpote1emo DC I B, ro4i DC - nepnengu-xy.nap i DC = 5 ct't, DN = 7 cM - troxr'ura.

CN - npoenqi" u]4!t""u?ry ua n:roquuv I (s)'

B tcDN: cN =^[DN' a'-UVz -rz =J4s-28=rE4=2J6 (cm).

Bidnoeido: 2rF cu.

5.54, AC L u, AC L 9, AC = 10 cvr. BD - noxlata.Ilporegeuo BK L F, rotl KD - npoexqia BDHa nJrorqrruy S. KD = 24 cw.3 IKBD: BK = AC = 10 cu; BD2 = BK2 i KD2i

BD ="tEK\ KD' = ^lTo\ r4' = vmo + 5?6 -= v6?6 = 26 (cm).

Bi1noeidu: 26 c*r.

5,55, ABCD - ,fiapa!'rerrorpatr, AC i BD - fioro 4iaro-na.ni. 3a yMoBoro AC c a, BD L a, 3a sracrfisicrm4iaroHa.refi trapaJreJrorpaMa AO = OC, ro6ro BO -

me4iaxa i sr,rcora aABC, lfe oanarrae, llqo aABCpinno6egpeHnfi. Tomy ABCD - pom6.

P*"r=6'4=24(e\c).Bidnoeida: 24 cv.

5.56, Hexaft B roqnr4 A 4o n;rorqranz cr npone4eno 4rinoxuni AB i AC rax, r4o AB : AC = 5 : 6. flponegemoAO L u, roai OB i OC - npoeHqii noxr.r.nzx. Bpaxo-Byroqfi Te, rqo 6i.nrrra noxr4Jra rrrae 6i.nrtry npoenqiro,

uaemo: OB = 4 clu; OC = 3t[ crt.flosHaqfiMo .r xoetbiqienr nponopqiftnocri, To,qi AB = 5x cm, AC = 6x cw.

Bupaeumo s rAOB i rAOC gonxczHyAO.'3 aAOB: AO2 = AB2 - OBz = ('x)z - 4z : 25x2 - 16.

a .AOC: Ao2 = ACz - oCz = 1ory, - (s"6)' = B6x2 - 27.Ilpupinnaenno orpuuaui BHaqeHHff: 25x' - 16 = 36x2 - 27;36x2 - 25x2 -- 27 - 16; l1x2 = ll; x2 = li x = 1 (r > 0).

Orxe, Ao ='lrb-' - tG = ^lrb - IG = rb = s (cM).Bi1noei)u: 3 cm.

5.57. 3 ro.rxu A Ha nJrorquuy ct npoBeJrr{ nepneuAr4nyJr,fipAO =12 cu i roxury-4-B = 15 cm.OB - npoeHqis noxr.r.noi -48 Ha nJronllrHy cL.

3 IAOB; AB2 = AO2 * OB2i

oB ,t dB' - ao' Jtn'iz' ,lr2b - r44

= ./ar :'s 1cuy.Onycrziuo a ocHoBZ nepileHArrdyJrflpa O OC L AB.AC - npoeHqifl nepreHArinyrapa AO ua noxnly -A-8,

B rypcy n,ranimerpii ni.qouo cninnigHorueuu,fi: Brrcora np,fiMolryrHoro rprrxyrur.rKa,onyuleHa ua rinoreHyay, c,cepe4uim nponoprlifiuum mirx npoexuinun xareris Ha

rinoreHyey, a xolrcsufi rcarer - cepeAHe uponopqifiHe ui:n rinoreHyeorc i fioronpoexqierc na rinorenyay.

Hexaft BC = r cM, roAi Ag = G5 .r) crr,r.

OC2 = OB, BC2; OC, = 92 , x2 = gI _ x2.

oc, = Ao2 _ ACz. OC' = 12' _ (L5 _ x)2 = 144 _ 225 + 3Ox _ xzi81 - x2 =144 - 225 | 30x x'z: 3Ox= 162; x =5,4.AC = 75 - 5,4 9.6 (cm).BiOnoai}o: 9,6 cttr,

5.58. 3 rouxu M, M e o., trpoBeru MO L d, MA - 37 cvi MB 13 cm - roxu,li.Ocrci;mr*z 6i.nrmiii noxra,riii ni4noei4ae 6i"nrua npoenqia,ro AO - npoenqia noxrz.nor AM, OB npoexqis BM,aa yMoBoro AO : OB = 7 :1.AO=7xcu,OB=rcu.3 aAOM: MO2 = AM2 - AO2 = 37' - (7x)2 -- 1369 - 49x'.3 r BOM: OM2 = BM' - BO' = I32 - x' = 169 x2.1369 - 4912 = 169 - x2; 4812 = 1369 - L69; 48t2 = L2OO; x2 = 25i r = 5 (r > 0).

Mo = Jt6s 4 = tT6s - % = tT44 =12 (cm).Bidnoaidu L2 cn.

iii{i:i1:ilti: TEOMETPI9 Sinsnixa O. 9 ra ix. Ara.4eruiqHrfr piBeHb il* 6836A2 ;:::::: FEOMETPIS EincHiHa O. I' ra iu' AxagevrivHuft piBeHb iii:iiili:i;iii

5,59. Hexaft y IABC lC = 90". DC L (ABC), DC = 1 ttn'BC=2tv,AC=3sNr.AB = ^tW + BC' =.,12' .g = G- I = ts (c*).

"s^..- = I AC. Bc =!.r. t= B (amz).22flpone4emo CK L AB, ropi

1 2S..^^ 2.3 6.S -_=:_4B.CK: CK =_:"q=

-=F (AM).

2 AB J13 J133a reopenoro npo rpn neprreHAhnyJrspn, ocxilsxraDC L (ABC), CK L AB, ro DK L AB-

--\2B .DCK: DK = JcD" +cK" =,/t.i*l = ^/t*9u =! lJ13l v 13

"m +=1=e,r (4ur,).J13 2

Bidnoai)a:3,5 Au'z.

5,60. Hexaft y IABC AB = I5 cwr, Bf = 44 cut, AC = 37 cu.V tpuxyrrrury nporra 6i.nruroi cropoxlr Jre)Rlrb 6i.nr'rrruir

rcyr, ro6to A - 6i.nrrrrait rYr.AD L (ABC), AD = ht= I c16.

Ilpone4erro DK L BC, rogi sa r:eopewrorc flpo rpr4 nepneH-

Arrrcyrrt{pf DK L BC,3a (fopuynoio lepona anaiigeuo nlorqy aABC:

75+37+44p= , =+x.

S*" =,[r{n - o[r - b)(n - ")= JI6 €. s ' 11.11 4 = 4.8.11 .2 = 264 (c*a2).

Are s..-^ =!.a11 .BC; AK=& =ry =12 (cu).-*'- -w( 2 --- BC 443 IADK: DK2 = AD2 * AK2;

DK -- ^[AD" * AK" = Js' -A' = !61 + 144 = JW = 15 (cm).

t o.s-- _2 66 _ 12 (cm).s*, = )AK- BC; AK =: ",

118 aADK: DK2 = AD2 + AI{2'

AD=.[DK" - AK' =$+" -tz' = Je4-rrlet.trl = tr€6 =

= J5.4 s6 = 2.6\6 = I z16 (cru).

Bilnoai\a: 12J5 crtr.

5.62. Hexafi aABC - pinno6e4penrafi. Ocrilr,nra B rpr{KyrHr{Kylrome 6ytu ri.nrrra o4uu upamlrrt Hyr, To qe rre Mo)fie 6yrr-r rcyrnpra ocnoni. Touy srrqo -4.C = BC, ro lC = 9O'. CF L (ABC),CF = 24 cv.

(gM). FD L AB, rogi ea reopeMoro npo rpr.r nepneHAlrnyr.ffpuCD L AB, CB.- sucora i ue4iaua a,43C,

AD = BD = \ o" =1.s0 = rs ("*).22t AC D - npauoxyrnrafi pinro6eApeuzfi Tpr{rcyruurc,CD=AD=18cu.

D

F{i

c

I

l

i

l

I

l

l

I

7

Jr3

"*, =i* "t =|

ijiui

tiHf,lffiii$i'ii:iirlfi

B dcDF; FD =,tEc\ cD' = Jr4\ 18' = Js?6 + sr4= v600 = B0 (mr).BiOnoaiOa: 3O clr.

5-63- B qerrrpa ripyra O psniyca OB = 4 cut trpoBeAeMogo ftoro nJron{rrur,r nepnerlAnrcyJrsp OA, BD - Aorr4rr-Ha Ao rcorra, rouy OB I BD, ro,qi AB L BD Ba reope-MOrO npo Tpr{ fiepneH,4rrnyJr,fiprr.

BC = alg cu;,4C = 10 cu.3 aABC:

Au=,171r',-Be= =fioo-48 =Jn=zJrg,

3 rAOB: OA= ^tW - OB' = !62 -16 = ^60

= c (cru).

Bi}noei)a: 6 cn.

$-5-4. Ilepnen,qrarynrpxitTb nroulr,rH- 3anexnicrr Mix napanenusicrpi nepnex4nxynnpHicrxl npcMlrx i nnoulum

Bi1noai}t: 15 cw. D

5.61, Hexafi y aABC AB = 2O cru, BC = 11 cin'AC = 13 cv.V tpnxyrnzrcy rvreumqfi Fcyr JrexrlTr' nporfi Meurrror

cropourr, rovy ZA - uenrurfi.Ilpoae4eno AD L (ABC) i DK I BC, toAi AI< L BCBa reopeMolo npo rpl{ fieprTerrAtlxyJr,spu. DK = 24 cw

3a yMoBoro.

f,Ix i e nonepe4sift sa,qaqi, cfiopucraeMocb norrrr4peHrrM upuiiolrou: 3uaftAeMo flro-rqy rpnr{yrHlr$aABC aa ogHiero $oprrny;roro, a norim a gpyroi @oprrnynra onaiigeuoBHaqeHHfi sucoru AK.

Ba goprvyaoro repona: o =LL * lin 20-

= zz.

s*" = J p@ - o\(p - b)(p - r) = @ ='t1'z' 1L s' 8 =

=.tfLJL r - r., = 11. 3. 2 = 66 (cru'z).

684 i{:l reouerplg Sincxixa o. 9. ra iH. Aralreilri'{Hrfi piBeHb ii;}iiiiii:$

5.64. B.

5.66. E.

5.65. r.

5.67. B, A.

- (+J5)'

,{ri n.ronlranra, rlo ileperrruarorr,cg, HagrrBalorr'cst nep'nen Ourcynapuu.lLz, axrqo rper,ff nJrorrlr{Ha, nepnenAilny-J.flpHa Ao lpRuoi neperuHy qax nJrouluH. neperunae ixIIo fieprreHArarcyJr,fi pHrrx npfi Mr,rx.Oanarca. Exnlo o4na B ABox nJrorqran [poxoAr4Tb qepe3

npfiMy, nep[eHgrftyJ]spHy ao Apyroi n.rouqr.rnr.r, ro qirrJr orq r.rHr.r nepne n4r,rx y,r.a pn i.BJracrfisocri1) flnqo a rorr&lr, qo Jrelrrrrb e o4uifi ie nepueu4uxy-JrspHHx rrJroulfiH, rrpoBecTr,r fieprreH4r,rxyJrfip Io ApyroirrJrorqr,rur.r, ro qeft nepnenAr.rnyJrgp reuturr, y uepuriftnJrorr{r4rrt.2) flxqo r o4niii s ABox nepfieuArrxyJrfipHlrx nJrorqrru npoBecrra rrepnenAzr{yJrffpgo ix .nixii reperrHy, ro nin 6y,qe trepnerr,4uxyJrffpnuft 4pyrift n.norqrani.

0(.\0=a,y.a=c,YaB=b,aLb.

[iifii{i{:::: TEOMETPI9 Sincxixa O. 9. ra ix- Axa,qeuivnrafi pieenu iililr 695

5.68.

5.69. (.4-BC) L (ACD),'roMy BM L MD' BM = MD'

BD =,IEBM, = ^lr.2b = rrE (cu).Bi)noai0t: B,

s.7o- AC = BCJ|= s.E (c.). AN - rqe * NC = Jlo.% = "lzr = rJEBidnoeidu: B.

s.71. BC =.tW - Ae : d00 -36 = nGZ = s (cM).

BP ,E( * PC' = 16" - 6. 1o (cm).

Bidnoaida: B,

' MN ,[MK'-KN" =VE'z+6' =lo (cv)'5.72. KN = :. AC = 6 cv.

BidnoaiOa: l.

5.73. B.

5.74.3a yMoBoIo cr I B i <p t B, roai o 1l <p.

5,75. Ocxim,xu ABCD - HBaApar' ro CD L AD'Ba yMoBoro (ABC) L (ADK), DK c (ADK\' ro*rv CD L DK'3 yADK: DK2 = AD2 + AK2 - 2' + 22 = 8'(cM)'

B rcDK: KC ="ED\ DK" = 'lz'lp = nEz = zJs t"*\.

5.76. 3a yMoBoro SB L AB, SB L BC, AB a BC = B'ro4i aa oaxaHoro neprreuAr'r(yr.fipnoc1i nparroi i nlo-rqunu SB L (ABC).fhorquua (5-419) npoxo.qurb r{epea npamy SB, rouy(s/B) 1(lBc).

AD5,77.HEXAft 1A:aO AABC, NA LAB, MC IAC, NA II MC'

BiAolro, qo srcrr{o oAHa 3 ABox napaJleJrr'Hr{x fipsMrxnepneHAr{nyJrspna 4o rperloi npauoi, ro i gpyra npauanepneHArztcyr.ffpua 4o uei. Touy NA L AC, MC L BC'

Koxna a nprmzx,4N i MC tepneugunyJlflpna uouluui(,4-BC). Toti trroquHa (ANMC), rqo trpoxoArrr, qepe3

npaui,4N L MC, neprl'enqrrlcylaspua nrorquui (ABC) (re'opema 7, uo4y;rs 5).

5.78, Hexaft MC - nepnetAr'rrcyJrsp Ao flJrort\vru ^ABC,MD L AB. Togi aa reoperrom npo rprr [epnel{Aur{yJTrpr4

DC LA.B.Ocxirrxu tnoryuna (MCD) npoxo4zrr vepea npamy MC(MC L (ABC)), ro 3a oor{ar{orl trepneHAnrcyJr,flpnocri u:ro-

quu (MCD\ r @BC).

5.79. Teopema 8, moAy.ns 5.

5.80. Hexaft gsi n.norqunu o i p neperunarorbcfl rro npauifi a i neprer4zxyJr,spui Aon;rorqunn y. To4i vepes 6y4r-axy rouxy npamor a npoBeAeMo fipflMy, nepleHAlxy-rflpHy trJrorrlnHi y. Bi4noxi4llo Ao Br'racrr4Bocri, c@opmylsoeauoi n reoperri g, 4anp,fiMa J'relrcl4Tb i n n.norqrani o, i n n.norquni B, ro6ro a6iraerrcs a npJrMoro a. Snig-cz a Iy,

5,81.3a yMoBoro nJrotr{u}ru o, P i y rolapuo nepneH,qnnyrrsp-ui, PoarrsueMo cx I B. Ilror4rzna y treperuHae qi rrorqNuu norrpfiMrax a i c. f,Irrqo trror{nHv c i B neperunarorr,c,r no flpfi-v'iir b, ay I cr i y I B, ro b I y.A ocni;rsnu npaua b nepneH,4r4nyr'rffpna n.norquni y, To BoHaneprreHAunyrspHa 6y4r-axifi npauifi nrorqnurr y, slca npoxo-AfiT6 qepea rx rouny rreperrrHy. Taxzm uuuou, b I a i b L c.Asaroriqno Moxtna AoBecrx, ryo i a L c.

5.82. Il.norquHu cx, i B nepnen4rHynapni. CD - npnua ne-perfiHy rrJrorqu:n. AB = 25 cm - 4aHufi ni4pisox, A e 9,B e s. AC L CD nigcranr, nig rovnr,A Ao nrrorrlrux o,AC=7cw.BD I CD - ni4craHs ei4 rovxu B Ao n.noulunz 9,BD = 15 cu. 3a reopeuon 8 AC L CB, BD I.4,D. ToaiCB i AD - npoexqii ei4piana -A-B ni4norigno Ha nrrotqu-HHcriB.3 rABC: AB2 = AC2 LBC2;

BC = ^lqnz - de; BC =.JEs" J, = J6% - 49 = G?6 = 24 (crvr).

3 r ADB: AB2 =..AD2 i BD2; AD = J AB' - BD" ;

Ap = J2b' - tb" = .JE2b - 22b = G00 - 20 (cm).BidnociOv: 2O av i 24 av.

5.83. Ba,qa.{a, o6epnena go N! 5.82. flnorqnHN o, i B nepneH-4uHylapHi. CD - npama neperuHy [Jrorrfrzn.

' AB = 25 cm - ganrfi rigBisoH, A e $, B e a.AC L CD - ei4cranr ni,4 rovxu A 4o n;rotluaz o. To4iCB = 24 cm - npoexqir A-B Ha nrorquuy a. BD L CD -ri4craxr, ei4 rounrz B 4o n,rou4unrn 0. AD - 20 crr -npoexqia.4,B Ha nrorquny P. 3a reopeMoro 8 AC L CB,BD LAD.3 IABC: AB2 = AC2 i BC2i

AC = JAa' - BC' = xEb' - 14' = l2b - r4\es. %)- Ge = T (cu).3 yADB: AB2 = AD2 i BD2;

BD=.tqn, An, =J;s, ,0, = [rb-roteb+ror--,li.4E =!669 =

= 5.3 = 15 (cm).Bi1noei1u: 7 crt i 15 cru.

Tecrr 4nn caMoKoHTponK)

9acrrsa 1

(cu).

'12;;v,"

1. f:2. A.

3. A.

7. B.4. B.

5. A.

6. B.

A

D

d

F

D

0

68O i::::: TEOMETPIS Eincxira O. 9. ra ix. Aragevri'{Hvti piBeHb iiiii:iiiiiii;:ii [t:iiitiliii FEOMETPI9 Einsnina O. 9. ra ix. AxaAervri.{xurfi pieexu iii;: 6A7

8. oB =.t@4 = "61- s6 = Gr = s.6

Bi4noaido: .1.

10. B.

11. ABCD - lrp,fiMoxyrHa rpaneqia.Ilpone4euo AK ll BD, ABDK * npa-MOKYTHI]IH.

AK=BD,CK=CD_KD==cD-AB=17-9=8(cm).3 IACK:AK = ^[Ae 4K' ='n0' - # = 6(cru).Bi1noei}u: B.

13, AO2 = AB2 - OB2 = L|'z - 92 =

= 225 * 8L = 144;

oc = ^lqe - Ao' =,lls'-t++ =

= Jt og - L44 = J25 = 5 (cu).

Bi1noei1a: B.

13. BD = 2Ji cvt; BD, = "[En" * n4 = J8.7 = Jn = 2Jd (cwr).

Bi0noeida: B.

14. t) 7, 3, 4.

(cu).

12. SB L (ABC), touy SB 1,43.tn = ,!i cw.

AS =.,tEB\ AB' = 'En = i3Bi1noaida: B.

16. A.I<2 = AT2 _ KT2 = !32 _ I22 == 169 - 144 = 25.

Ka = ^[Aa' - AK' = Ji" -% =

= J4e -% = J% = 2G (cu).

Bi0noaidu: [,.

A

ctuna2

17. Y aABC lC = 9O', rosi AB2 = BCz * AC2;

ACz = ABz - BC2 = a" -(2"!i)" = B6 - 20 = 16.

Y IAMC ZC = 9O" (MC L (ABC), a sgaqurb, MC I AB).AJ\I2 _. AC2 * MC2;

AM=JAc'*Mc" =Jifl.9 =JE=s @ttr).Bidnoai1u: 5 clu.

18, V nprnoxyrnuxy MNKL ML = 6 cu, Ktr = 8 cu,O - roqxa neperr4ny giarona;reii.SO L (MNn, rosi SO I MO.

3 aMKL MK = JML" + KL' =1F -8" =

= J5o + 04 = Jioo = 1o (cna).

Mo =:MK =+= b (crvr).223 aMOS: MS2 = SO2 * MO2;

so -*,[MS' - Mo'; so = Jrg' -f = JL6g - % = JI44Bidnoai4a: 12 cw.

19. ABCD * rcoagpar, AB -- 2 cv'. MA = I cwr,MA L (ABC\. Toai MA L AC.

AC = ABJ, = 2J2 rcv).3 IAMC: MC2 = AMz 't AC2;

MC=.{AM\ Ae,rA

rvc = 1lr +(zJi)' ="4.s = Jg = s rc').BidnoaiOu: 3 cryt.

2O. Hexaft ABCD - rna4par.SB r @BC), BC L CD, rorvry SC I CD

'34 TeopeMoro trpo Tpr{ neptreHArrKyJr.ffpfi,3 " SCD (zC = go'lt sD'? -'sc'? + CD'?;CD = AB = 12 cnn.

sD = ,lic' * cD' = ./16\ rr, = ,lrs6 - 144 =

= Goo = 2o (cu).Bi6noei0n: 2O cv-

21 . KO L a, KO * nepreHAnKyl.:tp, KM - roxr{rra,OM = lO cru - ii npoenqia.Hexaft KO = , cM, roAi KM = (x + 2) crvr.

3 aKOM (ZO = 9O"): KM2 = KO2 + aM2;(t + 2)2 = x2 -t IO2; x2 + 4x + 4 = x2 + lO0;4x = 96; x = 24. KM = 24 + 2= 26 (cu).Bi}noai}a: 26 cv.

22. MB L d, MB - fiepilenArarcyJrsp Ao rrJrorqr{Hr4 g,MA - noxu:;.a, AB -

'i\ npoexqia, MA = 17 ctr'.Hexafi MB = r cu, ro,qi A.B = (r + ?) crvr.

3 aABM: AMz = AB2 + BMj; 172 -- (x + 7)2 )- x2;289 -- x2 + lhc + 49 r x2; 2x2 i- L4x + 49 - 289 = 0;2x'z + l4x - 24A = O; x2 -t 7x - 120 = 0;D -- b2 - 4ac = 4O + 480 = 529 = 232;

(crur). M 6cu

= 12 (cu).

A--/oh \ /

/ a/a\

/"h \ // "/

A Ji"*

688 ii::: TEOMETPIS EincxiHa O. 9. ra in. Ara4euirHri piBeHb :iiii:liiifii: :ililt{ili:i TEOMETPIS EincxiHa O. 9. ra ix" Ara4enairxrafi pieexr l{:i: 689

tlr.*a.-..,..-.Frtr:.0:--r-r^.^.-.rr-'

filkft;i:::i:::::::

tiffil#i::i.-ii::il#ffii:ii:i:i:ii:i:irJtFrs-r-.r+.i....-.'',i iFl.'ii:::.-,-..:..L

-7 t 23 -7 -23xr= 2 =8; rz

AB=8+7=15(cm).Bir)noeida: L5 cw.

23, MN - cni.nrsa cropoHa pinnoctoponnix rpurcyr-ttut<is MKN i MNF.3a ynronoro (MNK) L (MNF).

KC L MN. FC L MN, KC = FC = 4n!i cm.

3a reopeuoro I KC L (MNF), ro4i ea oeua'rennauneprreHAr4nyJr,fipnoi npauoi Ao rrJrorr{rurl KC LCF.3 rKCF (lC = 9O'): KF2 = KC' i CFz;

KF = JrKC' = $ ta.z= 2.4 = 8 (cm).

Bidnoeidu: 8 cv.

24. O - rorrfta reperrtuy 4iarona;refi rcnagpata,4-BCD,

OM L {ABC\. AM = 53 ctv-, AB = 2aJ2 cm.

AC = ABJ, = 2sJ, .,8 = 56 (cu);

Ao =! AC = 1.uu : 28 (cu)..22Ocxi.nsxu MO L (ABC), ro MO L AO, rotia AIVIO npsmoxyrHufi.AMz = AO2 'l OM2;

oM = ^[tJn' - Ao' = J*",g'z = {53 -r8X53- r8) = -85€1 = 5'9 = 45 (mr).

Bidnoeida: 45 cttt.

27. aABC - pieHocroponniir, AO - uegiana.OD L (ABC), ropl AD - noxnJ'ra,AO .._ npoexqis noxuto\. AO uegiaHa,a Briaqlrrb, i nucora rABC.

B aAOB: Oa =!aC= ] a- 4 (cm); :48 = 8 cnn.2LoA = ,[AB' - oB' = ^18"4 = vC4 16 = J4B = aJ5 (cu).3 IAOF: AD2 = ODz + AO2;

AD -- joD'z + Ao' = ^[+"+"

+ ae= fi6 - 48 = Joa = s lcrvry.Bidnoaida: 8 ctr.

28, aABC - pinHo6e4penwfr, AC - 24 cv - ocHoBa,AB = BC = 20 cru. BS L (ABC), to,qi BS I BK,BS = 8!6 c.u. BK - megiana, orxe, i Brcora.

Y a KBC: fc = ! dC = 12 ctvt;2

BKz = BC2 - KC2 -_ 20' 122 = 4OO - 144 = 256.

3 a SBK: SK' = SB'z + BK2; SK = ,IEA' * nf';

= Jszo. 286 = Jr?6 = 24 (cyt\.

r{acrrxa 3

29. fl.notlurru cr i P - nepneagzxynapni.Hq - "niHia neperuuy trroquH, rovy AH L HQi Bq I HQ (reopema 8).3 rABH: A-82 = AlI2 r BH2.3 aHBQ: BH'= BQ2 * HQ2.Toai AB2 = AI{2 'r BQ' + HQ'.3a ymoeoro AH - a, BQ = b, HQ = c,

ro4i .43 ,E , b" . r'.

30. 3a yuonoro (x LF, KT - risis treperrHy n,'rorluH oi B, rorvry MK I KT i NT L KT aa uacrueictro nep-leHArrnyJr.fi pH14x IIroquH.3 rKMT: KM2 = MTz - KTt'z.

3 aKMN: MN'= KM2 * KN2.MN2 = MT2 - KT2 + MNz.3a yMoBoro MT = a,

KN - b, KT = c,ro4i MN = tE +b, - c,.

31. To.rHuA i B rerxars no pisHi 6oru ni,q rrJrorrIzHu o.AC L a, BD I s. AC = 3O cu, BD = 18 cu.,flxulo ani npaMi nepnen4uHy.napHi 4o 4auoi IrJrorqu-Hn, ro BoHr4 naparemui: AB 1l BD,IhouluHa, fllta rrpoxo.4rrrr, qepes napaJreJrbni npnrrni ACi BD, neperuuae trJrorqvuy c no npauifi CD, a ni4pi-aox AB - r rouqi O.rAOC * aBOD sa ABoMa r{yraMz, 6o ZAOC - IBODfiH Beprrr{aJrbHi, a tAOC i rBOD - npnuorcyrni.IlosHa.rrrvro CO = x cm, ro4i OD = (16 - r) cM.To4i e ro4i6uocri rpuxyrnraxi.e BurrxBae nponopqifi-nicrr, ri4loni4uux cropin:

q

..\

B

25. }JexairAB iAC - lr.i noxuri, npoee4eHi s rovxuA 4otrJrorqr,rHli cr. flpoBe,4eMo AO I oc, -ro4i OB i OC - npoex-qii noxzrux AB i -4C.OB = 27 ctt, OC = 15 cu. Ocxi,rr'xz 6iasura npoexqia niA-noei4ae 6i:muifi uoxr.r.nifi, ro AB > AC.Hexait AB = r cv, ro4i AC - (x 6) cnr.

3 rAOB: AO2 = AB2 - OB2 = x2 - 272.

B \AOC: AO2 =AC2 - OC2 = (x - 6)" 152.Maeuo pinnnuas: x2 - 272 = (x - 6)2 - L52;

x2 _ 729 = x2 - l2x + 36 - 225i L2x = 540; x = 45.

Ao -,17 -zn = J4b' -?r, = J2o25Jrs - Jrzgo = se

Bi1noei)o: 36 cttt.

(cu).

26, 3 roqxn A Ao tr,aoqr,tut4 o trpoBe.qeHo nepneuAunyJrsp AO i ,4ei noxuli: AB = 4 cttt'

i AC - 8 cv. OB i OC - npoercqii qux noxunux.Ocri.nrxz AB < AC, ro i OB < OC.Hexaii OB = , cM, roAi OC = 7x cv.3 aAOBt AO2' = AB2 _ OB2 = 42 _ x2 = 16._ x2.

B \AOC: AO2 = ACz - OC2 = gz - (7x\2 - 64 - 49x2.

Pisssrrug: 16 - x2 = 64 - 49x2; 48x2 - 64 16;48x2 = 48: x2 = Li x = 1 (r > 0).

or*te, OB = I ev. Ao = l{6L = JG (cu).

BilnoaiTu: r[E "*.

T

b

N

690 li::::: rEOMETPIq Eingxina O. 9- ra in. Arag,euisHerfi piBeHb iiiii::-..':iii:i:i: iili:iiil{ii{ TEOMETPI9 EinqxiHa O. 9. ra iu. Ara.qerurirnrafi pieexu iti: 691

AC BD 30 18

#= #, ; = #;t 30(16 - rl = !8x;480 - 30r = r8x; 48x= 480; r = 10'

CO = 10 cm.

B rAoc: Ao =,[AC'*oct =Jsa".Lo' = {500+100 = v{000 - r0",[0 (cu).

OD=16-10=6(cu).B r BoD: oB = .{aD\ BD' -- Je ;W = J:.:,6 + 3% = !660 = 6Jr0 (cu).

AB = Ao + oB = 10t0 + 6./i0 = 16d0 (cn).

Bilnoeila: 16f0 cu.

32. t - flp.rMa rreperrtrry rrJrorqr{rr rpaneqiir4SCD i rpr-*yrnurta ABC.IABC - npauoxyrnzft, ocxi.n*u nzronyerrcs pir-sicrr, AMz = AB2 t MB2, lDz = 82 * 62, 100 = 100.IABC = 90", to6ro MB L t.ToAi grs roro, rqo6 rrrrou{rrga (CBM\ 6vna reptren-

Ar,rKyJrlrprroro 4o npnuoi J, norpi6no, qo6 6y.nN nep-

neu,qlrxyJrspHltulr nparvri I i CB (aa osuaxoro treprreu-

Asryrgpsocri nparvroi i rrJrorqusr I L (CBM\, s$r\otLBMlILBC).Orrxe, I L (CBM) 3a yMoBn, qo rpaneqii -43CD - npruonyrua, ZB = 94"'

MOAynb 6. Kvrt4 I BIACTAHI v nPocroPl

$6.1. Kyrnynpocropi

6.5. F, r" A.

6.9.8) /ABO.

6.12. B) 10Jd crl.

6.6. r, A.

6.10. A) ZBDM.

6.13. I) 40 c*r.

6.11. B) 12 cn.

6.14. B) LMBC.

6.17. B)

6.20. B.

6.8.

6.1 5. A) flpsmoHyrnraft. 6.16. E)

6.1 8. B) 6-19. B, I.

ffi,ffi

,(,ra nrnanauenna ryrin y npocropi ra poee'asynanu,fl BaAarr e,qiei re*lra neo6xi,q-Ho cHarx osnaqeHH,fi rpr,rronoMerprrruNx (pyuxqifi y npsMoryruoMy rpariyrrrlrftyra ta6.nrzqro ix ana.{eur.Cuny c on rocrporo Kyra B rrpfiMor{}TrroMy rprrrcyrtl4riy Easr,rBaerbcrBiAnoruenn.s fiporr{Jretrcuoro ftarera Ao rinor:enyav:

sina=9: sinB=-b.cc[dociayc on rocrporo Kyra np,fiMonyruoro rprrr{yrur,rna Haar4Baerbcssi,qsorleHHs rrpnJrerJroro ]carera Ao rinorenyeu:

b^acosg = -; cosp = -.

Tatteer+cou locrporo Kyra [psMoryruoro rprrryrnr4rda rraorrBaerbc.ffnpoTr4rex(soro RaTeTa,qo ilprrJrerJroro:

teo=!; te1=L.'bu

6.3, l) 6 cu.

IIde 4enrci OopMyrur:sin a

sinz o + cos'o = 1; :::-: = 1961; sin(90'- o) = cos cr; cos(90'- o) = sin cr.cos cr

6.1.f) a = 90". 6.2. E) 90".

AE 2 3 4 a

B

Itymon nixtt Jt uno6iilcHunu npflltumu sasr'rBaerbcfi

rcyr uix npsMr{Mn, axi neperuuarorr,ca i uapa.near'-

ri ganuu nr,rlto6irxnuu [pgMr{M.

I{ymon nix nptLtrtuJttu Ea3ur-

BaeTr,cg nrenmr.rft ia rtyTis,yrBopeur4x nPu nePerrniqr{x np,EMrrx.

m

I aozpannun KA nt'ow uasfiBaetr'cr @irypa, yrBopena

4noua ninn.roulr{HaMll pasoM si cnimuoro fip.rMoro,

u1o ix o6mexcye. Bosa naarasaettca pe6pom 1aoepau''

noao nyma (m)..flrrqo s ^ 9 = m' a r m, a c a, b L m, b c F, roZ(a, b\ * fi:ri\ftrtuia fiyr ABorparruoro Kyra l(a, $.

Idym nitt np,nMon i n,Lour'u-Hon * \e nyr rvri*c rrpgMorora ii npoerc4iero Ha rlro rrJro-

quEy.

CT 0. 30" 45' 60'

srn cr 01

t "E2

t;T

cos fI G2

^,82

I,

tga 0t;T 1 ar6

ctg c .i5 1,6

692 i:.; reouefplg EincHiHa O. 9. ra ix' Ara,4euisHrfi piBeHb iiiil:{iitli :::ititlli:i:{ TEOMETPIf, SinsHixa O. fl. ra iu. Araseruirxrafi pieenu ill 699

6,21.AO L u, AB - roxrJra, OB - npoexqifi roxuror.A.B sa n.nourzuy cx,. To.qi IABO - uryranrzii ryr uirrloxrrJloro i n.norquHoro ctc.

AB = L2 ciu, OB = 6 cm.

cos.lABo=98 = 6 =1.AB122LABO = 6O".

6,22. AO L a, AK - roxuJra, OK - npoerqig noxuloi-4J{ Ha n.norqzny a. IAKO - xyr mixt noxutroto A-I{i n.nouqunoro ct.

AO = 5.tE cu, Ol( = 5 cu.

tE z,qxo = lo LE = Jr,'oK5IAKO = 60".

6.29, AO L u, AB = 4JE c* i AC - noxutti, BO i CO -ix npoexqii. IABO = 60'- ryr ni:+r npauoro-4J3 i o,IACO = 30" - xyr nrix AC i n.norqznoro o.3 IAOB:AO = AB ' sin .IABO = 4rE ' sin 60" =

It=4JB

==6(mr).2

AO63 " AOC: AC = "" = :, 12 (cu).sin30' I

2Bidnoeida: 12 cv.

6.24. AO L a, AB = AC - pinni noxu.ni. Ocxi.nrrcu no-

xnni pieui, ro i ix upoexqii pinni: OB = OC'Togi rABO = IACO Ba rpboMa croponauu. 3 pinnocrirpraxyrnuHin BrrnruBae: IABO = 1ACO.

M

6.26, ABCD - r{BaApar, O -

ro'rna reperrrny fioro 4ia-ronaaeii. SO L (ABC). S1(, Sr, SM, SN - noxz.ni;KO, NO, MO, NO - ix npoeritrii; K, L, M' N - ce-

peAlrHu cropin xna4para.Bonu e ro.rxamrr, B gr[x cropoHu Aortrrtarorbcfl AoBtr[canoro noJra 3 qeHTpoM y rorrqi O.Touy OK = OL = OM : OM r't< pa,4iycr4 BnlrcaHotou tragpar,4-BCD t<ota.

Ocxi,rsHz SO L (ABC), ro nprrMa SO nepnen4r.rxyrflpHa 4o 6y4r,-anoi npnirnoin n.norquni (ABC), qo rpoxo.4srb qepec roqxy O: SO I KO, SO L LO, SO I MO,so r No.a O,KS = aOLS = yOMS = a ONS aa ABoMa r{areraMz (OS - cnilruurl).B pinHocri rpzxyrnrHia BunJrvBae pinnicrr ni4noni4uux t<yria: ZSKO = ISLO == ZSMO = ISNO. To6ro noxra;ri Sf, St, SM i SN saxuresi go nnorlnnu ABCDtr1A OAHar(OBrrM nyTOM

A_____-/'\rl

6,25, O ..._ roqrta fleperuny 4iaroua"reii xsagparaABCD,MO L (ABC).

,{iaroHari xnagpara pisHi i to'moro neperrny gi.nxrr,-

c,s sanni.n: AO -- BO = CO = DO.C Tpltcyrrt:r't<uAOM, BOM, COM i DOM pinni aa 4so-

Ma KareraMu (MO - cni.nrHllfi).3 pienocri rpzxyrnrzria:ZMAO=IMBO=IMCO=IMDO.

6.27. ABCD - trpsMonyrHr.rx, MA I (ABC), MC - no-xura, AC - ii npoenqia. "logi IMCA = 60' - rcyrl.ti*t noxnloto i n;rorqr.rnoro npfl MoltyrHr{Ka.3 aACD:

AC =,tqF * DC' - JB' r 4' = Js+ro = b (cv).

3 aAMC: AM = ACLg6O" = SJI tcru). IBi|noai\t: 5JF cu.

6,28, Hexafi ABC - piruocroponniii rpr4nyrHrr{, AC c ct,i BC - noxlr.ni, a AD t CD ni4noni4uo ix npoenqii, xyruyrBoprororb noxz.lri a nJTotqt4Horo c{,.

B e cr, BD L o. To,qi 1-BBAD i BCD - xyru, arci

B

L

l

l

IlosHaqzuo cropony aABC - x.tACD nparroHyrnnfi sa yMoBoro i pinno6e4peHzfi,ocHirrnq,A-D = DC stt< npoexqii pieuux noxv.nzx.4Bi BC.

3 IACD:ACz =AD2 t CDz;2CD2 = x"; CO=j;.

3 a BCD: BD = ,lBc, cD' = F:-- = l;' ,Ocni,rsnrz BD = CD, ro r BCD npRnnoxyruufi pinHo6e4penuft B ocnoBoro BC, ro4iIBCD = 45". IABD = aCBD Ba rpboMa cropouaMt4 (a6o ea xarerou i rinoreuyeoro),tBAD = 45".

6.29. AO I a, AB = AC = 3J, cm - noxuri , OB =OC ix npoenqii, IBAC = 6O", IBOC = 9O".IABC pinno6e4penafi s ryrou npz eeptruni60', ro4i ir xyru npu ocHoei BC 4opinHrororr 60",

a a ABC - piruocropoH nik. BC = AR = Syli cn.B IBOC: BO'i OC'z= BC2;2802 = BC2;2802 = L8;BO2 =9; BO - 3 cu (BO > 0).3 t A-BO:

Ao=JAB" Bo" =t8-9=16=e (cu).Bidnoai)u: 3 cm.

6.30.AO Ltt, ABiAC - noxuri, OB = OC = 3 cm - ix' npoexqii. ZBAC = 60", ZBOC = 90". OcHinHn npo-ercqii noxz.nux pirni, ro i cavi noxn,ri pinui.3 aBOC: BC2 = BO' i OC2; BC'z- 3'? * 3'? = 18.LABC - pieHo6e4penzit B KyroM npr.r nepuuHi 60',a orrfie, pirHocropoHHifi: AB2 = BC2 = 18.3 rAOB: AB2 = F'02 * AO'z; AO2 =AB2 BO";

.ao = JAB' - Bo2 ="{s - g - Jg = a (cv).Bi0noei6u: 3 cv.

i

A

694 i:::i: TEOMETPIS Einsxina O- 9. ra ix. Axa4eruivHufi piBeHb iil;ii;iiiii;::i iiiilfi:t:i:lt rEOMETPI9 EinqHiHa O. L ra in. Arasenivxnr pieexu l!:i: 695

r

6.31 . ABC - AaHrrit rpr4ryr:g.vt1, AM L BC - ftoro nrco-

ra, AM =6.,6 cu. BCca,Aes,AMIa.AMLBCaa yMoaoro, rogi MH L BC sa reopeMor) rrpo rpr trep-negArrrcyJrspfi. IAMH = 6O" * .tiHifiuraii rcyr ABorpaH-r{oro KyTa.

3 aAMH: AII = AM.sin60'= 6JE (crr).

Bi1noei}a: I cttr.

6.32, oABC - pieno6e4penurl i np.auoxyrnr,ril:AC = BC, ZACB = 90", AB = 20 cu.AB.(1., C e cx, CM I cr, roAi CM L6ya;-stxifi npruifiu.norqrani o, qo [poxoArrrb qepe3 rovxy M.Ilponegeuo CK L AB, rogi MK L AB sa reopeMororpo rpr,r neprrenAr{Kyrrfipu. ZCKM = 30" - nyr nrirxn.norqnnanrz {ABC) i a.CK - nucora i nre,qiana ar43C.

1

'foai KB==AB= 20:2-10 (cu).2

Bracora CK gir.*rrl, IABC na 4na piano6e4peui npsuorcyrni rpzxyr:uvxn:CK=KB=10cm.

3 aCKM: Ctvt =!XC = 5 cu (sx Karer, rr{o reficrrrb rporr,r r{yra.30o),.2

6.33. alBC - pinnocroponnifi , AB c a, C e d, CO L a *nepneEAr.rRyJr.flp Ao rrJrorqr{Ha Ct. ToAi croponz ACi BD * noxu.ni, a niApierdn AO i BO - ix npoercqiirra nJrorqr{Hy d.Ilpore4euo CK LAB - Br^coay ^ABC,CK = 4JB cvr.OK I AB Ba reopeMoro npo rpr{ nepnellArzxyJrffpfi,ZCKO - nyxar.uit xyr uix flJrorqurraMn rABC i a.

s rCKO: sinICKO=co ='J2 =L. 1CKO=3r,.cK 4J3 2

Bi0noei1a:3O'.

6.34. ABCD * r{Ba,4par, AD c a, B e d, BO L a.BA - noxuta, OA - Ii npoexrlia Ha rrJrouluHy o,BA L AD src cropoua KBa,4para, OA LAD Ba reopeMororrpo rprr nepfieH4urcyJr.flpz. Togi ZBAO * ryr ui*cnJrorrlunaMu xsagpara ABCD i a.

- Y aABO: AB = 6 cvt, BO = 3G cM;

sin tBAo = L; =* = +, tBAo = 6o".

BiOnoeida:6O".

6,35. a-4BC - np,nMonyrrrrft (lC = 9O"),AC=BC=6cu.ACca.,Bea.BH La - neptregArrr{yJrxp. To4i BC iAB - no-xrtli, CH i AH - ix npoexqii Ha nJ'IouluHy cL.

IBCH = 45'- Hyr, ni4 arrau rater BC Haxr4Jre-

nraft 4o n.norqrurz. a, ZBAII * ryr rrix ri[oreuy-aoo i n.norqrnnoro o,

3 |ABC: ABz = AC2 + BCzi en=JdC\BC";AB = J6' + 6' =

"0.86 = 6.uE (crvr).

aBHC - nparroryrnufi (BH L cr, orx<e, BH L CH),

BH = BC.sin45. =* =aJz (crvr).2

B .ABH: sin zBAtr = BH'

= s:lZ

= ! . znu+ = 2,o..AB 6"12 2Bi0noei6t 3O'.

-V3=g2

6.36. a-4-BC - nprMoxyruklit (ZC = 9O'), AC = BC.AC c s., B e a. BH I cr - uepnenpurcynap, BCi AB * roxr,uri,. CH i AH - ix npoexqii rra rrrro-qrrny d. IBCH = 45" - xyr, axraft yr:noproe xa-ret BC s trroqzuoro u, IBAII - xyr mirx rino-TeHygoro i n.norqranoo o.Y rABC AB = 30 cm, AB2 =AC'I BC2;

AB2 = 2BC2; 2BC2 = 3O2; 2BC2 = 9O0:' BC2 = 450; .

lo3 'BCH BH = BC.sin4b. =|SJZ Y = rb t.*1.

B aABH sinZBAII = BH

= 15

= 1. tBAt{ = Bo..'A8302Bi0noai1o:30', 15 cu.

6.37. Ilnorquu,r o. i B fieperlrHarorbcfi no npamifi /,,Aeu,BeB,AB=25ctl.

AC L I, A9 = 5 cm; BD L l, BD = 8 cn.flporegemo CK LIi KB ll L To.qi IACK - nilt\it-ulrfi xyr ABorpaHrroro ryra uixr trJroulr,rHaMr.r o,

iF. ztcK=60..KCDB -- upfiMoxyrtr4n: KCDB - fiapaJreJrorpaM,6o CK I DB, KB ll CD aa no6yAonorc, ICDB = 9O' ,

rony KCDB * npaqorcyruiax.Orxe,CK=DB=8cm.3 aACK Ba reopeMoro rcocranycie:AK2 = AC2 + CKz - 2AC ' CK . cos 60o;

Ocxi,rsrcu ZBCH = 45', ro

BC =LI\E (or) (BC > 0).

C

@

fr411 =,ls' +8'z -2 s g I = J25+66+-ao = Jag - z (cm).!2

fllorquna (ACK) L I, 6o AC.L I1 KC L l, rouy AK I BK.B .AKB: KB= JAB, - AK" =,1;b" -f = J625-49 = Jb?6 = 24 (cnr).CD=KB=24cm.Bi0noei1u: 24 cyr.

6.38. Il.ror4zuu o i B fleperi,rHarorbcfl no npnrraifi l.Aea,BeB,-4.8=5cru.AC L t, AC = 4^tE cv.; BD I l, CD = 3 cw.flpone4enno CK L I i KB ll I.Toai /ACK = 30' - .irisiftnraft ftyr ABorpauuororyra uinc n.nouqr.rHaMrr o i g. KCDB * [p,fiMor{ylHrrn(o6fpynrynaNus Ar.B. y N! 6.37): KB = CD = 3 cw.I L (AKC), 6o I L AC i I I CK. KB I CD, rottyKB t (ACK), onne, KB L AK.

696 ii:i: FEOMETPI9 Sinsuixa O. 9. ra ix. Aragevi.{Hni piBeHb :;:*:iiti:i;ii ;i{::iiqi:il{ |EOMETPIC SincHiHa O. f,. ra ix. Araseuivnufi pieenu i{!: 697

3 t AKB: AK = 4 at (erurercrxlrft tpuxyrnur).3 aACK 3a reopeMoro xocr.trycie: AK2 --ACz + KC2 2AC'KC cos 30'.Ilosna.rrnuru 4na eplnrnocri ei4pisoH KC auissorc,, orpllMaeMo pisasHua:

+' = (+Ji)' + x' - 2. 4$. r. @,2'16 = 48 * x2 - l2xi x2 - L2x + 32 = 0; xt = 4, xz- 8.BD -- CK = 4 cn a6o BD = CK = 8 cv.BiOnoai?a:4 cu a6o 8 cru.

6.39, Hexafi-ABC - pirnocroponnifi rpzx;"ruux. AC cct,B e cx. BD I o, rogi 1, i CD - npoeHqii cropiH -4-B

i CB aa rrrorr{rHy o. ToMy ZBAD i ZBCD - xyru,yrropeHi ni4noni4iro cropoHaMz A-B i CD i n.noulra-Horo (x. flo3HaqlMo AJs apyvuocri cropoHy rpfiKyr-surca suissoro a.

'foti s tACD ACz = AD2 + CD2: a2 = 2cD'zi CD = +,12

6.50.

6.52. MN ll AC, rouy MN L BC; KM L (ABC),orx<e, KM L /tlN. MN - ruylcaua si4cranr.BiOnoeilu: !,

6.sg. MN =!o"-12 c*r;KM=t2c.]..2

KN = irivt, + MN" =,ilr, -rX' = jiT+s = tzjzBi0noeida: l.

6.54. B.

6.55. B.

6.56. A.

6.57. B, B, E.

6.58. A, r, E;

6.4s. A.

6.46. B.

6.47. A.

6.48. B, A.

6.49. r.

6.51.

6.59.

B.BCD cosZBCD=CD = o=,o-L=Q. turo=nu'BCJzJz2AHanoriqso Mo)fiHa AoBecrr, r4o IBAD - 45'.

(cu)

LAC, MN 1BC GTN).M

K10 cn

- cepeAHfi linir rpaneqii

h:^[ltnr. , ... .. .i1:r,lg::::i::::f:

{i,fiiEi:iliif:I /: i,l\rf .....!.........: /, ii:41:ii.'..-11!..; ii ii,ili::*:::::::*::_ /_ v?:rir!.'-r.....r.'...'/l i.t $a-'....-..'-'-... ?i I ir;7i1:Jljj.'Ji.,i rrlf ri'...-..r.r.......,:,?_{t:..i-.rtr....'...' ri irt4]'.-.-|:.^...-.-..

6,4O, BO L cx, BA - [oxr.rra, AO - npoercqis noxu.noi-B,4 Ha nrolluny o.ZBAO = 45" - xyr uixt noxuroro BA i nnorqlrHoro o.BA = a. AC c a, IOAC = 45", AC = a.Il.nouluua (BOC) npoxo4rzrr, vepea npauy BO, nepnen-

,qurcyJrspHy n,rollraui ct, roMy (BOC) L s, a BHaqnrL,(BOC) L AO (AO c a). To.qi AO - nepneu4rr(yrflp,a AB i AC - noxa,ri, npone4eai B rotlnu -4 go n.no-rqnuu (BOC), a OB i OC - ix npoexqiiOcxi.nsxn ,4-B = AC sa yMoBoro, ro OB = OC, 6o pinsi noxu,li uarctr pinHi rpoeHqii.Ile osuavae, qo aBOC npauoxyruufi i pinno6egpenufri IBCO - 45" - nyr Ha-

xr,r,ry eigpiaxa BC go nlorquHu ct.

a BOC = t BOA sa ABoMa rcareraMu, azrpcu Br4 rrJrrBae pinnicrp niApisrin BC i BA,ro6ro BC a.Bidnoaida: o:45".

S6.2. BincraHi y npocropi

Bi4crans uint 4eorvra roqr{aMu A i B - AoB}nI.rHa si,4pisrca.4-8.Bi4cranr nig rovrv A 4o npa*roi I goexcuua nepneHAraxyrspa, onyrqenoros roqxu A Ao np,nmoi l.Bigcranr, eig rovxu -4 Ao nJ'rorrlaHr,I o - AoBt{uHa nepneHArtnyJ'rspa, npoBeAeHoroB Toqr{u A 4o n.nou4rznrz cx.

Bigcranr, uir+c napa.ne.nrur.rMu rrpsMoro i n.nolluuoro a6o 4eoma [JrorqrrHaMu - AoB-xcrzna ix cni.nr,soro neptreEAuKylaspa.Bi4crars uix muuo6i**rr{Mr4 flpsMilMff - AoB}KhHa ix cni.nrHoro trepneH,Ithfiy,'Ifi-pa - ni4pisna s xinqauu Ha qux trpsMl4x, nepneHAr4xyJrffpHoro Ao xolxHoi a sux.

6.41 . B. 6.42. B. 6.43- f.

6,44, BC L CD sa yMoBoro, MC I CD 3a reopeMoro fipo rprr flepneHgrxyrspu, roMyMC rxyrcana nigcraHr.Bidnoai1u: B.

698 i:i:j; reOuerplg EincHixa O. 9. ra ix. Axa4enirHui piBeHb iiiillliii;ill:

6.60- .(o4arroea no6ygona: OK L AC, ON L BC, ropi MK1l

oR =;BC 9 cu; oN - raC=s c.v..

MK = ,tmoi\o( =,llr"*r' =

= Jn4+n = Jz% = 15 (cu).

MN=,lMO,'+ON'=rllr'E= Jl69 = 18 (cnn).

Bi0noeida: A,

6.61. Bi,qpiaox KM ue nepernnae rirorluHy o. flpone_pemo KK, L ct i MM r I cr. fllonllrna, fiKa fipoxoAr.rrbvepea npami I{l(ti MML(KKrll MM), uicrrarr ni4pi_sox KM,6o ro.mz K i M sanex<au r1ifi n,rorqr,rni,i neperznae rrJrorqrrHy cr no npalrift KrMr.Hexafi F - cepelr,rHa KM, npore4ervro _f'f, t u.To4i -F, e KrMri KKr, FFr MM1- oi4noni4no ni4_crani nig rovox K, F i M po nroqrrHr,r oc.

?.".:::p:*". (Da.neca F, - cepe4una K,M,. Toli FF,K$MM1.

KK. + MM-tIt= --T-, aei4xz MMr=2ppr-KKt=2.4-l,g =g_ 1,g=6,2(cu).Bi|noei\a: 6,2 cm.

:i:iitiiiti TEOMETPtq Eingxina O. f . ra ix. Axa4eui,rxr,rn pieexu i{:;: 699

A E B3 5 2

6,62. A-BCD - KRaApar, SB -t (/3C), SB = 7 clt'Ocrci;rutu SB I (/-BC)' to SB -L BD, orxce, nigcranrei4 rourcu S ao np,nuoi tsD 4opinuroe 7 cn.

,{iaroira.ni rcnaApata neperrrHarorbcs nig lpauuu xy-rou: BO I /C. Toai SO L AC 3a reopeMoro npo rpr'I

neprreH4lrnyJTspn, SO - rigctanr ni,q touxu S 4o

,qiaronari AC,

BD=ADJ'=8J'no =!no = +Jz

so =,[sn\ no';r--- . --.

so = J7' *(+Ji)' = J4s.n= Jsr = s {c'a).

Bi1noai0u:7crvri9cm.

6.63. Ilponegemo MO L (ABC) | OK L CD.'fogi MK L CD sa reopeMoro npo rpll nep-

rrerr.414r{yJr,'IplI.

MK * siAcranr ni4 rourra M 4o ctopoura HBaApara'

Ocri.rrrcu pinni noxra.ni, nporegeni a o4rdei rovnu, rvra-

rorr pinni npoerqii, a 3a yMoBoro nigcrani ni4 rouxra

M ,4o nomnoi croponnr r{Bagpara pinni, r:o i ix npoex-qii pinni, ro6to O - rleurp Brrucauoro fi onr.rcasoro

rraBRoJro KBai4paTa ICoJra.

AD=7ocrvr, OK =r=+={=r f"*1.223 yMOK: MKz = MOz * OK2;

Mo =.lEtr' - or" - Jfu" - 5' = JL6g - % = nEt'z-' = nBi1noeida: 12 ctt.

6.64. a..43C - piruo6egpenzft upaiuorytnsfi.: lC=90',AC = BC, A-El = 36 cm.

CK I (ABCI, CK -- 24 cn.Ilponegeno CF L AB, CF - ei'{cora i ue4iana pinuo-

6eApenoro aABC, txa B npsMor(yrnoMy rprrxyruflrty/BC 4opinnroe no.nosani rinorenysra:

taF = lAB = 18 cu.

2

Ocxirrru KC I (ABC\, CF LAB, ro KF LAB 3a reopeMoro rpo rpr4 neprerrArrryJrspi{'

KF - nigcranr nig rovxra K Ao rinorenyor 48.3 aKCF: KF2 = KC'+ CFz; KF =.EE'-te' =^li?6-3U =.i600 = 30 (crvr).

Bilnoei}a: 30 c*r..

6.65. Bi,qcranr niA tovrcv M Ao ecix Beprrrr{rr :crra/'palaABCD oAHarcoBa (M q (ABC))'

MA=MB=MC=MD=5cw.Ilponegemo MO L (ABC), ro,qi O - 4elrrp r{orra, oruca-Horo EaBrdoJro ntagpara ABCD, 6o OA = OB = OC = OD

ar npoenqii pinuux noxu.nax.1

V a MOC: OC = - AC = 2,5 cwr.2

MC2 = MO2 + OC2; MO = .{Me - OC, i A

Mo = ,F - z,s" = Jr,l '? 5 = z, rJF (cm).

Bilnoeida:2,5n8 cM.

6.66. AABC - rp,nMonyrur.ria, ZC = 90", AB = !2 cwt. S e (ABC\.Ilpooe4emo SO L (ABC) - nigcranr, oi4 rovrra S go naorqrznn (/BC).SA = SB = SC = 10 cn - nigcrani ri4 rorxz S go nepurr.tr a-A-BC,Togi, OA = OB = OC ax npoexqii pirHux noxra.nnx narrJ'rotr{t4rry (ABC}. Togi O - qenrp ftoJra, o[rrcanonoHaBr(oJIo \ABC, - cepeAr,rHa rinorenyara AB.

1OB==AB=12:2=6 (cw).2

3 a SOB: BS2 = OS2 't OB2;

6g=Jrs'-oB2 = J10,-6, =fi00-86="/6a=a lcru).BidnoeiOa: 8 clu.

6.67. IABC - rpaMonyrrrqit, lC = 90', AC = 3 cn, BC = 4 cm.S e (ABC). Hexafi SO L (ABC) - nigcranr ni4 rovnra S 4o nroiqtrur.r (ABC). Ocxins-rcr S,4 = SB : SC = 6,5 cu Ba yMoBoro, ro OA = OB = OC sx upoerqii piarux no-xtrJurx Ha nrorqr,ruy (zlBC).To,4i O - qenrp KoJra, ofificanoro HaBHoro aABC,O - cepe4r,raa rinorenyor,r AB. AB = 5 cu (a.A-BC -cnrnercrxr,rft),

oa=L*en=2.b cu.2

B aSOB: SB2 = SO, + OB2; SO = nlSA, - Onr;

so = J6,E' - Zs, = il65+r,11(65 - r51 = Jt 4 = B 2 = 6Bidnoaido: 6 cttr.

6168. V rcragpari MNPK O - roqr(a neperzny 4iarona-nefr. AO -E'(MNP\, nO = 2',12 crvr, MN = 4 cn.AK.= AM = Al{ -- AP - nigcrani ni4 rovxu A 4oBepruuu r(BaApara. Bonr,r pi*ni ar noxu.ni, sxrawr ai4-nonigarotr pierii npoexqii. KO = MO = NO = PO sxnoJroBr,rHr{ pierrax giarona.tefi rna4para.

KN = KpJ, = 4Ji @yr). on = lm,r = 2r!2 @nrt. K2

3 IAON:ANz=AO2 *ONz;r,-_.--

41't =^lno'+oNz = rl(zft)" *(zJi)'= G+8 = Jro =+ (cu).Bi0noei1v: 4 c*r.,

6.69, .(aao IABC, AC = L0 crra, AB = 17 crrn, BC = 21 cu.V rpzxyrnraxy HaBnporrr 6i.nr,nroi cropoHr.r ,rer$rrrb 6irrl tnuir xyr. Onne, ZA * naft-6i.nrruvft. DA L (ABC). DA = 15 cu.flpone4emo AK I BC - Brrcory rABC, roni aareopeMoro upo rpra neplerrArxyJrffpu DK L BC.DK - ei4craHr rig rovrcu D 4o cloponra BC.Br,rana.rzmo AoBlrcr,rny DI{ o npauoxyrHoro rprr-xytlelzxa ADK (DA L (ABC) , rowy DA L AK) . !,naqboro EorrepeAnro euaftgemo :4J(. s OopMyrr.r Ars

cwr (ax 4iarona;rr rcna,4Pata).

cM.

ffiffi*f;$ffif,ffi

(cu).

LO+L7 +21_-rL,2 --'

NJIOII{I TPIIKYTHfiKA:1 2S*.

S^"". =: AK CD: AK2cBflrorqy Br,r3HaquMo aa Qoprry,rorc fepona: p =

7OO i::::: rEOMETPI9 EincHiHa O. 9. ra ix' AxagewrivHrf piBeHb:i:::i':::i:i:C: :iiil*it:ii*i TEOMETPI9 EinqHixa O. 9. ra ix. Axa.qeuivmnfi pieenu l:ifi zll

s*" = @ = $+14 I -s = J+s.r 2.'? I . B =

= 4.3.7 = 84 (cn).

dx =2 8i = g t"*).

2t3 aADK DK2 AD2 - AKr:

DK = ^[AD\ AK' = "{F - # ='lrrb. 84 = .tEs8 =17 (mr).

Bi0noai}u: \7 cw.

MD = ^tmc" 'Ma, = lT?, ,Lr, = Jts? -:.:1rG: -14 = Ji{+g = s

Bidnoeida: 35 cv.7 = 35 (cu).

6.70. y aABC AB = 13 cm, BC = 20 cu, AC = 11 cm.

lB - lr.afi'mesllnufi, ocxi.nr,xu rerrtI'rrb HaBnpor[uesmor croponra AC.MB L (ABC\. Ilpoee4euo BK L AC - Br{cory*ABC, togr Ba reopeMoro trpo rprr nepneuAur{y-napu MK I AC, MK - ei,4craHr ei4 rovnu Mgo uaiimennoi cropoHu AC.3a yuoeoro MK = 15 cv'. MB ni4craun aigro'rxu M Ao nrrorqrrHtr (ABC). Suaftaevo MBB npsMoltyrHoro rputtyruutta MBK.

"logi ABCD - trpsMoxyrHun.

AC2 -AB2.t BC2 =AB2 i AD2; AC = JrO\rt' = fqll+44L = J841 = 29 (cm).

Bi}noei1u: 29 cw.

6.72. fleprenguxyrnpui trroilIlrHr o i B nepetttuarorr,cs tro np.sMifi c,

M@s,MeP.Ilpose4emo MB L o., MD L 9, MC L c.

BC .'._ npoexqis MC nannonluny ot,, roMy 3a reopeMolo

rrpo rprr neprreHAnftyJrfipu BC L c, a sa s.nacrunicrlotrepner{Aur{yrrflpHux rtJlorrlnu (S5.4) BC I o.Octti:rcxn MD L B, ro ea e:racrznicrro nepnenguxylap-nocri npamoii i.n.norquHr.r (55.2) MD ll BC, a asa.rrzrr,,MD i BC rexarb e oAHifi naouzui.Toai MBCD - flpflMonyr]rux.MC2=MB2*BC2=MB2*MD';

7O2 l::::: ;EOMETPIS Einnxixa O. 9. ra ix. Axa4ervrivHl4i piBeHb ii:ii::ii:i:i:::i:i i

6.73. ABCD - nBarpar, M e (ABC), MA = MB == MC = MD. Ilo6y4yemo MO )- (ABCJ.IAOM = aBOM = aCOM = aDOM sacninr;eumuIcareroM OM i rinorenyaoro.Toti AO = BO = CO = DO. O - rorrfta neperrHy4iaroHa"re* HBa4para.flronlraHrz (AMC) i (BDM) neperwHar)rbcn tro Anpamifi MO, MO L (ABC\ sa no6ygonoro, rogiMOLAOiMOIDO.Ilrorqzna (.48C) neperzHae nr'roqr4 Hr.r (AMC) i (BDM) no rpnMux AO i DO.AO L DO arc npsui, rqo uicrarr 4iarouali ftBa"qpara. Togi (AMC) L (BDM) Ba oBHa-rreHHsM trepleHA[KyJaEpHrrx nJlorqrrH.

6,74.3 roqnz O neperrruy giarona.nefi npflMoxyrHlrxa ABCD nponegeuo treprrenAuxy-.nap, Biar,lremo Ha HboMy 4osiar,ny lowy M i s'e4Haervro ii s repurHau[ npeMo-KyTHrKa.Ocmirsxn aiarosaai npfiMonyrHuKa pisui i rovnoro neperuHy 4irRrscn naenir,AO = BO - CO = DO. "Iogi rAOM = rBOM = tCOM = aDOM Ba ABoMa KareraMr4(rpzxyruznu npmronyrni, 6o MO L (ABC), or*.e, MO I AC, MO L BD),3 pinnocri rpuryruzxin nunnwnae:. AM - BM = CM = DM, ro6ro roqrca M pinuonig-AaJIeHa B1A BeplfirlH trpsMOKyTHlria.

6.75. ,{aHo ni4picox KM, au:aft He neperuHae nJaon{r4Hy

cx,. qepes Hinqi K i M ra cepe4zHy F ni4piana I{M: trpoBeJrr.r nepfienAraxy!'rsp[ Ha trroqqxy a: KKp MMr

i II, (nia.ctaHi ni4 rovox ,4o nrou{r4Hr4 cx). Ocxi,rrxuKKr L a, MM1 L cr i FFr -L u, ro KK,ll MM, I FFr.Ilpoee4euo uepea npaui KKri MMr nrorquxy, eonarreperr.rHae trroqzHy o no rpauiii K,M,.[ei rouxu npauoi KM Hare]rcarb (K$M), rot'ty KM e (K$M\. F, ua.neuczrt nro-l4lnai (K.KM) i o, romy Fr-F KrMr. Tanuu uuHom, rpavri KM, KrMr KKr FFrMM, temarr e o4nifi n,roruuui.3a reopemoro @a.neca I, cepeArrna KrMr. To4i FF, cepeAus :rinis rpanerlii

K,K M M ,. FF, = KK' t=M *' . lt\o i;i rpe6a 6y,ro .4onecrz.rr'2

6.76. Hexaii rovxa S, trIo Jaerfirrrb rrooa trJorr{[Horo trpflMo]ryrHoro rpraxyruuNa /BC(lC = 9O'), BHaxoArlrbcs na nigcraui 10 cu ei4 nontnoi fioto eepruzuu:SA=SB=SC=10cm.Ilpone4e*ro SO L (ABC), rogi AO = BO = CO nn npoen-qii pinnux noxzJrr{x, ro6ro ro.rxa O -

qeHrp onucaHo-ro HaBrtoro rpzxyrHzxa-43C ro.na, V npamoxyrHouyrpr{nyrHr.rxy eiH ler*curs ua cepe4nui rinorenyaa.

1

OB=1AB =6 cm.2

S aSOB: SBr=SO' + OB2; SO= JSB, AB' =t0' -6' =8 (cm).

SO - mynaua sigcrans si4 rovxrz S 4o llorqzHu (-4BC).

6.77. ABCD pom6. M 4 (ABC).Ilponesemo MO L (ABC) i MK L CD. MK BiA-craHr nig roqxv M Ao cropoHr CD. MK = 8 cm. OK L CD Ba reopeMoro npo rpzn epneHAr4KyJspu.

:iiii;:ll{l:ll TEOMETPIC Eingnisa O. 9. ra ix- Ara4ervrirHr4fi piBeHb iii{i 7Og

M

BK anaft4erro B (DopMy,au n.irorqi rpHrcyrHuHa: S*" = 1o" BK; BK =2

fkorqy aABC snaitgeuo 3a OopMyroro lepoua.

o =lr + rB + 20 - 22. s*o" = @ = ^tEr -1L Y, ='2= $.'r.1L1Lr, = 2.Lr-B = 66 (cu').

"K =2::6 = 12 (cu).

11

3 t MBK MK2 = MB2 r BKzi

MB = ^[MK" - BL: = jrb'z -1r'z = tEE -\44 - rfal= s (cu).Bidnoaida: 9 an.

2S*"AA

6,71. Il.norqunu o i 0 neptreHAuxyaspni i neperunarors-ca no npauifi c. A e a, A e F.Ilponegemo AD L a, AB L9, AC L c'BC - npoexqis AC Ha n,rouluny B, touy 3a reopeMor)

npo rprr neptrenAur(yrJrpu BC L c, a aa o.nacrunicrronepneuArny;r.ffpHilx nJrorqrn (S5.4) BC I g' Ocxial-xtt AD L cr, ro ga e,Tacrusictro nepnen4unyaapHocrinpauoi i rrrlorqr{Hrr (55.2) AD 1l BC, a a+taqurs, AD i

D CI

a"

BB

BC .nemarr n o,4Hiit nJrorqrrHl

D B

cM

C

Ba

l1;,Ii.lr,i;o.':.:.i:r.:,:

,1r r-nski-r^.rq.ir-.-..

'Jlisk.:t !t.c.r.:.

Bi,4cranr OM s\F,ro,txu M uoxna snafirz 3 np.fiMoxyrHoro rplrrytuuxa MOK'Ilonepe4nro exaftgeuo AoBrnl4Hy ni4piaxa oK. lde uoxna apo6urv, BI,IROprcroBy-

rorru nnouly a COD.

,{iarora.ni pou6a nepneugrrcylapni, ro'{rca O gi.nunr ix naenir'

3 tcoD: oc =f,ac= f; = a 1"*1;

oo--!no-r]=a <"*t.

cD = Joc' * oD' = J8' + 6' = Jroo =.ro (cm).

s^"oo--!oc oo=t 6.8=24 (crvr'z). A

s*- = tco. oK; aK ='* =# =4,8 (cu)'

3 tMOK: MK2 = MO2 * OKz;

Mo=.tMK'-oK' = l7 4,8'= Ji8-4fX8+4s) = JtTLr,s --

= Co, 16 = 6,4 (cm).

BiinoeiOv: 6,4 cw.

6.78.. ABCD - pinno6iuua rpaneqia. AB = CD. P *"o = 48 cw' 1A = 6O"'

To.rra S He Jre)fir4r6 y n;rorqrzni rpaneqii. flporegemo so I (ABC). OCni.nrHra toqxaS piBnosiAAalena aig cropin rpaneqii, ro i tovrca O pisuoniA,qarena ri4 nzx, 6o

pioni noxu.ni uarort pinni upoerqii. Tomy o - qeHrp noJra, Bnr{caHoro n rpaneqiro.

To.*ca O - cepeAfiHa ni,qpioxa MN, qo e'eAnye cepeArrHr'I ocnon rpaneqii'MN LAD, MN 11 BK, MN = BK, pe BK - nzcora rpaneqii.SM - lryxana eigctaHr ui4 touxu S go cropoxu rpaneqii, 6o SM L AD sa reo-peMorc npo Tpr{ ilepfieHArrrcynnpu. ocxi.nr,ria n rpaneqiro Mo]r{Ha Brrr4caTI,I I{OJIO, TO

cyMa AoB)r(uE ii ocnos gopinnroe cy*ri 6iunux ctopin: .

pAB + CD = BC + A-D = -T = 24 (cwl,

AB=L.24=12 c*r.2

3 ,ABK: BK = AB.sin 6o' = n g= 6.,6 (cu)'2

NM = BK = 6V'B cru; OU =\Nttt = 3175 civr.2A3 aSOM: SM2 = SO2 + OM2i

| . -.tSttt=.ISO'+OMz = ya'z*(a./a) --Jg+zt =J36=6

Bi4noeidu: 6 cra.

6,79. y IABC AB = 26 em, BC = 28 cw, AC = 2O ctt.M e (ABC).Ilpone4emo MO I (ABC), MO - nia-craHr nig toqnra O Ao flJrorqurrt4 Tprtryrnr{na.flo6y4yeuo MK L AB, ML L BC, MF I AC, MK,ML, MF - ni4ctani oig ro.rrcra M 4o ctopin tpu-rcyrnuna. OK, OL, OF * ix npoerqii. 3a yuonoroMK = ML= MF = 17 cu. ToaiOK -- OL= OF, rouxaO - pa,4iyc r{oJra, Brrr,rcaHoro n rpr'rxyrnux -ABC.

,[onnruny ri4pisrca MO zuaftgewo s tMOL'

Ol snafigemo s (popMyrl4 . = { , g" r - pagiyc Brucanoro KoJIa, p - ninnepu-

pMerp rpnl{yTHffKa.

7o4 iii: fEoMETPlg SinqHixa o. 9. ra ix' Axa4ervrirHufi piBeHb ii::.:.r:{;ii:ii

Ba rpopny;roro fepona: o - 26 + 29 + 30 = 42 (cn\;

2

s*" = @ = tE.r6:r+n = #1, sr6 = 886 (cr,r,).

ot=Ssa =8 (cm).42

3 a MOL: ML2 = MO2 + O.L2; MO = ,lllt" - Ot';Mo = ^m" -8" = Vi1? -8)O?+8) =,lg.r5= 3.5 = tb (cM).

Bidnoeida: 15 cn.

6.8O. tABC - pinnocropouNlit, P*"" = 36vE cm. S e (ABC). flpore4euo SO L\ABC\,SN l/3, SK I BC, SM IAC. To.4i SO - ni4cranr ri4 rovrcr S no nnon\uau (ABC).SN, S,M i 51{ * nigcrairi ni4 rourcu S 4o cropin IABC. ON, OM i OK - ix npoex-qii. Ba ynnororo SN = SM = SK = 10 cn. ToAi ON = OM = OK src npoexqii piniriaxnoxr4Jrux. O - qeurp Bnrrcanoro s aABC no,na, ON, OK i OM - fioro paAiyctr.

AB=BC=46=Pwc - 36'FB = g =12V3 (cu).

^ AB"J8 L44.8 t;S*. = '*Z*" = 108J8 (cur). A

oK =21n.*:= ,l1ot--Jt

= 6 (crvr).P*"" 36J3B a SOK SK2 = SO2 -f OK2;

so= Jsr, -oK, =Jro, -6, =Jr00-g6 = Jca = g (cu).Bi)noaidu: I cm.

6.81 . ;ABC - pinuocroponuifi, Smc = 27JB cm,.Toqxa S piuHori44a.neira nig nepurran a-43C.AS=BS=CS=10mr.Hexafi SO L (ABC), rosi OA, OB i OC - piniri npo- A

'ercqii pinulrx noxr,rJrux, O - qerrrp KoJra, on[cauorosaelcoao a A-BC.8nafi4euo cropony rpr4nyrnr,rra a @opmy,ru fioro n;rorqi:

"^=#, AB2 = az = fr. =#= r08, ,48 = u1* = 6Jd (crvr).

Cxopncraemocs tpopuynoro ca,rencrrocri cropoHrr rrpaBHJrhHoro rpuKyrHr.rfta i pa4iyca

AB 6vE=-= ._=o (cM).

V3 J3

so -,[N - Ao' = t0' -6, = n{00 -86 = 162 = a 1cu;.Bidnoaida: 8 cyr.

6,42. ABC - nparr-rmHuft rpr4nyrnrx: M

onr{caHoro KoJra @ = RJ5 ,ars sHaxoAXceHHs oA: oA

3 aAOS: ASz =AO'z* SO2;

AB = BC = ,4,C = 8Jd crvr

AMr@BC),AM=5cm. 5cm

flpone4eno AD I BC, rogi AD - Br4 cora, a Bnaqrrrb, Ai rue4iaua IABC, CD = DB = I BC = 4JT cv.

23a reopemoro npo rpr4 nepnen,qrrxyJr.fipu MD L BC,MD - eigcraar rig rovnlr M 4o croponu BC,

:itfilfiili TEOMETP|9 Einsnina O. 9. ra in. Axaseruivnrfi pieenu {!} 7Oa

23 y"i rAg lo M., q. I

3 tACD:r-L

-.' -=-=,-AD = t@ ID, = Jta.F)'-(+S)' = Jta,n aG)(sJs '

aJs)=

,FJi tif,s , r2 (cM).

Morxsa taxo:rc suafirvt.A'D inanme.t;

-C = 60', AD = AC sin 60' aJs Y = 12 (cu).2

3 tMDA: MD2 = AM2 + AD2; MD = E +lr' =

BilnoaiOu: L3 cv.

6.83. ABCD trpsMoxyrgr'rn, AD = BC = 12 cwr,

AB--CD=9cw.

,lrb + 144: .,[ 69 = 18 (cu).

BM L (ABC). MD = 39 ctt. MD - noxlrJra'

BD -1i npoen4ia uan;totqwuY (ABC).3 "ABD: BD2 = AB'? AD"'BDz = 122 + 92 - 144 + 81 = 225.3 a MDB: MD2 = BM2 i BD2i

MB = "tfio" an" = ^tEs" ,% = .,T5rl - rr5 -= Jr296 = 36 (cm).

Bidnoeidu: 36 cwr.

12 cu

6,84. flpaui aib - unuo6ir+rni. aca, b ea,b La- K e b.

Ilpone4euo n n,lorquui cl OM I a. "logi KM I a aa reopeuororrpo rpn rreprelr,quxyrrspu. KM - ei4cranr ni4 rovrcu I( go

npauoi a.3 aOKM: KM2 = OKz * OM2;

oK - ^[KM' - oM' = JB" - 5' = ^fi 6, - % ='1L44 = t2 (rltt\.Ridnocida: 12 ctt.

6,85. flprui m i n unno6ixtni; n c a, m Lo', A € m.

Hexafi O - roqxa neperrrny npamoi m e nnorqlrnoro tx. To4iAO - ni4crans ei4 rovnrz -4 4o n.norquurz o. OA = 6 clt.Ilpone4emo OK L n, OK - ei4craur mirx mumo6ixnrauutrpsMuMrt m i n (ix cni.nsszfi nepneH4rzxylcp).3a reopemoro npo rpu neptreHArrnyr.fipu AI{ L n, A-K BiA-

craHr eig ro.{xz -4 4o lpamoi n, AK = 1O cxr.

3 yAKO: AI{2 = AO2 r OK2;

oK =,tqx' - do" = v[0' -d - .,[00 -36 =.,62 = s 1cu).Bidnoaida: 8 cn.

g 6.3. OproroHarbHe npoe(qitoBaHHe

Opmoaouanana npoettutin - onpeuufi BfltraAofi napa,le.nsuoi npoexqii, noJru Ea-

npau npoertqiroBaHH,tr trepfieHAuxy napuuit Ao nroilIuHtr npoeHqii'Teopeua. Illorqa oproroualrHoi npoexqii ,qoninsuoro MHOTOHyTHIIICa Ha ITJIOIqIIHy

4Opinnroe 4o6yrxy n,rorqi cauoro MHoronyTHuKa Ha ROCIIHyC Hyra uirx trJIoIqIIHOIO

@irypn i n.nourunoo nPoexuii'S'=S.costp.

6.86. A. 6.89. r. 6.91. B, tr. 6.93. B. 6.95. B.

6.87. B. 6.90. F. 6.92. A. 6.94. .U.

S^*" - S^*. cos cr; s-r" = &le

A) swr = #h = Tr, = + = r2Js (crur,):

E) s,,,* - t:," -18;2,r2y[2 (cvzr:

cos4D" \1218

= ra:1 = 36 (cu2).cos 6Oo 2

84cos0 = - = _:

10 5

6.98.

6.97. A.

S.or" = S"o"" cos o;

"oro=ffi

B) cosa = I = 1,' 15 3

93R) cosn=l=:.'21 7

_61ll cos0= = :'122'' 1t t

lll cosa = =183

B) Swr =

A)

fifi,ffi

6.1OO. .41{ L (ABC), AD I DC, rouy KD -L DC (aa TTII).

S.-. = S"no, . cos 45'; {-- = "flTr".

=V=*f =B2li @nt).

6.101. S€'* = Sor"r'cos60o. S*"o- AB . CK = 15. 12 - 180 (cu'?).

1S*,,, =180 t=90 (cv'?;.

Bidnoeida: B.

6,102. aABC i IA.BD - pirnocroponni.

AB ' AD - BC l',6 cm.flpoee4euo CK L A-8, T oai DK L AB ea reopeMoronpo rprr nepflneH,4rrxyJT,flpu. ICKD = 6O" Hyr mi:HnJrorIZHaMu rprHyruzxin -4-BC i ABD.PosrrsneMo rBCK. KC - Bucora, a BHaqurb,i ue4iaua tABC; IKBC = 6O'.

-f3KC = BC.sin60'=+Ja *= 6 (cu).

,ABC = aABD aau*o"on1, uo = KC = 6 cvr.3 tCKD aa reopeMoro xocuuycir: CD' = CK'

CD2=62+6"-2 6 6 1=36+36-36=362

BiOnoeiOa: 6 cm.

iiiiiiill::li TEOMETPI9 EincxiHa O. 9. ra ix. Axa4euivxnfr pieenu ii;i: 7O7

23*

("

v "r An. nc

"uo" - ,!i2

Bi}noai0u: B.

+ DK2 - 2CK DK.cos 60o;

CD=6cw(Cr>0).

l

A B B4 5 'I

(\

7OO itli reOnnefplg Einqxixa O. 9' ra in. Axa,qevrivHnfr piBeHb:iii:*i:::i:i:il

6.103. aIB,Ae cx,,Be B.fkorqranu o i B neperr,rnarorBc'a rlo npsrrifi c.

AC L c, BD L c. AC = 3 ct*r, BD = 4 cll, AB = LB u.3a sractr4sicrrc nepnengtxy;raptocrirrron{rru (55.4) AC L BC.3 aABC: AB2 = ACz 't BC2;

BC2 -- AB2 - AC2 = 13' - 3' = 169 - 9 = 160.3 r BCD: BCz = CD2 * BDz;

oA

" F-\DC

B .B

f = ao <.*'1.

6.108. A e cr, B e a, C e AB, AC : BC = 3 : 5.Ilpouegeuo CC I ll BBr Togi AB, - npoexqiaei4piexa.43 Ha trJrorqury ct". Cre ABrOcxi.rrxia CCr ll BBr ro aACC. o al3B,.AC : AB = AC, : ABri 3 : (3 + 5) = AC, : 48:'

AC. = 3' 48

= 18 (cut.'8

Bi}noeida: 18 cpr.

6. 1 09, Poarr nHeMo AJts cpylnocri oproroHa,rb-Hy npoexqiro 4aroi rpaneqii ABCD ua nao-rr$4rry o, gxa micrurr ocnorylS i napa;re.nr-Ha ocnosi CD.Ilporegeno DD, L a i CC\ L a. ABCTD, -oproroHaJrbua npoexqia rpane4ii ABCD uanrorllrHy d.Ocni.rrxr,r DC ll a, ro DC ll DtCr DIDCC. -rrpffMonyrnrrx, Dpl= DC = 3 c1a.Ilpone4errno DE L AB -- Br{cory rpaneqiiABCD. DP LAB o,a reopeMoro rpo rpu nep-rreHArarcyr'rffpfi. Togi IDED. - ryr mixc n,no-ulrruoro rpafferlii i nrorqznorc lpoexrlii.Ana.noriqno CF LAB, Cl IAB.

cD = ^[BC' - BD' = Jtoo - 16 = ,1144 = 12 (civr).

Bidnoai1a: 12 cv.

6.104. Hexaft ABCD - 4anui napa;renorpaM, aAlBlCrDr - ftoro oproronaJrbua rlpo'

exqis.ToAi S4e,c,D, = S-",'cos3o'.

t;s*", = 6.8.sin60" = u, + = 24JB 'rcvt'). saa",r, = 24.6

Bidnoai1u:36 cvt2.

6.1 O5, Hexaft ABCD - gaua rpaneqia , A\BrCrDl - ii oproroua,'ii'na npoerqis'

sr,u,.., = ss(,, cose. "o",1

= !rre' .* acD

a+b - 7+9S*ro=if h=;.5=40 (crvr'?). cosg=

Bi4noeida:60".

6.106. frcrilo 4eorpauni xytu npz ocnoeinipaui4r.r pieni rvix co6oro, ro n ocrioni .ne-

IcI{Tb MHoro}cyrur,rx, y .arcufi Mo}rcHa Bn}rcaru

KoJro, a ocnoBoro urcoru nipaui4h e qeHTp

qr,oro KoJra. ,{onegelro qe.

Hexait SO - arrcora niparui4ra SABCD,SM IAB, SN I BC, SP LCD, SK LAD.ToaiOM LAB,ON LBC,OP LCD,OK LAD3a TeopeMoro npo TpIr fiepneHA]',rrcyJrspu.

ISMO, ISNO, ISPO i ISKO - ABorPaHHiKyrll nptr ocnoei. 3a yMoBoIo nonu pinni.Tod aSMO = aSNO = aSPO = a SKO

<P = 60".

Ba r{areroM (SO - cni;rrnxfi) i uporu.ne*car{M roc'lrpr'rM nyroM.

Snaqrrr, oM = oN = oP = oK, ro6ro 'ro.rxa o pinnoaig4a;rena ni4 crnpirr tlorupu-

xyrwxa ABCD. O - rlenrp KoJra, Btrr{cauoro s ABCD.

6,1O7. Ba yrroBoro (/3S) L (ABC\. SO - sncoranipaui4u. SO L (ABC\, rorray SO c (ABS).flpone4ervro SM LAC i SK L BC, roaioM LACi OK I BC aa reodeMoro npo rpr{ nepnenAil-KyrI.fipu.ISMO = ISKO - pinni 4aorparHi ny'tu, y"reo-

peui rpannur 3 ocnoBoro. ISMO = a SKO sa

KareroM (SO - cni.xlnr'rft) i nporunerxuramrocrpr{M xytort. Bnigclr OM = OK,tMOC = aKOC sa KareroM i rinoreuysoro.To,qi OC - 6icetrpuca rcYra C..

Taxunr vuuom, ocgoBa Bt corrz O - to'rxaHoro r43.

leperrruy 6icexrpucu xyra C ai cropo-

3a e,racrilsicrro pinHo6ivHoi rpaneqii AE = BF = *;o' =T =B (cm). To4i

rADrE = dBC.F aa ABoMa KareraMu (DrE - CrF ax ni4crani minc napalemuramlrnpsuzuia ,43 i Clrl). 3 qboro Bz rrrraBae, ryo AD, = BC r, ABCTD, - pinuo6iuna rpa-neqin. Ocriarxr,r Ba yMoBoro e sei MoxHa Brtucarvt rcoJro, ToAB + DtCl = ADt 1- BCr 3 + I = 2ADr,,4), = 6 cm.

S rADrE: ADI - A5z * Or[zt

D,E = ,[AD: * AE' = J6' -s'= JB6_o = Jn =8.,6 (cu).

B a DED,: .o". onn, = lE 3"8 "6 rr""o. Jt .DE= i=:. IDED,=t

4

20 1

40 2'

6.110. 3a yMoBoroS ll cr. AB nCD = M,CD n o =N. ICMB = 4b".rlepea npaui AB i CD, ulo neperrruarorr,cfl, [poxoArrrb rrj'rorquna (ABC\, ax,a ne-perrrHae nrrorqr{Hy cr no npnuift PK, PK ll AB. IMNK = 45" strt ni4noeignnfi xyryCMB lrpw napaJrerrbHr4x npsMrzx PK i AB i ci,a:r.ifr CD.flporegeuo MO L cr. To4i NO - npoercrlia nparroi CD Ha nJrorlzny cx,.

IMNO = 30' - xyr uirrc npauoro CD i n;roulunoro o,Ilpone4euo MF L PK, OF L PK sa reopeMoro npo rpt4 neprrenAzlcyJrffpu. To4i xyrMFO -;risiftnzji rcyr ABorpaHHoro Kyra, yrBopeuoro firou]rrHaMrr (ABC) i a,flosHaqfiMo MF = a. Ocni.nuru s d MNF IMNF = 45", ZMFN = gO.,

ro i INMF = 45", ro6ro a MNF - pirHo6e4penzfia ocgoBoro NM: NF = MF = a.

NM = ,[MF\ Nr' = ,17 + o' = oJz.

SaNMo: Mo=Luu=ofr2 2

sH HaTeT,

rrlo Jre)firrrb HaBnporrz Hyra 30o.

3 t MOF: sinlMFO = MO - a'r2 . - -'',tE'MF= 2'o= 2'

ZMFO = 45". II{o fi rpe6a 6y,ro 4oeecrz.

7oa iii: TEOMETPIS EincxiHa O. 9. ra ix. Ara.qeuivHxfi piBeHb :iilii'.::.fi::r iiili:ltiitil TEOMETPIC EincHixa O. 9. ra ix. Axa4euivnrafi piaexu;ii:fi 7Og

6, 1 1 1 - Posr,T,aHeMo Arfi 3p)rllHocrl oprorouaJrb-

ny npoexqiro Aairoi rpaneqii ABCD sa nno'rrluHy ct, ,srta uicturs ocuony /-B i napa,'re.irr-

na ocnosi CD.flpore4euo DDr L cr i CCr L s- ABCTDT -oproroHaJrbHa npoenqir rpaner{i ABCD saIIJIOIIII4HY d.Ocrcirr,xu DC l1 a, ro DC ll DtCr DtDCCt -trp.fiMonyru[n, DrCr= D( = ]' sNt.

Ilpoae.4evo DE I AB Br4cory tpaneqiiABCD. DP LAB 3a reopeMoro npo rpr4 rep-neHAnHyJlflpr4. 'IoIl IDEDt - Hyr mint n.no-

rrlrHolo rpanerlii i n,ror4r'ruoro npoexuii.Asa.noriqno CF LAB, Cf IA-8.

Ba n.nacrneicrro piaHo6ivHoi rpaneqii AE - BF =

D. t: 3,13 .23aDED.: DE.--t.-: -- =b (cM).

' cos 3U" J3

ABTDC -- t'o

.s --- ='*':- .DE="'-1 .6=36 (cM':).-MtD 2 -- 2

6.112. IABC i aABD pinHo6e,4peni, ,4-B -cri,tr,sa ocHoBa.flpose4ervro CK L AB, roai CK - Brrcora

i ue.qiasa pieso6e4peuoro \ABC, rowty K -cepe4nua A,B.

V tABD KD - megiaua, a auaqurs, i snco-

ra: DK LAB. zCKD = a nYr uinr n.no-

rrtrrHaMv (ABC) L (ABD).Onyctuuo nepneuAurdy!'r,flp CO na n,roqrzny(ABD). "AOB oproroHaJlbHa npoexqintABC sa nrourrauY (,4-BD).

OcHirtxz CO L (ABD), (KCD\ L (ABD\' CO c1

3 IAKC: AK -:AB=24:2=12 (cv).2

1Bi1noai1a: o. = arccos-

30s: - cM,

7

6.113. IABC i rABD - pieuo6e4peni, AB -cnimsa ocHoBa. AIfpone4euo CK I AB, roxi CK - Bhcorai megiaxa pinno6e4penoro IABC, rouy K -cepe4uua AB.y

^ABD KB - me4iaHa, a sHaqrrrb, i nHcora:KD - AB. ,zCKD = 0( - Kyr vix n,rorqznauz(ABC) i (ABD).flpone4euo CO L(ABD), IAOB oproroHarb-na rpoercqis aABC aa ntot4uny ABD.Ocxi.nsrcu CO L (ABD), (KCD) L (ABD\, CO c (KCD'), O

*#=+=B (*r). ro4i

tADrE = dBCrF ea.4BoMa rcareraMvr (DP = C_r-F ax^niAcrani uirx napa'neJrr'Hr{Mr4

npai.mu AB i C1r1). 3 rlT'oro Brtrrrr4Bae, t4o ADt = BC r ABC rDt - pinno6ivna rpa-

neqis. Ocni.nsnr4 3a yMogoro e sei moxna Blr4 caru KoJro' To

AB + DtCl -- ADt + BCl, 3 + I - 2ADrl,1 = 6 cM'

3 aAEDr:

Mi = Aa" + D,E2i D,E = F4 -;A =.t6' -l'z =-vE6f = "m

= sr6 (cru)'

B ;AKC: 'dx = ! an = 16 cru.2

AC2=AK2 + KC2; KC=,tAC - AK, =^tab, -16" = Jaf .+O-OS.

S |AKD:AD" =Al<'+ KD2: KD= JAD,, _ AK, =lTO, _16" = G6.4 = 12 (cu).3 t KDC:, DC2 = KC2 + KD2 - 2KC . KD . cos a;

63' +lz2 - 682 L44 2 2COSd =

-

= aTCCOS2.63.12 2.63.12 2r', 21

s^,,. = 1 AB. KC = 16.68 = tooS (cMr).2

s-o, s*. . cos c = 1oo8 . :. - 6:2 rc*'\.21 7

Bidnoeidt,: o ,.""o.2, 5=9I? "*'.2l 7

6.1 14. O6fpyrryBaHns Mararouxa - 'aft y N!N, 6.112, 6.113.

B oAKi: ax =Lln=L2 cn.2

AD2 =Af{2 + KD,; KD=^tqn, - ax" =lffi, -Ir, =,lr|.49 = 85 (cu).

3 iAI(C:AC2=AK2+ KC2; KC=^ttrC -dX'=1T5'z -n" =5 (cu.).

3 t KDC Ba reopeMoro HocrzHycie : DC' = KD' + KCz - 2KD . KC . cos a;KD' + KC2 - DCz 352 l- 5' - Bb' 1

2.35.5 742.KD.KC1

cI' = arccos-.t4

KD - 72. 35 = 420 (crr,r').

s-uo = SLrr'cos0 = 42o' L= 3o (cu'z).74

Bi}noai}o: o = arccosl: S - 30 cu'2.74'

6,1 15. IloscseHHfl AuB. 4o Jr&i\! 6.112, 6.113.3 IAKD: AD2 = AK2 * KD2;

ax =!ln= 16 cm.2

KD =,7AD'- AK' - Jro' ta' =

= !@o - 16)(10 - 16) = J4 €6 =

= 2.6 = 12 (crr).

i{li{l:{iii{t rEoMETPle EincHiHa o. e. ra ix.

e KD.

(KCD\,roOeKD. ,*" =**

ACr=Af{2 + KC2; KC=.tAe -AI<" ='ll}'-lr' =1DS =f 1cn).

3 IAI(D: AD2 = AI{2 'f KD2i

KD = "tqn', dx" JrF -rr" : JGr ' rrr(vt +rD = Jr* 4s - 5'7 =35 (cu).

3 aKCD Ba reopeMolo rocunycie: CD2 = KC2 + KDz - 2KC'KD'cos u;

KCr+KD,-CD2 ,2b+35', 35" =!, o=u"""o"l.cosc, = - 2EC . KD 2.5.35 14 *'"""" 14

'

1 Ao rnv2r s =s .cosfl=Go -L =!9 (c*t).S*. - iAB KC = 12'5 = 60 (cv'?). S*or = S*, coou - -- 14 Z ."'

710 iii: rEOMETPIS Einssixa O. 9. ra ir. Axa4ervlivHrf piBeHb :i::::::;i:i:::i:::::Ara4ervlivxvrfr pieenr {:]dl 711

B yAKC: KC2 't AK2 = AC2;

KC = ^[AC' - AK'= {CL 1d = j(65 - 16X65 +16) = J4r €1 = 63 (cu).

3 r KDC Ba reopeMoro nocrnycis: DC'z = KDz + KC2 - 2KD ' KC ' cosa;

KD2 + KCz - DC2 122 + 632 -632 144 2

"o"a= ,.1qp.1{c = z'L2'GB = 2.12'f3=^1

s..^- = -'AK.KD= 16.12=192.2

s*" =s*" coso(= rcz +=X.

2o= arccos 2l

2BiqnosiAb: ct = arccos

21 i

Tecrgnn caMoKoHTPonK)

1. A.

2. [,.

384s=-2t

r.lacrnxa 1

3. B.

4. B.

5. A.

6.oB=R=*=LE=Js (cM).JsB

B aMoB: Mo = ^[MB' - oB' -- J4-3 = 1 (cru)'

Bi1noei1a: A.

10. E.

11 . Arr = "ltz'; ta' -- J144 + 256 = J4o0 = 20 (cu).Bidrtoaido: l,

"o = s!2

2

F "Jo= A-l\t r I

h,li:n,n.,..,-....-' 7' B'

Flfl.$H*iii,li '. "B:l?'xlri!.:'.'.'c:

i{ln\S;iiiil' n. I A-T-tltlllii,:i:;ii'lii t---f-t_l'/;ri$'.,ri.IJ.... ll|'illliir,l;'i:l.ii lel t I

,r;ii..:r:J:::::::::lfl6l

f A--fl_lt--T-t--l

12. BD = ali:f^2

oB, = .IBB: + BO' = ,la' " !- =-_r v__, \ 2

Bi1noai1t': l.13. B.

14. Bacora rplrr{yrnt4fta: h = ^,82' - 6' = JT44 - 36= r/i08'

. AII 3.F 1Srn@ =: =

- =:: tP = 30".,h6J32

Bidnoei0u: B.

15. A.

16. B.

712 i::l: TEOMETPIq EincxiHa O. 9. ra ix. Ara4ervrirHui piBeHb iill:i{i;ii::i

9acrrarla 2

17. Posn'saanua i4enruune 4o Jrl! 6,69.

18,Il;rorqaga"o*ottp"*o*yrnr{RaS=4.6=24(cvr,).flrorqanpoexqiisr=12"*2.s,1211

cosQ = J = Z+= Z; Q = arccost = ti0".

19. MO I a, MA i MB - noxra.ni, OA i OB - ixrpoercqii. MO = 4 cw, ZMAO = 30", ZMBO = 45",IAOB = 150".

BaAoM: lo= Mo =!.3 =126=+,8 r"*r.te30" JB 3

B a BOM: OB = OM = 4 cu (npsuonyrHzft piano-6egpenrafi rpuxyrnzx).3 tAOB 3a reopeMoro rcocnHycin:

AB AO'z + BO2 - 2AO . BO. cos 150'

-

V48 + 16 + 2 413.4. cos30" =

(cnn).

Bidnoai|u: 4J7. "*.

2O. fl.norqznr,r o i B neperzxa"rorr,cs no npamiii c. flpo-ne4emo -4D L a, AB L I, AD i AB - ni4ironi4no ni4-crani ni4 roqnu A 4o nnorquir cr i B.Il.noiqr.ria (/.BD) npoxo4lrrr, qepeo lpav.i AB i AD,rotly (ABD) I a i (ABD) I B, ro4i (ABD) L c.1"DCB + 6O' - xyr uirx n.norqruarru cx i B.dABC = aADC salg"anerorr i rinorenyeoro (-4C - cni;rl-na rinorenyaa, AD = AB = 10 cel).VTABCZC=60.:2=3O'.AC = 2AB = 20 cu (-4-B - xarer nporz xyra 30o).Bidnoaidu: 2O cwr.

21. fllorqunu cl i B neperuuarorr,c,ff flo npruift c. flpo-ne4emo 8.4 I u, BD L P, BAi BD nignoei4no ei4cranini4 rovrcrz B 4o n.rrorqun cr i B.ffurorquna (ABD) npoxo4urb rrepea npaui 243 i BD,rouy (,43D) L a i (ABD) -L B, ro4i (ABD) L c.IACD = 60" - rcyr uirx naorqunamu cr i p.

d BAC = t BDC aa KaaeroM i rinorenyaoro (BC == 16 cm - cni.nsua riuorenyaa, BA = BD sa yMoeoo).

A

ZBCD =! ^co=

30., BD = !ac --8 *r. AB =BD = 8 cv.22Bi0noai}u: 8 cttr.

22. a LB,Ae o, B e B, o. F=c.AA, I c, AA, = 3 cur; BB, L c, BB, = | sNr.

44 =.,,[T "*.AA, e a, AA, L c, rouy,4-4, J' B,orxte, AA, I ArB.3 aABrB:

ArB = JTq * BBi = "41-

16 =.@ = sJ5 (cm).

Ga+s2r6 \=[a-aJV

3a2

2

fi{iii{il:ii TEOMETPI9 EincHixa O. 9. ra in. Axa4euivxr,ri piaexu ii{:;i 719

3 .AA,B Au = rtr4 - 4n', =,tl',; n = t6;n = 16o = o (cru).

Bi4noai}u: 6 cv.

23. fl;rorqnnz o i B nepernrarorbc.a no npnulifi c. A eflpoee4elno AD LaiAB LF.Il;rorqrzua (/3D) npoxo4urr uepea npsui-4-D i-4-8.

Touy (,4-BD) I c.tABC - aADC sa KareroM i rinoreiyaoio (AC -cni.nrsa rinoreHYea, AB: AD = a\.IACB : IACD = 120" : 2 = 60".

Aa = a :sin 60" - a'-2 - 2a'13 .JaB

2a,liRidnoeida:: cM.3

24. (ABC) f 6BC), rABC = r KBC - pinHocropouni.AD L BC, KD I BC - Brtcora (ronn x me4iaHir) aABa i tKBC' /ADK - nisifi-nuii nyr ABorpaHnoro xyta e pe6pou BC, IADK = 9O".

.A,I{ - pe6po ABorparrHoro ryra uixc nrrorrlrrraMu (AKC) i (ABK)'flponegeno BM LAI{, rotii MC LAI{,6o IABK i *ACK - pieHo6egpeai a ocHo-

r'ow AK. IBMC - ryr uil+c nrroilInHaMlr (AKC) i (ABC).

flosHaqumo cropoty piaHocropoHnr,olo rpl'rnyrsuna a'

B ^ACD: co = ! sc -- o., zACD = 60";22

^ti BAD - AC.sin60" - "v" .

2

DK = AD= of

r"*1.

3 IADK: AKz = ADz * DK2i

26. ABCD - r{Ba,qpar ei croponoro 6 cu. F e (ABC).AF-BF=CF=DF=7cM.N - cepeglrna FC, M - cepe4zua-4-B.IlossaqnuoP - cepe4zxy MF i K cepepnay CD.Poar.n.sHemo qortrpuryrHzn MPNK.

pN I BC, pN =l=eC=B cu, Mp. A-F,2

vr =f;dr = 8,5 crvr, NK FD,

*" =;FD - 3,5 cu nn cepe4ui .rriHii

MK )l BC ax ni4pisox, rqo a'e4Hye cepeAuurz rrporr4r're)fiHl4x cropiH xra4para.MK = 6 cttr. MPNK -

pirno6ivna rpaneqia, MN - li 4iarona,rr,..flpoee4evo NT L MK - Bwcora rpaneqii.

B a NrK: rx - )tu rc pN) =t (6 - B) = 1,b (cM).

NK2=TNz+TK2iNr =,[NK' - rK" = !Eb' L's' = !fu.5 l's)(B's - 1,r) =,1,i =.Eo.Y tMNT: MT = MK - 7:K = 6 - 1,5 = 4,5 (cm).

MN' = MT2 + NT'; MN= laJl'F10 = Jn,rb +tO = ^FO,%= 5,5 (cu).BiOnoei1t: 5,5 cm.

27, IABC - trpflMor(yrHrfi pinHo6egpenuin: lC = 9O',

AC = BC = 'E c*. M e (ABC), MB I (ABC), ntB = tli ,

Ocxirrrz MB L {ABC), ro MB L BC; MB L AB.aCBM - npamoxyrHufi pieno6egpeuufi. IMCB - 45'.BC LAC, rovy MC LAC Ba reopeMorc trpo rpr treptreH-

Arnyrspu. To4t /MCB - niniiinuii lcyr ABorpannoro AKyra, y'rBopenoro trrou{ruaMu (AMC) r (ABC).3 roro, n1o MB L(ABC), BlrnrrlrBae, u4o a.4BC - op-ToroHarbHa npoenqia rpurcyruzxa AMC.

a,A€8.

BTATIOBIAHIZX TPIAfi YTIIIIITlB.

(crnr').

YZ cu

D

lZ c*

!g1 =o& .*.24

a2 5a" 1

445

Eo"' ou/ioVto 4

1ct = arccos;.

c

S"*. = S"*. : cos 45o. .-

s*"=|dc ""=;

tE tE-1 (cu,). swc

Bilnoei\u: nli .*'.

t;=r:;= J2

3 IAMC: AC2 = AM' + MC2; MC =.tIC' , AM' =

3a reopemoro Hoczuycin o

BM'+MC'-BC'soso"= 2 EM .M-

aBMCI

25, AtBl - oproroHa,'lbna npoexqia nigpiarca/-B Ha

n.notqrzny u: AA, L a, BBr L a, rofi AA, ll BBr.C - cepe+hsa AB, CC, L a, C, - cepeguua AtBt(ea reopeuoro (Da.neca).

Ba yuoeoro AA r: BBr= 3 : 5, AAr= 3x, BBt - 5x.Ocni;*,nu CC, - cepe4ua,'rinia rpaneqii frBAt'

ro cc. = 4 1 Bq , 24 =3t !1" ; 4x=24;x=6.n2AAt=3 6= 18(cna); BB, = 5 6= 30 (cm).

Bi}noei}a:18 cn, 30 cn.

28. MO L (ABC), MO = 3 cu - ni4cranr ni4 rouxr M go ntot4uuu (ABC).Ocrci.nr,rcra ro.rxa M pinnoni4garena eig cropiH IABC, ro O - 4enrp noJra, Bnuca-Horo n a-4-BC, - Toqxa neperr{Hy ue4ian (nucor).Hexaft AD - ue4iaua i grcora IABC, rogi MD L BC aa reopeMoro trpo rpr.r trep-neHArrnyJr.rrprr. MD - ri4cra:ur ni4 rovru M go cropouz BC.

MD = 2J3 cv.B t MOD: MDz = MO2 + ODzi OD = JMD" _ MO, .

r.-on = lbJs)" -l'z = h2-e = Ja 1cuy.

A;le OD - paAiyc BnrzcaHoro HoJ'ra.

3a $opvyroro aa,rexHocri paaiyca BTrr.{caHoro KoJra

i cropoHn npaBl,rJrr,Horo rpr{}cyrHuxa a 2rJ3, roilBC = 2. rE "6 = g (cu).Bi)noei0u: 6 c*r.

roa2 loaz+ -a'16 16

;r@!@44

CI

c

D

A

b0

6a2-16

714 ::::::: rEOMETPI9 Sincxixa O. 9. ra ix. Ara,qelurirHufi piBeHb liiiii:;iiliiii [itilitilli{ TEOMETPI9 Finsnina O. L ra ia. Axaseuivxun pieenu l{:i Z1S

AB 5

B 1 2 4

{acruxa 3

Ilpaaunonutt. mempae7p - rpraryrHa nipauiga, y axoi nci rpaui - pinuocropon-

ni rpzxyturaxz.

29. SABC - npanz.nrnnft rerpaeAp a pe6pou a.

Ilponegeuo SO L(ABC\. O * tovrca neperr'rny meAian (uucor) aABC' SO - niAcraursi4 rourcz S go rpani r4SC. S

OB - pa4iyc onrrcauoro r(oJra.

3 oSOB: SBz = SO2 l- OB2:

MOAy,nb 7. ySArAIbHEHH9 I C]4CTEMATt4SALllg B]4BqEHO|'O

S 7.1 . Ocnoeni Sirypn reoMerpir'ra ikne posniu4eHHe y npocropa

o"=#'

7.1.A.7.2.t.

7.3.

7.4.8.

7.5.4.

7.6.8.

7.7.8.

7.8.r.

7.9.r.

7.10. B.

7.11. r.

7.12. A.

7.13. A.

7,14. L.

7.15. E.

7.16. f.7.17.8.so = JsB' - oB' =

BiAnosiAb:

30. MO I tr - nepner,qlarcyJr,Ep. flosua'ifiMo fioro gonrxi4ny a'MA i MB - noxu.ni, OA i OB - ix npoenqii.Hexair IMAO = d, roAi ZMBO = 90' - ctc.

B 'AMO: a6 --*L.sln fl,

""16

-

cM.7.18. Hexa* giarona;ri AC i BD vorrapuryrsaxa ABCD

fieperr4rrarorbc,s n rou4i O.r{epea ABi npaui AC i BD, qo fieper]rHarorbcs, Mo]rd-Ha npoBecrr,r e.4r4Hy nJrorrlzuy o (axc, IIr). Tovxrz .A,B, C i D,'relr(arb y r1ift n.norquui,6o A e AC, C e AC,BeBD,DeBD.

7.19.CeAB,DeAB.{epee 4ri roqrcra C i D sa a}cciolroro I, morxna npo-BecTr4 eAHHy rrpgMy,r{epea nparvri,4S i CD, r1o neperr4Harorbcs B roqqi C,aa arcciouoro II, uoncna npoBecTr4 eArrHy nJroulr{Hy.

7.20. Ilpone4eMo qepeo 4aui npavi a i b n;rorquny o, Ileuorrtna-apo6rarz sa arciol{oo IIr. flpanna c, sra rre-perr,rnae 4aHi npaui, Mae 3 nJron{ilHoro c 4ri cni.nsuiro'ur'u M i N (rovru fieperr,ruy B AaHnMn npaur-ruu).3a uac,ri4xou s axciom (S2,2), sxiqo 4ri rouxr,r npa-moi na.nexarr n.norqrni, ro i scg npsMa c Halrelx[TF,

- n.noulr,rni ct.

7.21. Ilprrvri a i b, axi neperr,rnarorhcs n roqqi O,oAHO3HarrHO Br,rSHaqaroTb fiJrorqrrHy cI.

,{ai rouxz A i B npannoi AB .ne*carr y n,ror4rania,6o A e a, B e b, roMy 3a uac.nigxolr a axcioru(52.2) AB c o. Ananori.rao ArB, c cx,.

Ao = Jill' - Mo" =a'.-, -a" =

s1n- cx

E1l - "t"' ", 4 cos o(=\/ fi'il - ""."

'

3 rBMO: BM = ---:- = o ,

sin(90" - 0) cos o(

Bo = JEM' - Mo" =

AO a cos cl a sin o d cos cr cos oc acos' ct. =___, .BO sinc cos0 dsinq'slnct dsln-cx

,(omnoxrzlto qlrceJrr'Hun i sHaNreHHu( Ha o:

Ao o" "o"b

d et cos'd a" a" AM2

Eo= "'"i"m = .in'o

*T = "itf "

t *"'" = Effi'31. ,{Nn. N, 6.78.

32. 3a r:eopeiuorc npo oproroHaJrbHy npoefiqiro S"o = S. ' cos <p'

s3ni,4cra cosq = +.

,5^

3uafi4enno nrrorqy npoercqii ea Soprry.noro repoua: p = 14 * 1^6 * 6

= rs;2

q" = v[(18li4x1r16X1s - 6) -- ^tf8 +2'n = J9'24'r'+3 =

= 3'2. 4h = zals @tr).

24\E J3cosq=::r:= f ; t0=:lO'.

7.22. Saaacni4norvr a arccioru crepeouerpil uepec rpr4 rorr-xn A, B i C, ari He Jrerfiarb na o4nift npnnnifi, rroxnanpoBecTr,r eAfiHy [Jrorqr,rHy g.To.rru A i B tencarr y n.noulrzui c[, roMy Bcs trpffMaAB c a. Ana;rori.rno AC e a,M e AB, K e AC,rouy M e u i ff e o,, romy.MK c crc,

Xe MK, rouyXe cr.

7.23. Ilpnrvri aib, axi neperr,rHarorbca n'rovqi O, o4uo-SHaqHO Bt4SHaqarOTb nJrotqrrHy 0.Ocxi.nrrra roqrcra ,4 i B ira.nexarr np,fiMrrM a i b, roBorrr Jrell{ar}, y n.norquni o, ToMy npa*ra.A.B Jre*cr,rrr,y n,rorqzni cr..To'rira Y ,'rerrcrrrb na npamifi.AB, rorvry Y e cx,.

DCL

A Bc

M

N

O(

716 i::::: rEOMETPIS Sincxixa O. 9. ra ix. Ara.4emirHufi piBeHb ::#:i:i:::iii:ii [r:{l{iiiii{l$ TEOMETPI9 Eincxixa O. 9. ra in. Ara4euivnrafi pieenr fiii; 217

7.26.Hexafr ABCD - 4anrzft npocroponzfi uotrapzxyr-srax. Hexafi Ar Br Cr D, - cepe,4zHn fioro cropin.ToAiArB, cepe4tre,'riuia rpurcyruunaABC,-ArB, AC,CrD, - cepeAHrr il:nia IACD, Cpr J) AC.3a oenaHoro napa:re.nrnocri npaltzx ArB, ll CrDr. Alralo-riuuo 4ono4umo, u1o ArD, ll BrCr. A

Ornre, voruptnyrrvr\ AlB rC tD1 Jrex(rzrr, e ogHifi n,rouluuii iioro rporrz,rerxui croponra napanent'Yi. ABCD trapa-

JreJrorpaM.

lly lll)llMtrx ll I I'x llJllrll(rrr!l It llltnMv I t, rlrrlr!rr1'rhrrY/1. Artr',tr.ltr,:r'trr,ttcv ('M(,;l(llrr lllx,ll(( tll Il.Jll.l\ll (r.llltV

npfiMy, napaJlenrHy b. A 3l{atlurb, ltl)tlMtl ('r6itrttlrl,-

cf 3 rrpgMoro cr, a orlrce, re?tcffTb y uouluui u.Taxult uuuolt, 6y4s-arca np,gMa c, nap€rJrerbHa b, axa neperuHae npfiMy a' Jlo)lcn'rl'

y n,rorqrni c.

7.25. [us. aa.qaqy J\! 3.38.

7.32. All(:l) ulll,l./tr'.lot,l,rM, () ,11y,11q11 rr.l,rrr,nlrv iiorrr 2lirrrrOr:rrirn,nn MA - M(:, ,r,o r AMC pillo6c4perrnii.lltr u.nacrueicrro trapalerorpaMa AO = OC, rcgi MOnc4iana, a BHaqrrrr,, i encora IAMC. MO LAC.Anaaoriuno, MO * n:,z,cora pieuo6e4penoro rpr{rcyrnr,rftaBMD (BM = DM, BO = OD\. MO L BD.Ocxi,rsxr.r npxwa MO nepnenAzxyJ'rflpna 4nou npairnum BDi AC, rqo nepeurnarorbcs, rrJrorr{zHn (ABC), ro Ba osnaxororrepneHAuxyJ'rflpnocri npnuoi i n.norqnuu MO L (ABC).

7.33. [iarona,ri xnagpara ABCD neperranarorrcs n rourli O:AC I BD,AM I (ABC), AO L BD, rogi OM L BD sa reopeMoro nporpu nepneHAr,rKyrsp[.OcHirrrcrr npama BD neptreHAr4nyrspHa ABoM rrpflMxM,rqo fleperrrHarorrcs, rrrrorqr{Hr (AMO\, ro BD L (AMO) sa .

oJHaKoro fleptreHAhKyJrspnocri npamoi i naorursn.

7-34.Hexaft piaui n.rrorqzuz o i p nepneHgzlcy.napni go npauoi a. Ocximxu uepeaToqr{y npoxoArrrb eAlHa trJrotquHa, neptreuArrnyJrspna 4anifi npamifi, trJrorqrrH[ di B neperuuaorr fipffMy a n pianrzx roqrax A i B. flprznycruMo, fiIo firoquxlr o i Buarorr, cni,rrHy routcy M. flpoaegerro rrJ'roquHy y qepea rrp.rrMy a i.,rouxy M . Ocxi,rr-n[ trpsMa a nepleHA[KyrspHa nror{uAaM o i F, To y leperr,rnae flJaorqtrHrr d i Fno trpsMilM MAi MB, nepneHnrrcyrapHnm npslriri a. TarcNm qrruoM, y nrorqzni yr{epea roqry M npone4eni 4ei piaui npami, nepneu4lrryaapHi o4uiii npauifi a, rqoHEMOXJ hBO.

Orrxe, nrorqznrt c[ i B He uarorr cni.ntnux roqon, ro6ro napa.ne.irrni.

7.35. ABCD - rerpae4p. M cepe4uuaBC,AB=AC,DB=DC.dABC - piniro6e4peHufi, AM - fioro ue4iana, ro1i AM -Br4cora, AM I BC.r CDB rasotx pinuo6e4pennii sa yMoBoro i DM - fioro me4ia-na, a ehavr.rrr,, i eficora. DM L BC.Taxulr unHonr, nparva BC treptreuAr.rlcy,'rflpHa ABoM rrprMr4MAM i DM rrnouquwu(ADM). Ba ogHarroro nepneu4ur<ynaprocrinpauoi i nrrorqr?rHn BM L (ADM).

7.36. {nn. Ni 4.15.

7.37. AN L(ABC). fl.norquna (/DN) npoxo4nrb qepeo np,n-ny AN. Toai (-4rN) L (ABC).FD L (ABC). If.norquna (FDC) npoxogurb qepea rpfiMyFD. Toai (FDC) t (ABC\.Hexaft BK - Bucora ilapaierorpaMa.BK L AB, EB I BK Ba orHaqeHH,sM hepneHAnKy,r.spHocrirpffMol i trrorrluHu. To,qi BK L (ABE).BK L DC, FD L BK sa oBHarreHHsM nepneHArzrcyJrspuocri npar,roi i naorquun. TogiBK r(FDC).Il;ronlnuu (ABE) i (DFC) nepneH4zxyanpHi npauift BK, roti (ABE) I @FC\.

7.38, [ue. N! 7.37.

7.39. !rs. nnacrunicrr a $4.2.

i;;iittlii{ti TEOMETPIC EincHixa O. L ra ix. Axa4ervrivHufi piBeHb i{d 719

7.27. SABC - rerpae.4p."lo,txn M, N, P i K - cepeAllHu pe6ep AS, BS, BC i ACnignoni4Ho. I{i rourcr - Bepilu Hu flpocropoBoro qorrrptr-

KyrHuKa. V sanaqi 7.26 6ytto Ac:BeAeHo' rqo MNPK - na-pareJrorpaM, MP i NK - fioro giarouali, roMy BoHu rre-

perilHarorbcff i ro.rxoro ueperuHy 4irarr,c,s nasni,a. AHaro-riqHo AoBoAr4Tr,ca, u1o ri4pisxz, rqo e'e4uyrorb cepe,4[Hupe6ep,4-B i SC taAS i BC, neperruarorr'cff B rii rrce rouqi.

7.28. !ns, MoAyrrb 3, Tecr 4ra camonepenipnrz Ne 31.

7.29. Muuo6ixni npaui a i b,rexcarr y piaHux nJrorqrrHax d i F.Hexaiic( d,bcB.Ilpone4emo upflMy c ll b, srca treperr4Hae trpsMy a e rovqi C.

ToAi np,sui a i c o,quooHaqHo BlrsHaqarorb nJ'roquHy.Biasmeuo ua npauifi b 4oni.nury rouHy B i npone4euo-ue-per Hei npauy a, ll a.3a osnaxorc napane,rr,uocri rrJrorqr.rn o ll P.9epes rouny C wroxua npoBecr[ eArzHy nJrorrluHy, irapa-reluy p.

Ilpone4euo inury npsmy d, rqo6 nona 6y.na napale,'rrnanpflMy o. Ilparui c i d ,rerxars n o4niii n,rouqzHi. Ilpama a

roqrtrr, a orr(e, Jlexcrrrb r o4nifi n.norquHi.

7,3O, BepulznuAi B napanenotpauaABCD, a orxce, i npa-ua -4-B ;rexcuri y nrorqvni cr.

BepurrHa C ne .nexurr y uiii.CD I AB Ba oaHaqeHHEM rraparerorpaMa."foli. CD ll ,43 sa oauariorc trapaJreJrr'Eocri npcmo'i i n.no-

lrlrrHt4.

7.31. flplrnycruMo, rqo npaui a i AB ne napa.nemni. To4i moxc.nrani ABa Buna,qftn.

1) Ilpsui aiAB - uuuo6imsi. Togi sa osnaxoro uuuo6i*cnnx trpflMux ($3.1) lpa-Ma a neperrruae nJron-lrrny, s sxifi rexurs npaua.A3, rqo cynepevzrl yuoni.

718 ::i::: rEOMETPI9 EinqxiHa O. 9. ra ix. Axa4enisHrt piBeHb iii:i;rLiii;it

npsuifi b'i neperr.rnanaMae B HrrMn ABi crrirbHi

7.41.

7.42.

7.43.

7.44.

$7.2. flepneHAlrKynep i noxrra Ao nnoulllHll, ei4craxi Ta Kyru

y npocropi7.40. t.

7.57. MO L a, MA, i MB - noxlr.ni, AO i BO * ix npoexqii.MO = 1,2 cu - ei4craur rig rovru M go urorqrzuu cx,.

AM = 37 clr, BM = 13 cu.3 "AOM:Ao = 1m'- Mo' -- li?'z -Lr'z =,fu? -rrrw -14 =

= ,lr{As = 5.7 = Bb (crvr).

B tBoM: Bo = ^[BM" -fro' = JB'z -tr'z = b (cu).

AO:BO=35:5=7:L.Bidnoai}u: 7 : 7.

7.58. ABCD - porvr6, AC a BD = O.AC = 16 crw, .BD = 12 cru.

MO I(ABC). Ba yuonoro rourca M pinnoeiAAaiena niA cropin polr6a, rogi O - qenrpBIII4CAHOTO IIOJTA.

F, N, P,1( - r:oqru Aoruxy rio,aa ,qo cropiu pou6a, ON L AB, OF L BC, OP L CD,OK LAD.To,qi MN L AB, MF L BC, MP L CD, MK L AD sa reopeMoro npo rprr flepreHArr-KyJrspH,MN=MF=MP=MK=8c}.tr.Poar.ngneuo ICOD. OC I OD (4iarouaai pon6anepnen4rzny,rapni).

oc=L,+c=8 cu: oD=LBD=6 cu.22cD = Joc\ oD'=

"61 6' - u/ioo = ro "*.

11s^"^"=:oc-oD=:.8.6=24 (cm2). A22

t =

2So,oo -2 24 =4.g k*\.So.oo = : DC .OP: OP = O" lO3 a MAP MPz = MO2 * OP2;

Mo =.rmp' - op" = SL 4's' = v6- 4-8X8 -4-8) = Js,%trg= 6,4 (cnn).

Bi|noaida: 6,4 c*r.

7.59. ABCD - poiu6, AC a BD -- O, AC = 16 cu, BD = 12 cyr.MO f (ABC).3a yuosoro rouxa M pinaorig4a.neua ni4 cropiH pou6a, ro4i O * ilenrpBnucaHoro Kora.F , N , P, K - ro.rxlr ,4orlrxy rcoJra Ao cropin pom6a,

ON LAB, OF I BC, OP LCD, OK LAD.Toai MN LAB, MF L BC, MP I CD, MK LAD3a TeopeMoro npo Tpr{ treptreHAurcyrspE.MN=MF=MP=MK;MO=2cll.PoarrsHerro aCOD. OC LOD (piaronasti pona6a

nepnen4unyrnpni).

oc=7.ac=8 cu: on=l-nn=6 cu.22cD = ^{oc' * oD" = Js' + 6' = .r[oo = to c*. A

A.

B.

t.

7.45. A.

7.46. D.

7.47.8.

7.48. B.

7.49, Bilnoaila: 3.6 cm.

7.50. B.

7.51. A.

flrcrqo roura 3naxoAr{Tr,cg nooa rrJrorr{[Horo MHoror{yruuxa i piaxori44aaena ni4ycix ftoro cropin,. ro ocuoBa rreprrerAtrr.yJrfrpa, npoBeAeHoro a 4iei rovxz Ha [Jro-rrIuEY, - qeHTp KoJra, BrrncaHoro B MHororcyTHr4n.

7.52. MO L (ABC), rowry MO LAC, MO L BD.ZMAO=tMBO=6O".DB = ABJ, = 8.,0 crvr; OB = 4r[i cm.

Ma= o!,, =j3= a.,E (cru).cos 60o I

Bi0noaidv: [,.

7.53, {un. J\! 6.69.

7,54, AC = 5 cir,r (err.rnercrxuft rpuryrnux).

3 .AMC: twC = AC.-.

= 10 cv.cos ti0"

3 tMBC:MB = ,[MC, - BC' - JIn - a'?= Jioo - 16 =

= 1EI = zvET (cu).

11^DS...-, = 1 MB. BC = i +.zrln= +rDi (cm').22Bi}noeilr': !,.

7.55. (|x)'z - 4'z = (6x)2 - (aJF)' ; 25x2 - 16 = 36r'? - 27; ILx' =

(5 . 112 - t6 = 25 - 16 = 9 (cu).

AO=\6=B cu.Bi1noai0a: E.

7.56, MO L P. MA i MB - noxra.ni, OA i OB - ix npoexqii'MA = 73 cv', MB = 37 clt.Ocxi.nrnr.r MA < MB, ryo OA < OB,OA=xcviOB=7xcv'3 yAMO MO2 = \32 - x2 = 169 - x2.

3 r BMo Moz = i72 - (7 x)2 = 1369 = 49x2.7369 - 49x2 = 169 - x2i 48t2 = l2OOix2=25ir=5(r>0).Mo = tl6s -2b = ,1144 = 12 (cm).

Bi4noei1t: 12 cm.

!1;x=l(r>0).

AE 1 4 5

B 2

72O ii::: |EOMETPIS Sinsnina O. 9. ra ix. Ara.4ewriqHui piBeHb iiliiiiiiiii i:iaiiii:;i{f::i FEOMETPI9 Fincxixa O. 9. ra iu. AxaAeMi,{xr,ril pieenu tt!; 721

"^".^ = 1o" -oo = 1.8.6 = 24 (cu,).22

s^"^" = 1 DC .op; op = 2s-"-:o ='-4 = 4,8 (cM).

2DCIO3 tMOP: MP'= MO2 i OP2;

vp = ,[rwo' , ap' = Jr\ 4"8' = $ * zs,o+ = Jr? fi4 = 5,2 (crrn).

Bi1noei)u: 5,2 cv.

7.60. ABCD - pinno6ivua rpaneqia (AD ll BC, AB = CD).P*"r,= 48 ev, 1A= lD = 6O".MO I o. - ei4cranr ri4 ro.rxu M 4o nnorquuu o,. Toqna M pisuosi4Aarena ri4 ycixcropiH rpaneqii. Ilpoeegeuo MF IAB, MK IBC, MP LCD, ML LAD.- ri4craniui4 rouxu M Ao cropin rpaneuii.OFL AB, OK I BC, OP L CD, OC L AD sa reopeMoro npo rpt4 nepreuAarcyrrspu.OF = OK = OP = OL nrc npoexuii pingnx noxranx.Tarrau vranom, O - qenrp HoJ'Ia, BlucaHoro n rpare4iro, OF = OK = OP = OL -fioro pa4iyclr.

V tBOC IBOC,= I8O" - 2IOBC = 180. - 2 . 80" =3a reopeuoro xocr,ruycis:BCz = BO2 + OCz - 2BO . OC. cos 120..flognaqlruo BO = OC = x cw.lQz = yz t, ,z 2 . x . x. cos(180" 60.);IOO - 2x2 - 2x2 cos 60";

Ioo = 2x2 + z*' .!; Bx, = tao; x =10

F (r > o).

cvr. 3 IAOB AB2 = AO2 -t OB2;

Ao -- 'IAB" - oB' =

Bo=oc=#

EooVa =10 (cnr).

1o!6

I{exrp nora, orrJ4canoro HaBKoJro pinno6ivuoi rpaneqii, ire}fiI{Tb na nepetzui ocicrauerpii rpareqii B cepeAr,rHrruM nepnen,qraKyJrEpoM, npoBeAerruu go 6ivnoi cro-poHr4.

7,61. Tpaneqis ABCD Btrzcana B (oJro. Ocni;ruclr 4yru,gaxnoqesi uix naparelsnuMlr xopAaMrr, pinni, ro pin-si i xopAra, axi ix crnryrorr,.Ocriarxr.r BC ll AD, ro AB = CD. Tpaneais ABCDpieHo6ivna.Hexafi O - qeHTp onacaHoro xora. 3a yMoBoro ocrroBaBC crarye 4yry n 60", ro6ro 4enrpa.nsnwfi xyr BOC

4opisuroc 60". OB = OC stu paaiycn orrucaHoro KoJa. A

Pinno6egperrzfi rprrnyrrir.rx BOC a rcyrou.60" rpu nepurzni e pinnocroponuiu:BO=OC=BC=16cu.Ilponeseuo MO L (ABC), MO = 12 cv.MC - r.i4cranr eig rovxn M go eeprrunr.r rpaneqii. 3 aMOC:

MC2 = MO2 + OCr; MC = "lTrz -*" = tT44+r56 = J400 = 20 (crvr).

Bidnoaida: 2O c*r.

7,62. AO L a, AB i -4C - pinni noxrari, BO i OC - ix npoexqii Ha nrorqr{Hy cx.

Ocxittxu AB = AC, ro BO = OC. 3a yuororc BC = lO cv, ZABC = 6O", lO BC = 30".V pisHo6e,qpesouy a, -BC rcyrrr npr4 ocnoni 4opienrororr 60', rouy aABC - pie-HocropoHnift. AB = AC = BC = 10 ctu.

722 iitl rEoMETPlg Einqxixa o. 9. ra ix. Ara4evrivHr,ri piBeHb i:;l!ii;i;i,!iiii

flpone4enro BH LAD - Bucory rpaneqii. BH ll KL, BH = KL.Ocninsxu 3a yMoBoro r rpaneqirc Mo,fiHa Btrrrcaru KoJo, To

BC + AD = AB + CD =!r*"o = 24 cv. AB = 24: 2 = 12 (cu).

B .ABH: BH = AB.sin 60. ="f= 6Jd (crvr).

oL =: KL =! ", = 6f

= eJe t"*1.222t-

xav =,!ont' u oL' = f,l' t (lG)" ='t0. n = 1136 = 6 1cm;.

Bilnoei}a: 6 cm.

Bionoeiou: to-6 "*.o

7.63. B ro.mz-A Ao nJ'roilIr,rur,r o npoBeAeHo nepleH,4anyJr.flpAO i noxzli AB i AC, OB i OC - npoexqir quxrroxnJrux rignosigno..48 = 13 cm, OB = 5 cw, BC = 19 cu, ZBOC : I2O".3 aAOB AB2 = AOz r OB2:

Ao = ,IEB, - oB, = u{FJ' = v!69 ,t =

= JM4 = 12 (cu).3a reopeuoro xocrznyciri s a BOC:BC2 = BO2 + OC2 - 2BO .OC. cos 120.;192 = 52 t x2.t 2. 5. r. cos 60"; 861 : 2b r f * bx:rc2 + 5x -336 = 0; r, = 16, xz= -21 - rre aa,4oBoJ,rbHfie yMoBy zagati. OC= L6 clr,3 ^Aoc Ac, = Ao2 r oC2; AC = "[r-1d = Jn4 + 2s6= J400 = 20 (cu).BiOnoai1'a: 20 cm.

7.6.4.V yABC AB = 15 cru, AC = lB cu, CB = 14 cu. ^tAM I (ABC), AM - 76 cw. .-'

flponegeruo AK I BC, roni MK I BC aa reopeMoronpo Tprl neprreHAr,rriyJrspE.3naft4erro S"*" B& OopMyroro lepona: A

15 + 14 + 13P= '-=21's*. = v'5ttzt - trxzr L<xzr - ts ) = ,i21.6 17 .B =

= rt.l.s.2.,.2.4:7.g.4= 84 (cru).

Sinuoro6oxy S*" =it" AK, saisxu AK =25*"" - 2'84= 12 (cru).

3 aAMR MK = JAM" + AK, = ,!tg * t* = zOBiOnoei1a: 2O cn.

7.65. y yABC AB - 17 cryr, BC - 8 cu, AC = lb cw.Hannporu rrenuoi cropoHr4 BC lerrrzrr uennlrftxyr A. flpone4emo MA L (A_BC). MA = 2O cw.172 = 289, 152 + 82 = 225 + G4 = 2a9.Maewo: AB2 = AC2 + BC2.I-{e oaaavae, u1o a ABC -

Anpauonytnzri. AC L BC, rogi MC t BC aa reope_MOrO trpO Tpu uepfieHAnr(yr.flpu.

BC 14

(cn).

M

2O cyt

100 - 100

=

12 (cu).

i::ii:t{i*ii FEOMETP|C Einsrixa O. 9. ra ix. Ara4eruivxui piaexr i:ji: Z2g

B .AMC MC2=AM2 +AC2; MC=JrO" -15" =J40o+2%=16t5= 25 (crvr).

Bidnoei}a: 25 cw.

7.66. dABC - np,nMoryarvft (ZC = 9O"), AC = 18 cru, BC = 32 cru'

O - cepeguua rinorenyau, MO L (ABC), MO = L2 cw'

IlposeAelro OK LAC, OL L Bc,roai MK LAC i ML L BC

Ba reopeMolo fipo rprr rrepflexAr{r(yJr.fipu' MK i ML -ni4crani rig ro.tcn M 4o rcarerir rprrriyrrrrrfta.Ocnimnu ABi flpsMi, trepneHAllltyJrspni rperifi, napa- A

.ne.rrni uix co6oro, ro OK ll BC, OL ll AC.To4i aa reo-perroro (Dareca OK i OL - cepeArri trittli rABC.

ox=Lnc=?2=rc (cm); or =\o"=4=s tc*t.2-"2'-"22MK2 = MO2 1- KO'; MK = Jle .n' = [44+256 = J400 = 20 (cru).

MLz = Moz + Loz: ML = ^l{r\r" = Jt44-el = tTrs = 15 (cu).

Ocrci.n*rz as = 2rJ3, 4e r = SO, naeuo: -4C = 2. SO. JZ = Z.OJE = fZJS (crvr).

BiTnoei\a: 1216 crvr,

7.7O. !un. N! 32 s Tecrie Arrs caMonepeBipxra, Mo4y.rr 6.

7.71. S.= S'o

^ cos60oSnai4euo nJrorqy npoexrlii aa $opuy;roro lepoHa.

^_!5*4+1Q_o,.'2s,".= V?r.fzl -rsx21-1aX21- 1b) =,1rr.8., $ = js.?. +2:r.T2 =

7.67.V yABC lC = 90', AC = 15 crr, BC = 8 cru.

AD L (ABC), AD = 12 cw.B yABC AB2 =AC2 + BC2;

Au = .,85? 8' = Jns + a+ = Jrse = 17 (crvr).

3 aABD BD2 =AD2 * AB'zi

D

L2 cv

A

=3.7.4=84(cm).s. = 84

= 1G8 (cu'z)." 0,5

7 .72. Pinso6etpeni rpranyrnrasu ABD i ACD uarorr cnilrnyocrony -4-D, AD = L6 cm, AB + 17 cm, ZACD = 9O'.flpone4euo qepes rrpsMy BC n;rorquny (BCE\ L AD.Toai BE I AD, CE L AD, ZBEC = 60' - ryr mirxurorqrzuaMrz (ABD) i (ACD).Ocximxu n pinuo6egpenouy rprftlnrrnr{y Bucora, npoBeAe-

Ha Ao ocHoBrt, e rrne4ianoro, ro AE = ED =! AD = 8 crvr.

3 ^ARE: 2

BE- JB' - AE" =Jl?, - 8' =ft?-8)(1?.8) =J9.2b =B.b =1b (cu).! aACE ZA= 45',6o rpraxyrnrax pinno6egpenrzfi npanoryrHuft.

EC= M =9=8 t".r.ts45" I3 r BCE Ba reopeMoro xocranycin: BC2 = BE2 + CEz - zBE . CE .. cos 60o;

BC= = Jnb + 64 - rn = .i{og = 18 (cu).

Bi0noei6a: 13 cv.

7.73, Pirno6e4peni rpnxyruaxw ABC i /J)C nnarorr cni.nrnyocuonyAC,AC = 16 cm,-48 = 17 cm, BD =13 cw, ZADC =90".flpoxegenro qeper rp,flMy BD n;rol4zuy (BD-O), nepnen4ra-nyrr,rrpry go npruoi AC. Toai BE L AC i DE L AC.IBED = cx, - nyr ui}lt u,rorqunannz rprzxyrnnrcin. AOcxiruxu r piauo6egpenoMy rprrr{yrHuxy Bzcora, npoBe-

E

BD = $z'm -- JT44+289 = n[43 (cNr).

Ocxirrnra AD L (ABC\, AC L BC, ro DC L BC3a TeopeMoro ripo Tpr,r nepilerrAr4rcyJr,fiprl.

3 yADC DC2 = AD2 * AC2i

Dc ='ltz' * ts' = Jl44 + 2%= 166, = Js - 4t =3141 (cm).

Bidnoaida: Ge3 "* i 3Gi cm.

7.68, yABC - pionocropounift', AB = BC -- AC = 12 cur'

To.nca M pinnoeig4a.nena ni4 cropin aA-BC. I{e oaua-qae, trIo ocHoBa rrepneunrryJrfipa MO - qerrrp Rora'BrrucaHoro B Tpzrcyrurrrc' Toqxa O nencr,rrr na neperuuiue4ian i nrzcor \ABC. OS = OP = ON - pa,4iycr KoJIa. aMS = MP = MN - ni4crani nig tourclr M go cropinTPITICYTHIIICA' 6O MP L AB, MN L BC' MS I AC SA

TeopeMoro npo Tpr{ flepnerrAr4rtyJrfipi'r'

AB 12 f t;(rs=r= =9!9=2J5 (cu).

2.13 2J3 3 .rreHaAoocgoBn,cMeAiaxorc, ro AE = I oC = * = 8 (cu).22 c

MS2 = MO2 + OS2; MS =

Bi\noei\a: 4Jd cu.B aABE BE =.tEz" - an" =,lt l" - e, =.@g 44 = Jzx = uY aADE IDAE = 45", DE = #= AE = 8 (crvr).

16 cu

l3 cu

cr = 60'.

(cu).

7.69, (Arae, pfic. A9 )l! 7.68) aABC - pinuocropoxnifi. Toqxa M pinnoei,44a:reua ni4

ftoro cropin. flpone4eIvIo MO L (ABC\ i MS L AC, MP L AB' MN L BC, AoB'r{rHr'I

qnx niApisris * qe ri4crani aig tovrcra M 4o cropin a-A-BC' Ocxi'nu(r4 3a yMoBoro

M P = M N = M S, ro OP = NO = OS src npoexqii pirnr'rx roxr{rrr{x' OP L AB, ON L BC'

OS LAC Ba reopeMoro rrpo rpr fiepfleuAtrrtyJrfipra' O - qenrp Ho'ta' Bilucarroro B a'A-BC'

IMPO=ZMNO=IMSO=60".

3 aoMS Mo = 6{3 cru, so = +2= 6f = o t"*l'' tg60' Ja

724 lilj rEoMETPlg sincrixa o. L ra ix. AxaAeuiqHrlfi piBeHb i:{iili:iii*i

3 a BED Ba reopeMoro Hoczuycin: BDz = BE2 + DE2 - 2BE . DE - cos u;BE2 + DE2 * BD2 15' + Bz - ].B2 228 + 64 - i,69 L2o 1

2.15 I 15.16 15.16 2'Bi]noeiOa:6O'.

7.74,lltrots1uwa cr i 0 - nepneHAtrr{yrspni, c - npamaixneperuHy. Ae a, B e p,3 rinqin ni,4piexa -A-B npoBeAeMo BC L c, AD L c. Toni BC I a i,4-D 1 B.

AB = 24 cM, BC = 12 c;ltt, AD =l2Ji clltr.

i:ii:;ii:*i:;ii fEOMETPI9 Einsnixa O. 9. ra ir. Axaseuivnui pieenr iil;; Z2S

COSO( =

-

2BE.DE

152+82 -2.15.8

r" aJo' * (26)' = .,60 - 1, = J4s = 4rE (cu)'

3a osna.resnnrvr trepfieHAultyJlspnocri npauoi i nnorqr.rHrt BC LAC iAD L BD. IBAIi ZABD - ftyrr, yrnopeni nigpiarort.4-B a n.norlunaMra o i P'

B IABC AC =,[Au" - Bc" = J;4' -Lr' =

= Ff-tzl<z+.rr) =,M.86 =6.2Ji = 12J3 (cM).

tgZBAC, X+= ,:rru=5. ,'"o"=ro".

3 .ABD an = Jl,n" - l,n" =

= ,,i76 - 288 - J2s8 = t2Ji @v).

tp .ABD = AD- =t'9 - r. IABD = 45".. BD 12J2

Bilnoei1a:30'; 45'.

7 ,75, (\uv. puc. ,4o sag:aqi 7 .74) fhorquHu o i B - nepneugunyrapui, c npflMa rx

rTeperlrry.At o,Bc P.3 xinqin ni4piena /-B rpoBeAeMo BC L c, AD L c, rol-i BC L a, AD L $'

BC = 4Ji cwr, AB = 4 cm, DC = 4 cm.

AC - npoexqis oi4pieHa-4-B na nnorqraxy crc, BD Ha n'norqr'rny B'

llyryn BAC i ABD - nyru, yrBopeui ni,qpiaxou -43 a n'noqznauu cr i B'

B IADC AC = ^[AD\ Dc' - J4" + 4' = ali kw).BC

= 4'[?

= 1: IBAC = 4b".3 "ABC LSZBAC= * 4J2

B tcBD BD = ^l6c\ BC' = ,l+' +(+"li)' = JI6* n = J48 = a.J5 (cu).

AD- - 4- = l-, IABD - 30 .B \ABD tezABD= B- 4Jg JB

Bi|noei|u:45'; 30".

Tecr Ans caMoKoHTPonK)

r{acruna 1

1. E,1..(. V suna4rcy I rpn rovrcu Jre]Karb Ha ognifi npgviii, oA"u " n'* - rorrxa

IIepeTr4Hy 3 Apyron fip.sMom.

2. E. 5. r. 8. B. 1 1. E. 14' r.

3. B. 6. E. 9' A. 12' f. 15. B.

4. R. 7. t. 10. E. 13' E. 16. A.

Hacrl,/,Ha2

17. rABC - rpnMoltyrHrfi, lC = 90", AB = 10 cu.Ocrcixrrr roqxa S pinnoei44a.nena nig neprrzn 4,43C,To ocHoBa rreprrenArrnyJr,fipa SO - qeHTp r{oJra' olr4-canoro HaBKoJro nroro. V flpsMoxyrHoMy rplrx)nH[KyO - cepe4uua rinorenyau. SO - nigcrats nig ro'rxuO go n,'rou.1zHz rpzr{YTHrlxa.

Y ^Aos ao =l 'qr- - 7?

= o cv; AS = l0 c:vr.22

so =.lTs' - ao' = Jlo' 4'-./oa - a c..Bi0nosi}u: 8 cw.

726 ii::i rEOMETPIS EincHiHa O. 9. ra ix. Axa4ervri'rHxf piBeHb i!iiii;:i:':+ii

lB. IABC - pinnp6eApennil,AB=AC, BC= 4 cm,AD - nucora,AD= 4 cyt'.SO L (ABC). 3a ynonoro ro.rxa S pirnonigga,reaa ni4 nepruzn a ABC, rogi AO = BO = COsr npoeHqii pinuux noxz.nzx. To,qi O - qenrp r(oJra, orucauoro nasnoro a43C.SO=6cm.

AD - Bttcorai nregiana, Zn =!SC = 2 cut,2

3 aABD AB =,,8D, * BD? = TV;,, _

= .,[6+4 =Jn=2r,6(cu).s^,"" 1 Bc . AD = l.n n= 8 (cm,).22A;re S*" ='ff , o" a, b, c

-cropoHrr rprrrcyrurrca, a R = OB- paAiyc Jrr""uro"o

*o'u.3"i.q"" R - .:0"-='*^Uf o=2,b (cm).' 4S-"'

B a OSB SB2 = OS2 + OB2; SB = .,!62 + 2,b2 = !86 + 6,rb = J4r,% = 6, 5 (cu). .

Bidnoaidu: 6,5 cn.

19. 3 roqxr A Ha rrJrorqrrny cx npoBeAeuo neptreHAr4nyJrff p AO i noxnny AB.AO = 12 cv, AB = 13 crr.flpone,4emo OC I AB, rogi AC - npoerqia nepreEArxyJrflpa AO na noxutry AB.B aAoB oB =,tqn, - nO = lTB, -n, - b (cu).Hexafi BC = x cv, roni AC = (18 - r) cu.3 a BCO CO2 = OB2 _ BCz; COz = 52 , x2 __ 25 _ x2.3 IACO CO2 = AO2 - ACz; QQz = 122 - (13 - x)2

== t44 - 169 + 26x x2 = -x" + 26x - 25.flpapieaacmo o4epxaHi Bftpaar4:

25 - x2 -x2 - 26x - 25: 26x = 50, * = 50.

.26AC = AB-.BC= 13- 50

- tt 1,"*t.

26 13

Bi}noaiila: 11 I c*.13

20. ABCD - npsMoryrHlrn DK L (ABC).DA IAB Ba yMoBoro, rogi KA I AB aa reopeMoro rpoTpr4 fiepneHAr4nytspu. KA - ri4cranr nig rou:cn K4o croponra A3. KA = 2,4 cm.Ana.noriuno, /{C - ni4cranr ni4 rounu 1( 4o croponuBC, KC = 2,8 c;ltt. I{D - 3,6 cu.3 IAKB AB2 = KB2' KA2 = 3,62 - 2,42 == (3,6 - 2,4)(3,6 + 2,4) = 1,2 . 6 = 7,2.DCz = AB2.

B rDKC DK =.tEC - DC' = $,8" -7,, = Ji,84JJ = J0"64 = 0,8 (cu).Bidnoai4a: O,8 cv.

K

2JB a*21. Y rABC ZC = 90", 14 = 3O", rcai lB = 6O'.

AC - 6un;smfi Harer, 6o aexzrr npotz 6i-nrurororocrporo xyra. AC = 6 c*r. KB f (ABC),

KB = 2J6 cut.Ocxi.nrnlr BC L AC, ro KC L AC aa reopeuorotrpo rpu neptreuArrxyJrspu, KC eigcranr ni4ro.rxra K 4o rarera AC.

-ltzJz)-

l{lilit:;ilit TEOMETPIC EinqHiaa 0. 9. ra ix. Ara,qenirxrail pieenu iiii: 227

BacBK KC=^[KB\BC' -JGtr;G.fa'r =$4.n=./so=o 1cu;.Bi0noai4a: 6 cw.

22. [:,as. ].1! 6.79.

23. Y yABC AB = 65 cm, BC = 75 cw, AC = 2O cw.BC ..._ 6i.nrrua cropona, rovy /A - 6i.nrrrr,rft xyr'MA I (ABC), MA = 60 cv.flpoae4ervro -A1{ L BC, M K L BC aa reopeMolo rpoTprr nepneHAr4Ryrtrpr4.AK i,MK - siAcraHi siA xiaqin nepnen4rrxyrrspa AMA 4o 6inl-rtoi croponra BC.3uafi4erro,41{ a tpoprty.nz n.nor4i a/BC:

s .^- = l-qx .BC. Ar( =2s*'2BCIlnoqy rpt'txyrurrr(a snaftgerro Ba OopMy,roro lepona:

E

s*"=@=J8o.1b.b.6o==.ll,r. 5. B - * b., "B r. s = 4. E. b. 2. B = 600 (crra).

,a,x =2 '6oo

= 16 ("*).75

B .AMK MK = "[MA' * AIC = Js600. "56

= ft856 = Jl6'% - +J241 (cn).

Bi|noai\u:16 cu i 4J241 cru.

24. ff.norqa pov6a ABCD 12O cutz, a fioro ctopoua - 12.cw.S =,4-B . KP, ge AB - cropona, a KP - ancora pou6a.

KP=+ =ry =lo (cm).AB 12

To.ma M piruonig4arena ai4 cropiu pom6a, rolry oc-HoBa nepneHAuxytapa MO - rIeHTp Bnr4caHoro KoJra

O. Padyc rco.na r 4opianroe no.noelrHi Br,Icoru: A

op-!xp=5 cu.2

MP LCD {ea TT[I) - ni4ctanr ri4 rouxra M 4o croponu pott6a, MP = 13 cu.

,B aMop Mo =.[Mp' -op' =,lls' -b' = [6, -% = IA = 12 (cu).Bidnoaida: 72 cm.

25, llz.a. ].1b 6.81.

26. 3 ro.rxnA ua n;rorqrEny o npoBeAego rreprieHAr4r(yJrspAO = 4 cra i Aei noxra.ni:

,43=5 cu i AC=4JB cla. OBiOC- npoerqiiquxuoxu.nr,rx' Z.BOC=6O'.3nafiAe*ro ix Aorxrnr.r Ba reopeMoro Ili(paropa s r:pranyrnzrcin AOB i AOC.

Bo = JAB' - Ao' =,lrs -t6 = J9 = s (cn{).

oc = "t@ * Ao' = J8o -16 = uGZ = a lcru).3 t BOC aa reopeMolo rcooanycin:BC2 = BO2 + OC' - 2BO . OC. cos 60o;

-1Bc =,ls + 64 - 2.3.8. + = Jno = z 1cu;.\l tBi1nosi]a:7 cvr.

27. IltrouTunu cl i.B nepueu4uxy.napni, c - rrp,fiMa rx nepernny. A e s, B e F.3 rcin4ir niApiaxa,4B npoaegemo BC L c, AD L c, rogiBC L a, AD L B. Bilpiaxir AC i BD - npoex4ii ri4piexani4nori4no Ha nrrorr{r4nn a i $, AC = 3rE cm, BD = }JScrvr,CD=3cru.Ityru BAC i ABD - rcyru, yreopeHi nigpierorvr:43 s nno-uquaamlr c i B.

B aBCD BC = ^tED" JD, - Jn4 = v[S = B.,E (cnn).

3 o ACB te tcAB = E9 =3+ = Li ZCAB = 45a." AC 3J2AB' = JAC'z.'+ Bc'z =

"q8.l- ts = ."6a = e 1cu;.

3 a ABD cos.lABD = BD

= 3J5

= -\6 : IABD = to'-AB6Bi0noai0a:45'; 30".

28. flaorqranr,r oc i B nepnen4ury.napni, CD - nparraa fiepert4ny nnorqnn.,43 = 25 cm -Aaruft siApiaon, A e B, B e c.AC L CD ..._ nigcranr nig rouxrz A ,4o fiJrorqr,rHra u, AC = 1"5 cvt.BD L CD - nigcraar eig rouxra B 4o naorqr.rnlr B, BD = 16 civr.3a reoperroro 8 (uo,4yJrr 5) AC I CB, BD L AD.Togi CB i AD - npoexqil ni4pisra -43 oi4noni4no Ha nJro-qunlr o i p.3 *ABC: AB2 = ACz 't BC2;

nc =,[nn' - aCi BC: Jrs" -rb" = JZ00 = 20 (cu).

B oADi: ABz = AD2 + BD2; AD = "[AB" _ BD,,

AD = Jrb' -16' = J625,286 = .,,66e = sJal (cu).

Bi|noeidu:20 cnn, 3i,[T cu.

r{acrrxa 329. Tpareqis ABCD - npsrror yrma, ZA = lB = 9O". AB = 24 cryr, CD = 25 cvt. BD -

6icexrpuca Hyra B. CM L (A.BC), CM = 7JG cv..Ilpoee4euo CK ll AB, toxi CK L AD.ICBD = ZBDA sr< euyrpiurni pianocropouni.To4i IABD - pi'nno6egpeuraft s ocHosop BD.AD=AB=24cw. B3 aCKD (CK = AB = 24 cv): CD'z = CK2 i KD2;

KD = JcD" -cK" -'lzs" -r4' ="v[g =z (cu). 24 cv

AK = AD - KD = 24 * 7 = 17 (cm).BC = AI{ = 17 cvr (ABCK - nparrrrorcyrnran).3 rABC (lB = 9O"): ACz = AB2 -t BC2 = 24, -t 172 = 576 + 289 = 865.

B aAMC: AM =.tmc' o Ae = v?Bb + 86b = vt600 = 40 (cM).Bi1noeidu: 4O etr.

3 aABC BC = ACtgSO" =6 = 2JE (cu)..63

M

65+2O+75p=-=au.

30, Hexafi ABC - 4anzi pirnoctoponnifi rpltryrHr{r{,cropoua rxoro 4opienroe 8 cu..43 ca, C e a.CD L cI, - nigcranr ei4 nepnrraHr.r C 4o nnorqunra cr,

CD = 2JB cnt.Ilpoae4e*ro CK LAB, ro4i DK LAB sa reopeMoro nporpr,r reptreHArrrcy.nspr.r. ToAi ICKD - .niuiftnr.rft xyrABorpaHHoro rcyra ui:*c IABC i rrJrorrlrtuoro c. floana-qlruo fioro g.na apy.rnocri g.

:iiii:;ilt:l! rEoMErPIe Eincxixa O. e. ra ix.

c2JF cu

D)

g

Btx

728 t::# FEOMETPIS SincHixa O. 9. ra in. Ara4er"rivHrfi piBeHb ::.:iiti::.:i::::i Axageuivnun pieenu if{ 729

.t;3 ;ACK (1A = 6O"): CK - AC sin60' = T = a.,E (cu).

srKCD: "inw=ff=2"b 1.4.rE 2' a -- 30'.

31 . ;ABC - pinHo6eApeHnit; AB - AC. Po o",, = 128 cm.

AD - megiaaa (i nucora), AD = 32 cm,

M e (ABC), MA = MB = MC = 65 cw.Ilpoeegemo MO L (ABC) - Bi.4craHb nig rovnu MAo nroulr,(Hrr (ABC). To.qi OA = OB - OC nn npoexqii Apinurax noxurux. O qeHTp Kora, ofir{caaoro HaB-

xoro a,43C, AO = R.IloaHaqr.rNro CD = BD - x crl.

124 2r'loti AC = AB '-"- - --' = 64 r (cM).2

3 "ACD: AC2 = AD2 + CD2; (64 - x)2 - 322 + x'!; 4o96 - 728 x i x2 =l28x=3O72:x=24.CD = BD - 24 cm, AC - 64 - 24 = 40 (cm).

1S.," = - BC . AD -- 24 .32 = 768 (cu").2

3 rfropvvnH s = ob:

rHaiiAevo Ao . R:"4RAB . BC . AC 40 .40 .48

AU = -- . Zb (cMr.4 S.-.u, 4 .768

3 yAOM: AM'1 = AO2 + OM2;

aM =,[AM" -Ao'=,[ss' z'; =J6b-r5X65+r5)=v?0 ,0 =

- Jsooo = 6o (cm).

Bi)noai)a:60 crt.

32, fhoquua, flKa trpoxo,ql4Tb qepe3 npflMy a i rtapa,rensHinpaui,4A, i BB,, nepernua€ nJronluHy tu no npauiri A,Br.A1B1 ll AB, AAt I BBt 3a yMoBorc. Ton' ABBI, trapa-reJrorpaM.AA) -- BBr - 14 cu, Ats = AtBr - 13 cpr, ArB = 15 cu.Srr,"r, - 2S*,r^. SHafigemo S*,u' 3a @opuyaoro lepona.

l3+14+15n= _=27-'2s^^"" = 1Fr1n -rslzr -raXzl-rsi - Jrlfl?€ -.tF.,vr:,.r4 =

=7.3.4-84(cu).S-,r'n= 2 84 - 168 (clr'z)'Bi)noai}a:168 cm2.

lD24 )' tc'!:.

ir

7gO ;:::::: FEOMETPIS EincHiHa O. 9. ra in. AxageMi.{Hufi piBeHb i:i:i:i:::i:i:i;::::i