8th Maths samacheer complete tamil medium PART 5
Transcript of 8th Maths samacheer complete tamil medium PART 5
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toéaš
5
toéaš
5.1 m¿Kf«
ô¡ë£toéaè‹ jªij
“khbgU«fâj nkij
ô¡ë£toéaèš j®¡fmo¥gilæyhd
Áªjid¡Fé¤Â£ltuhth®.
ô¡ë£ »¿°JÃw¥gj‰F 300 M©LfS¡FK‹ng toéašg‰¿a gšntWjftšfis¤Âu£o 13 ò¤jf§fshfbtëæ£LŸsh®.
Ï¥ò¤jf§fŸ
ô¡ë£vyk‹£°v‹W miH¡f¥gL»wJ. ô¡ë£,‘KGik mj‹vªj gFÂfisélΫbgçajhF«’ v‹wh®.
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ÂU¥òjš gæ‰Á
gl« 5.1
gl« 5.2
gl« 5.3
gl« 5.5
gl« 5.4
.
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toéaš
5.2 K¡nfhz¤Â‹ g©òfŸ
5.2.1 K¡nfhz¤Â‹ tiffŸ
g¡f§fis¥ bghW¤J:
nfhz§fis¥ bghW¤J:
gl« 5.6
_‹W« FW§nfhz§fŸ xU br§nfhz« xU éçnfhz«
_‹W g¡f§fS« rk« ÏU g¡f§fŸ rk« mid¤J g¡f§fS«
bt›ntwhdit
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> >
gl« 5.7
KoÎfŸ
5.2.2 K¡nfhz¤Â‹ nfhz§fë‹ TLjš g©ò
nj‰w« 1
xU K¡nfhz¤Â‹ _‹W nfhz§fë‹ TLjš 1 0° MF«.
‰W huz«
(i) BC XY< , AB xU FW Fbt o XAB+ +ABC
(ii) YAC+ +BCA (iii) YAC+ABC +XAB=
(iv) CAB+++ABC =
CAB+++XAB (v) CAB` +ABC + =1 0°
x‹W é£l nfhz§fŸ.x‹W é£l nfhz§fŸ.(i), (ii) T l
ÏUòwK CABI T l.
ne® nfhz .
(i) _‹W g¡f§fis¡ bfh©l gynfhz« K¡nfhz« MF«.
(ii) vªj xU gynfhzK« mt‰¿‹ _iy é£l§fis Ïiz¡F«nghJ
gy K¡nfhz§fshf¥ gF¡f¥gL»wJ.
(iii) gynfhz¤Âš c£nfhz§fëš TLjš = (n – 2) 1 0°.
ϧF, v gJ g f§fë v©â if MF
cjhuz«
gl«
g¡f§fë‹v©â¡if
3 4 5
if¥ghL nfhz eh fu I§fu
nfhz§fë‹TLjš
(3 – 2) 1 0° = 1 0° (4 – 2)1 0° = 360° (5 – 2) 1 0° = 540°
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toéaš
nj‰w« 2
K nfhz  VnjD xUg f ij Ú odhš V gL K nfhz Â
të¡nfhzkhdJ mj‹ cŸbs®¡
fhz§fë TLjY F rkkhF .
T W huz
(i) ABC Ïš, CABABC ++ =180
(ii) ACD+ + = 180
(iii) CAB+ABC ++ = ACD+ +BCA
(iv) CAB+ABC ACD
(v) të¡nfhz« ACD+ , cŸbs®¡
nfhz§fŸ ABC+ , CAB+ »at‰¿‹
TLjY¡F¢ rk«
K nfhz  nfhz§fëTLjš.
ne®¡nfhz«
(i), (ii) ÏèU J
(iii) Ïš ÏUòwK« BCA+ ¡bfh©L fê¡f.
ãWt¥g lJ.
(i) xU K¡nfhz¤Âš rkg¡f§fS¡F vÂnuÍŸs nfhz§fŸ rk«.
(ii) xU K¡nfhz¤Âš Ú©l g¡f¤Â‰F vÂnu cŸs nfhz« bgçaJ.
vL¤J¡fh£L 5.1
Ô®Î
vL¤J¡fh£L 5.2
Ô®Î
gl« 5.9
KoÎfŸ
gl« 5.8
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rk g¡f§fS¡F vÂnuÍŸs nfhz§fŸ rk«
x ya =c c
vL¤J¡fh£L 5.3
Ô®Î
vL¤J¡fh£L 5.4
Ô®Î
gl« 5.11
gl« 5.10
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toéaš
vL¤J¡fh£L 5.5
Ô®Î
(K¡nfhz¤Â‹ _‹W nfhz§fë‹ TLjš)
T Ï‹tçir v©
a(br.Û)
b(br.Û)
c(br.Û)
(c+a) > b
rçah / jtwh
(a + b) > c
rçah / jtwh
b + c) > a
rçah / jtwh
T1
T2
T3
T4
T5
T6
nj‰w« 3 K¡nfhz¤Â‹ rkå‹ik¥ g©ò
xU K¡nfhz¤Â‹ VnjD« ÏU g¡f msÎfë‹ TLjš _‹whtJ g¡f
msit él mÂfkhF«.
gl« 5.12
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KoÎfŸ
rçgh®¤jš :
xU K¡nfhz¤Â‹ VnjD« ÏU g¡f msÎfë‹ TLjš _‹whtJ
g¡f msit él mÂf«
vL¤J¡fh£L 5.6
Ô®Î
12 10 252+
9 7 16, 9 7 162+ = +
(i) c a b ( b c a+ ( b c a
(ii) b c a2 ( a b c+ ( a b c
(iii) b c ( c a b ( c a b
xU K¡nfhz¤Âš VjhtJ ÏU g¡f msÎfë‹
é¤Âahr« _‹whtJ g¡f msitél¡ Fiwthf ÏU¡F«.
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toéaš
gæ‰Á 5.1
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5.3 r®trk K¡nfhz§fŸ
gl« 5.13 gl« 5.14
x‹¿‹ nkš x‹W bghU¤J« Kiwgo 1 :
go 2 :
go 3 :
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toéaš
r®trk«: ÏU js cUt§fŸ x ¿ ÛJ x W rçahf¥ bghU Âdhš mit
r®trk« vd¥gL«. Ïij ‘ v‹w F¿p£o‹ _y« F¿¡fyh«.
5.3.1 (m) r®trk ne®nfhLfŸ
ÏU nfh£L¤ J©Lfë‹ Ús« rk« våš mit r®trk« MF«.
(M) r®trk¡ nfhz§fŸ
rk nfhz msΟs ÏUnfhz§fŸ r®trk« MF«.
(Ï) r®trk¢ rJu§fŸ
rk g¡f msÎila rJu§fŸ r®trk« MF«.
(<) r®trk t£l§fŸrk Mu msÎila t£l§fŸ r®trk« MF«.
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r®trk K¡nfhz«
x¤j c¢ÁfŸ x¤j g¡f§fŸ ¤j nfhz§fŸ
A P A = PQ P+A
B Q C = QR Q+B
C R CA = RP R
5.3.2 r®trk K¡nfhz§fŸ
ÏU K¡nfhz§fëš VnjD« xU K¡nfhz¤Â‹ _‹W g¡f§fS«
_‹W nfhz§fS« Kiwna k‰bwh‹¿‹ _‹W g¡f§fS¡F« _‹W
nfhz§fS¡F« rk« våš mit r®trk K¡nfhz§fŸ vd¥gL«.
F¿¥ò: ÏU K¡nfhz§fë‹ r®trk¤ j‹ikia¡ F¿¡F«bghGJ, c¢Áfë‹
tçir rçahf mika nt©L v gJ mtÁa .
∆A C/ ∆PQR v‹gjid ∆ AC / ∆QPR, ∆ A ∆RQP vdΫ vGjyh«.
fofhuKŸ R Wtj v® Âir tçiræY mj c Áfis vGjyh .
5.3.3 K¡nfhz§fŸ r®trkkhf ÏU¡f ãgªjidfŸ
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toéaš
(i)g-g-g mo¥gil¡ bfhŸif
nkY«, AB , QR , CA RP= = .
ÏijPQAB
QR BC
RPCA 1vdÎ vGjyh .
mo¥gil¡ bfhŸif:
c©ikahf ã%áf¥
glhkš V‰W¡
bfhŸs¥g£l T‰W
mo¥gil bfhŸifahF«
Ϫj é»j¤Â‹ msÎ1Mf Ïšiy våšv‹d ãfG«?
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vL¤J¡fh£L 5.7
Ô®Î
(g–g–g bfhŸifæ‹ go)
vL¤Jfh£L 5.8
Ô®Î
(bghJ)
(g–g–g bfhŸifæ‹ go)
( RSPD k‰W« PSR D Ï‹ tçir kh¿ cŸsJ)
(ii) g-nfh-g mo¥gil¡ bfhŸif
xU K¡nfhz¤Â‹ ÏU g¡f§fS« mit cŸsl¡»a nfhzK« Kiwna
k‰bwhU K¡nfhz¤Â‹ ÏU g¡f§fS¡F« mit cŸsl¡»a K¡nfhz¤Â‰F«
rkbkåš m›éU K¡nfhz§fS« r®trk K¡nfhz§fshF«.
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toéaš
(iii) nfh-g-nfh mo¥gil¡ bfhŸif
xU K¡nfhz¤Â‹ ÏU nfhz§fS« mt‰whš Ïizªj g¡fK« k‰bwhU
K¡nfhz¤Â‹ ÏU nfhz§fS¡F« mt‰whš Ïizªj g¡f¤Â‰F« rkkhdhš,
m›éU K¡nfhz§fS« r®trk K¡nfhz§fshF«.
F¿¥ò:
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vL¤J¡fh£L 5.9
Ô®Î
(juÎ)
(juÎ)
(vÂbu®¡ nfhz§fŸ)
(g-nfh-g bfhŸifæ‹ go)
(x¤j g¡f§fŸ)
vL¤J¡fh£L 5.10
Ô®Î
(gl¤Âš cŸs go)
(juÎ)
(nfh-g-nfh bfhŸifæ‹ go)
f®z«
br§nfhz¤Â‹ v®¥ g¡f« f®z«
gl« 5.13
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toéaš
(iv) br-f-g mo¥gil¡ bfhŸif
xU br§nfhz K¡nfhz¤Â‹ f®zK« br§nfhz¤ij cŸsl¡»a¥
g¡f§fëš x‹W« Kiwna k‰bwhU br§nfhz K¡nfhz¤Â‹ f®z¤Â‰F«
br§nfhz¤ij cŸsl¡»a¥ g¡f§fëš x‹W¡F« rkkhf ÏUªjhš m›éU
K¡nfhz§fS« r®trk K¡nfhz§fshF«.
(juÎ)
(juÎ)
5.3.4 r®trk K¡nfhz§fŸ mika¥ nghJkhdj‰w ãgªjidfŸ
(i) nfh-nfh-nfh
V‹
(ii) g-g-nfh
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´AC, AD M»ad xnu t£l¤Â‹ Mu§fshF«
vL¤J¡fh£L 5.11
Ô®Î
ABCD XU ÏU rkg¡f K¡nfhz«
mik¥ò
br-f-g bfhŸif
ãWt¥g£lJ
ÏUrkg¡f K¡nfhz¤ nj‰w«
vL¤J¡fh£L 5.12
Ô®Î
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toéaš
mik¥ò
juÎ
nfh-g-nfh bfhŸifæ‹go
x¤j g¡f§fŸ
ÏU rkg¡f K¡nfhz¤ nj‰w¤Â‹ kWjiy
vL¤J¡fh£L 5.13
Ô®Î
juÎ
juÎ
g.nfh.g. nfh£ghL
vL¤J¡fh£L 5.14
Ô®Î
rk g¡f§fS¡F vÂnuÍŸs nfhz§fŸ
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gæ‰Á 5.2
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toéašwww.errsm.org
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5.4 K¡nfhz¤Â‹ xU òŸë tê¡nfhLfŸ
Ïiz nfhL
bt£o¡ bfhŸshkY«
ÏU òŸëfëš
bt£o¡ bfhŸ»‹wd.
_‹W òŸëfëš
bt£o¡ bfhŸ»‹wd.
xnu òŸëæš
bt£o¡ bfhŸ»‹wd.
xU òŸë tê¡nfhLfŸ
rªÂ¡F« òŸë
5.4.1 K¡nfhz¤Â‹ eL¡nfh£L ika«
xU K¡nfhz¤Â‹ X® c¢ÁæèUªJ mj‹ v®¥g¡f¤Â‹
eL¥òŸëia Ïiz¡F« nfhL eL¡nfhL vd¥gL«.
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toéaš
xU K¡nfhz¤Â‹ eL¡nfhLfŸ rªÂ¡F« òŸëahdJ,
m«K¡nfhz¤Â‹ eL¡nfh£L ika« vd¥gL«.
F¿¥ò:
5.4.2 xU K¡nfhz¤Â‹ br§nfh£L ika«
xU K¡nfhz¤Â‹ c¢ÁæèUªJ mj‹ v®¥g¡f¤Â‰F br§F¤jhf tiua¥gL« nfh£o‰F
br§nfhL v‹W bga®.
K¡nfhz¤Â‹ _‹W F¤J¡nfhLfŸ rªÂ¡F« òŸë
m«K¡nfhz¤Â‹ br§nfh£LikakhF«.
br§nfh£Lika¤Â‹ mikél§fŸ
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5.4.3 K¡nfhz¤Â‹ cŸt£l ika«
xU K¡nfhz¤Âš xU nfhz¤Â‹ ÏUrk
bt£o v‹gJ m¡nfhz¤ij ÏUrk¡ T¿L« nfh£L¤J©L MF«.
xU K¡nfhz¤Â‹ _‹W nfhz§fë‹ ÏUrk
bt£ofŸ rªÂ¡F« òŸë m«K¡nfhz¤Â‹ cŸt£l
ika« vd¥gL«.
5.4.4 K¡nfhz¤Â‹ R‰W t£l ika«
xU K¡nfhz¤Â‹ g¡f¤Â‹ ika¡F¤J¡ nfhL v‹gJ m¥g¡f¤Â‰F
br§F¤jhfΫ, mnj rka« m¥g¡f¤ij ÏU rk¥ ghf§fshfΫ Ãç¡F« nfhL
MF«.
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toéaš
xU K¡nfhz¤Â‹ _‹W g¡f§fë‹ ika¡F¤J¡ nfhLfS« rªÂ¡F«
òŸë R‰W t£l ika« vd¥gL«.
F¿¥ò :
MŒy® nfhL OG : GH = 1 : 2
rkg¡f K¡nfhz¤Âš O G
H I xnu òŸëæš
5.5 Ãjhfu° nj‰w«
5.5.1 Ãjhfu° nj‰w«
xU br§nfhz K¡nfhz¤Âš f®z¤Â‹ t®¡fkhdJ k‰w ÏU g¡f§fë‹
t®¡f§fë‹ TLjY¡F¢ rk«.
a b c2 2 2
+ =
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a b c2 2 2
+ =
Ãjhfu° nj‰w«
Ãjhfu° nj‰w¤ij ãiwÎ brŒÍ« v©fS¡F¥ äjhfuì‹ _‹w‹
bjhFÂ v‹W bga®.
I II
IIIIV
c
c c
c
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toéaš
vL¤J¡fh£L 5.15
Ô®Î
vL¤J¡fh£L 5.16
Ô®Î
juÎ
vL¤J¡hf£L 5.17
Ô®Î
PT br§nfhL
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toéaš
gæ‰Á 5.3
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5.6 t£l§fŸ
t£l¤Â‹ tiuaiw
t£l« v‹gJ xU js¤Âš cŸs ãiyahd
òŸëæèUªJ rk bjhiyéš efU« òŸëæ‹ ãak¥ ghij
MF«.
ãiyahd òŸë t£l ika« vd¥gL«. rk bjhiyÎ
Mu« vd¥gL«.
F¿¥ò:
eh©t£l¤Â‹ ÛJŸs ÏU òŸëfis Ïiz¡F«
nfh£L¤ J©L eh© vd¥gL«.
é£l«
t£l ika¤Â‹ têna bršY« eh© é£l«
vd¥gL«. t£l¤Âš tiua¥gL« äf¥ bgça eh© é£l« MF«.
é£l« = 2 × Mu« Mu« = (é£l«) ÷ 2
F¿¥ò:
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toéaš
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fâj k‹w brašghL
r®trk¤ j‹ikæ‹ K¡»a¤Jt« ekJ m‹whl thœéš, r®trk¤ j‹ikia gy Ïl§fëš ga‹gL¤J»‹nwh«.
ekJ Å£oš cŸs miwæ‹ Ïu£il fjÎfŸ x‹W¡F x‹W r®t rk«. bgU«ghY« ekJ Å£o‹ K‹ thr‰fjÎfŸ x‹W¡F x‹W r®t rk«. gwitfë‹ Ïw¡iffŸ
x‹W¡F x‹W r®t rk«. kåjå‹ clyik¥Ãš iffŸ, fhšfŸ ngh‹wit x‹W¡F
x‹W r®t rk«. ÏJnghy gy cjhuz§fis eh« Twyh«.
thåš gwitfŸ gw¡»‹wnghJ mit xU
K¡nfhz tot¤ij mik¡»‹wd. Ïš K‹dhš
gw¡F« gwitæ‹ têahf xU ika¡ nfh£il tiuªjhš
mJ r®t rk¤ j‹ik bgWtij m¿ayh«. Ϫj mik¥Ãš
r®t rk¤j‹ik Fiyªjhš bjhl®ªJ tU« gwitfë‹
ãiy¥ò¤ j‹ik FiwªJ mt‰whš gw¡f ÏayhJ.
Ï¥nghJ, Ïa‰ifæY« ekJ m‹whl thœéY«
r®t rk¤ j‹ikia ga‹gL¤J« totik¥ò¡fis¡
f©l¿a Ka‰Á brŒf.
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