Chuyen de Hinh Khong Gian Co Dien Full
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Transcript of Chuyen de Hinh Khong Gian Co Dien Full
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
CHᝌ Äáť:
THáť TĂCH KHáťI ÄA DIáťN
1. Háť thᝊc lưᝣng trong tam giĂĄc uĂ´ng : Cho ABC â vuĂ´ng áť A ta cĂł :
Äáťnh lĂ˝ Pitago : 2 2 2 BC AB AC = +
CBCH CA BC BH BA .;. 22==
AB. AC = BC. AH
222
111
AC AB AH +=
AH2 = BH.CH
BC = 2AM sin os tan cot
b c b c B c B B B
a a c b= = = =
! = a. sinB = a.cosC c = a. sinC = a.cosB a =sin cos
b b
B C =
! = c. tanB = c.cot C!.Háť thᝊc lưᝣng trong tam giĂĄc thĆ°"ng# " Äáťnh lĂ˝ h#$ s% CĂ´sin: a2 = !2 & c2 ' 2!c.cosA
" Äáťnh lĂ˝ h#$ s% (in: 2sin sin sin
a b c R
A B C = = =
$. CĂĄc cĂ´ng thᝊc t%nh &iáťn t%ch. a) CĂ´ng th*c t+nh ,i-n t+ch ta$ gic:
1
2S = a.ha
( = 1 . .
. sin . ./ 0/ 0/ 02
a b ca b C p r p p a p b p c
R= = = â â â vi
2
a b c p
+ +=
'(c )iáťt : " ABC â vuĂ´ng áť A :1
.2
S AB AC =
" ABC â 34u c5nh a:
2 6
aS =
!) 7i-n t+ch h8nh vuĂ´ng : ( = c5nh 9 c5nh c) 7i-n t+ch h8nh ch nht : ( = ,#i 9 <ng
,) 7i>n t+ch h8nh thoi : ( =1
2/ch?o ,#i 9 ch?o ng@n0
,) 7i-n t+ch h8nh thang :1
2S = /3 ln & 3 nh0 9 chi4u cao
) 7i-n t+ch h8nh !8nh h#nh : ( = 3 9 chi4u caoD) 7i-n t+ch h8nh t<En : 2( . RĎ =
*guy+n H,i H- /0$$!2$/1
MáťT Sáť CĂNG THᝨC GIẢI TAM GIĂC & CĂNG THᝨC TĂNH DIáťNTĂCH
1
A
B CH Ma
!c h
!FcF
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
A.QUAN Háť SONG SONG
1. ÄĆŻáťNG THẲNG VĂ MáşśT PHẲNG SONG SONG
ÄL1: Gu 3IJng thKng , LhĂ´ngn$ t<>n $N/P0 v# song song vi
3IJng thKng a n$ t<>n $N/P0th8 3IJng thKng , song song vi$N/P0
d (P)
d/ /a d/ /(P)
a (P)
â â â
d
a
(P)
ÄL2: Gu 3IJng thKng a songsong vi $N/P0 th8 $Oi $N/0ch*a a $# c@t $N/P0 th8 c@t thogiao tun song song vi a.
a/ /(P)
a (Q) d/ /a
(P) (Q) d
â â ⊠=
d
aQ!
"!
ÄL3: Gu hai $Qt NhKng c@tnhau cRng song song vi $t
3IJng thKng th8 giao tun cSachTng song song vi 3IJngthKng 3Ăł.
(P) (Q) d
(P) / /a d/ /a
(Q)/ /a
⊠=
â
a
d
Q"
2. HAI MẜT PHẲNG SONG SONG
ÄL1: Gu $N/P0 ch*a hai3IJng thKng a ! c@t nhau v#cRng song song vi $Qt NhKng/0 th8 /P0 v# /0 song song
vi nhau.
a,b (P)
a b I (P) / /(Q)
a/ /(Q),b/ /(Q)
â
⊠= â
I#
a
Q
"
ÄL2: Gu $t 3IJng thKngn$ $t t<ong hai $Qt NhKngsong song th8 song song vi $Qt
NhKng Lia.
(P)/ /(Q)a/ /(Q)
a (P)
â
â
a
Q
"
ÄL3: Gu hai $Qt NhKng /P0 v#/0 song song th8 $Oi $Qt
NhKng /U0 3V c@t /P0 th8 NhWi c@t/0 v# cc giao tun cSa chTng
song song.
(P) / /(Q)
(R) (P) a a/ /b
(R) (Q) b
⊠= â
⊠=#
a
$
Q
"
%.QUAN Háť UĂNG G'C
1.ÄĆŻáťNG THẲNG V!NG G"C V#I MáşśT PHẲNG
ÄL1: Gu 3IJng thKng , vuĂ´nggĂłc vi hai 3IJng thKng c@t nhau
a v# ! cRng n$ t<ong $N/P0 th83IJng thKng , vuĂ´ng gĂłc vi$N/P0.
d
a#
"
*guy+n H,i H- /0$$!2$/2
QUAN Háť SONG SONG ( QUAN Háť UĂNG G'C)
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
d a , d b
a ,b mp(P) d mp(P)
a,b caĂŠt nhau
⼠âĽ
â â âĽ
ÄL2: /Ba 3IJng vuĂ´ng gĂłc0 Cho3IJng thKng a LhĂ´ng vuĂ´ng gĂłc
vi $N/P0 v# 3IJng thKng ! n$t<ong /P0. Xhi 3Ăł 3i4u Li-n cYnv# 3S 3Z ! vuĂ´ng gĂłc vi a l# !vuĂ´ng gĂłc vi h8nh chiu aF cSa at<>n /P0.
a mp(P),b mp(P)
b a b a'
⼠â
⼠â âĽ
a*
a
#"
2. HAI MẜT PHẲNG V!NG G"C
ÄL1: Gu $t $Qt NhKng ch*a$t 3IJng thKng vuĂ´ng gĂłc
vi $t $Qt NhKng Lhc th8 hai$Qt NhKng 3Ăł vuĂ´ng gĂłc vinhau.
a mp(P)mp(Q) mp(P)
a mp(Q)
âĽâ âĽ
â
Q
"
a
ÄL2: Gu hai $Qt NhKng /P0 v#/0 vuĂ´ng gĂłc vi nhau th8 ![tc* 3IJng thKng a n#o n$t<ong /P0 vuĂ´ng gĂłc vi giaotun cSa /P0 v# /0 34u vuĂ´nggĂłc vi $Qt NhKng /0.
(P) (Q)
(P) (Q) d a (Q)
a (P),a d
âĽ
⊠= â ⼠â âĽ
d Q
"
a
ÄL3: Gu hai $Qt NhKng /P0 v#/0 vuĂ´ng gĂłc vi nhau v# A l#$t 3iZ$ t<ong /P0 th8 3IJngthKng a 3i \ua 3iZ$ A v# vuĂ´nggĂłc vi /0 s] n$ t<ong /P0
(P) (Q)
A (P)a (P)
A a
a (Q)
âĽ
â â ââ
âĽ
A
Q
"
a
ÄL$: Gu hai $Qt NhKng c@tnhau v# cRng vuĂ´ng gĂłc vi$Qt NhKng th* !a th8 giao tuncSa chTng vuĂ´ng gĂłc vi $Qt
NhKng th* !a.
(P) (Q) a
(P) (R) a (R)
(Q) (R)
⊠=
⼠â âĽ
âĽ
a
$
Q"
3. KHO%NG C&CH
1. K'()*+ ,-,' / 1 0 4 1 056*+ '7*+ 8 09* 1 ;'7*+: XhoWng cch t^ 3iZ$ M 3n 3IJng thKng a /hoQc 3n $Qt
NhKng /P00 l# LhoWng cch gia hai 3iZ$ M v# H t<ong 3ĂłH l# h8nh chiu cSa 3iZ$ M t<>n 3IJng thKng a / hoQc t<>n$N/P00
,/_; a0 = _H; ,/_; /P00 = _Ha
H
O
H
O
"
*guy+n H,i H- /0$$!2$/6
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
I C&C C!NG THC THáť TĂCH CEA KHáťI ÄA DIáťN :
1. ?H@ ?ACH BHD EF*3 ?G : V F .'/B: (3 ; h: chi4u cao0
?háť t%ch khIi hJ8 ch5 nhKt :
?háť t%ch khIi lK8 8hĆ°Lng :vi a l# 3 ,#i c5nh
V F =..,/a!c l# !a L+ch thIc0
V F =3
/a l# 3 ,#i c5nh0
2. ?H@ ?ACH BHD CHM: : VF1
3'
/B: (3 ; h: chi4u cao0
3. ?N O ?H@ ?ACH ?P =DQ* fff
fff SC
SC
SB
SB
SA
SA
V
V
C BSA
SABC =
C
G
A
C
G
A
S
$. ?H@ ?ACH BHD CHM: C? : 0f.f/6
B B B Bh
V ++=B A
C
A'B'
C'
J. KHáťI N"NĎ 21 1
V= Bh= r h3 3
Ď xqS = rl
. KHáťI TĎ 2V = B h= r h
Ď xqS =2 rl
. KHáťI C 6Ď 4
V= r
3
2Ď S = 4 r
*guy+n H,i H- /0$$!2$/
TH+ TĂCH ,HáťI ÄA DIáťN ( M-T N'N M-T T$/ M-T C0U
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
ChR S#1 ÄIJng ch?o cSa h8nh vuĂ´ng c5nh a l# , = a 2
ÄIJng ch?o cSa h8nh lN NhIbng c5nh a l# , = a 6
ÄIJng ch?o cSa h8nh hN ch nht cĂł 6 L+ch thIc a ! c l# , = 2 2 2a b c+ +
2 ÄIJng cao cSa ta$ gic 34u c5nh a l# h = 62
a
3 H8nh chĂłN 34u l# h8nh chĂłN cĂł 3 l# 3a gic 34u v# cc c5nh !>n 34u !ng nhau /hoQc cĂł 3 l# 3a gic 34u h8nhchiu cSa 3nh t<Rng vi t$ cSa 30.$ jkng t< 34u l# lkng t< 3*ng cĂł 3 l# 3a gic 34u.
*guy+n H,i H- /0$$!2$/m
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
II2 CĂC D3NG TOĂN
Loấi 1: TH+ TĂCH 45NG T$/
1. D*+ 1: K'Q R*+ U 0*+ ,W ,'XB ,=( '=Y ,*' 0-Y
T% &U 1# Ä cSa lkng t< 3*ng ta$ gic ABC.AFBFCF l# ta$ gic ABC vuĂ´ng cn t5i A cĂł c5nh BC = a 2 v# !it AfB
= 6a. +nh thZ t+ch Lh%i lkng t<.
jJi giWi: a cĂł ABCV vuĂ´ng cn t5i A n>n AB = AC = aABC AfBfCf l# lkng t< 3*ng AA ' ABâ âĽ
2 2 2 2AA 'B AA ' A 'B AB aâ = â =V
AA ' 2a 2â = = B.h = (ABC .AAf = 3a 2
T% &U !# Cho lkng t< t* gic 34u ABC7.AFBFCF7f cĂł c5nh !>n !ng a v# 3IJng ch?o a. +nh thZ t+ch Lh%i lkng t<.
p
J=$=
D C
A
D C
A
jJi giWi: ABC7 AfBfCf7f l# lkng t< 3*ng n>nB72 = B7f2 ' 77f2 = a2 B! 3aâ =
ABC7 l# h8nh vuĂ´ng3a
AB2
â =
(u <a B = (ABC7 =2"a
# = B.h = (ABC7.AAf = a6
T% &U $# Ä cSa lkng t< 3*ng ta$ gic ABC.AFBFCF l# ta$ gic 34u c5nh a = v# !it ,i-n t+ch ta$ gic AFBC !ng q. +nh thZ t+ch Lh%i lkng t<.
A' C'
B'
A
B
C
I
jJi giWi: eOi r l# t<ung 3iZ$ BC .a cĂł
ABC 34u n>n
AB 66
2Ar 2 Ar BC
Afr BC/,l6 0
== âĽ
â ⼠âĽ
AfBCAfBC
2(1( BC.Afr A fr 2 BC
= â = =
AAf /ABC0 AAf Ar⼠â ⼠.
2 2AfAr AAf Afr Ar 2â = â =V
: ABC.AFBFCF = (ABC .AAf= q 6
*guy+n H,i H- /0$$!2$/
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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A'
D
B'
C'
A'
C
D'
C
D
A
60
D C
A
D C
A
(Z
C
A
C
A
ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
T% &U <# Mt t[$ !8a h8nh vuĂ´ng cĂł c5nh c$ ngIJi ta c@t ! 3i áť $i gĂłc t[$ !8a $t h8nh vuĂ´ng c5nh 12 c$ <ig[N l5i th#nh $t ci hN ch nht LhĂ´ng cĂł n@N. +nh thZ t+ch ci hN n#.
D'
A'
C'
B'
D
A
C
B
eiWi ho 34 !#i ta cĂł
AAf = BBf = CCf = 77f = 12 c$ n>n ABC7 l# h8nhvuĂ´ng cĂłAB = c$ ' 2 c$ = 2d c$v# chi4u cao hN h = 12 c$
thZ t+ch hN l# = (ABC7.h = qddc$6
T% &U # Cho h8nh hN 3*ng cĂł 3 l# h8nh thoi c5nh a v# cĂł gĂłc nhOn !ng md
d
ÄIJng ch?o ln cSa 3 !ng 3IJngch?o nh cSa lkng t<.+nh thZ t+ch h8nh hN .
jJi giWi:a cĂł ta$ gic AB7 34u n>n : B7 = a
v# (ABC7 = 2(AB7 =2a 6
2
ho 34 !#i B7f = AC =a 62 a 6
2 =
2 277fB 77f B7f B7 a 2â = â =V
= (ABC7.77f =6a m2
2. D*+ 2: L*+ U ,W +W, +<= 056*+ '7*+ >? ;'7*+
T% &U 1# Cho lkng t< 3*ng ta$ gic ABC AfBfCf cĂł 3 ABC l# ta$ gic vuĂ´ng cn t5i B vi BA = BC = a !it AfB hNvi 3 ABC $t gĂłc mdd .+nh thZ t+ch lkng t<.
jJi giWi: a cĂł AfA /ABC0 AfA ABAB⼠â ⼠l# h8nh chiu cSa AfBt<>n 3 ABC .
$ ogĂłcwAfB/ABC0x ABAf md= =dABAf AAf AB.tanmd a 6â = =V
(ABC =21 aBA.BC
2 2=
= (ABC.AAf =6a 62
*guy+n H,i H- /0$$!2$/q
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
T% &U !# Cho lkng t< 3*ng ta$ gic ABC AfBfCf cĂł 3 ABC l# ta$ gic vuĂ´ng t5i A vi AC = a $ACB= md o !it BCf
hN vi /AAfCfC0 $t gĂłc 6dd. +nh ACf v# thZ t+ch lkng t<.
=(Z
(3Z
C
A
C
A
jJi giWi: o a 3ABC AB AC%tan& =â =V .
a cĂł:AB ACAB AA ' AB (AA 'C'C)⼠⼠â âĽn>n ACf l# h8nh chiu cSa BCf t<>n /AAfCfC0.
gĂłcwBCf;/AAyCyC0x = $BC'A = 6do
o
ABAC'B AC' 3a
tan3â = =V
=B.h = (ABC.AAf2 2
AA 'C' AA ' AC' A 'C' 2a 2â = â =VABCV l# nza ta$ gic 34u n>n
2
ABC
a 3S
2=
= 3a &
T% &U $# Cho lkng t< 3*ng ABC7 AfBfCf7f cĂł 3 ABC7 l# h8nh vuĂ´ng c5nh av# 3IJng ch?o B7f cSa lkng t< hNvi 3 ABC7 $t gĂłc 6dd. +nh thZ t+ch v# t{ng ,i>n t+ch cSa cc $Qt !>n cSa lkng t< .
(3Z
=
DI
CI
AI
GI
D
C G
A
eiWi:
a cĂł ABC7 AfBfCf7f l# lkng t< 3*ng n>n ta cĂł:77f /ABC70 77f B7⼠â ⼠v# B7 l# h8nh chiu cSa B7f t<>n ABC7 .
gĂłc wB7f;/ABC70x = $ d7B7f 6d=
d a mB77f 77f B7.tan6d6
â = =V
= (ABC7.77f =6a m6
( = (A77fAf =2a m6
T% &U <# Cho h8nh hN 3*ng ABC7 AfBfCf7f cĂł 3 ABC7 l# h8nh thoi c5nh a v# $BA! = mdo !it ABf hN vi 3/ABC70 $t gĂłc 6do .+nh thZ t+ch cSa h8nh hN.
=
(3Z
(KZ
DI
CIGI
AI
D
CG
A
eiWi
AB!V 34u c5nh a2
AB!
a 3S
#â =
2
ABC! AB!
a 3S 2S
2â = =
ABB'V vuĂ´ng t5iB oBB' ABtan3 a 3â = =
3
ABC!
3aV B%h S %BB'
2= = =
*guy+n H,i H- /0$$!2$/
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
3. D*+ 3: L*+ U ,W +W, +<= 2 ;'7*+
T% &U 1# Cho lkng t< 3*ng ta$ gic ABC AfBfCf cĂł 3 ABC l# ta$ gic vuĂ´ng cn t5i B vi BA = BC = a !it /AfBC0hN vi 3 /ABC0 $t gĂłc mdd .+nh thZ t+ch lkng t<.
CI
GI
AI
C
G
A
(KZ
jJi giWi:a cĂł AfA /ABC0 BC AB BC AfB⼠⼠â âĽ
$ ogĂłcw/A fBC0/ABC0x ABAf md= =dABAf AAf AB.tanmd a 6â = =V
(ABC =21 aBA.BC
2 2=
= (ABC.AAf =6a 62
T% &U !# Ä cSa lkng t< 3*ng ta$ gic ABC.AFBFCF l# ta$ gic 34u . MQt /AFBC0 t5o vi 3 $t gĂłc 6dd v# ,i-n t+chta$ gic AFBC !ng q. +nh thZ t+ch Lh%i lkng t<.
[
(3Z
I
C
A
C
A
eiWi: ABCV 34u Ar BCâ ⼠$# AAf /ABC0⼠n>n Afr BC⼠/3l 6
⼠0. gócw/AfBC0;0ABC0x = $AfrA = 6do
eiW sz Br = 9 62
62 x
x AI ==â .a cĂł
x x AI
AI I A AI A 26
62
6
26dcos:f:f d
====â
AFA = Ar.tan 6dd = x x =6
6.6
ABC.AFBFCF = Cr.Ar.AFA = 96 6M# (AFBC = Br.AFr = 9.29 = q 2=â x
7o 3Ăł ABC.AFBFCF = q 6
T% &U $# Cho lkng t< t* gic 34u ABC7 AfBfCf7f cĂł c5nh 3 a v# $Qt NhKng /B7Cf0 hN vi 3 /ABC70 $t gĂłcmdo.+nh thZ t+ch Lh%i hN ch nht.
*guy+n H,i H- /0$$!2$/1d
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
=
ZKZ
O
AI
DI
GI
CI
C
A
D
G
eOi _ l# t$ cSa ABC7 . a cĂłABC7 l# h8nh vuĂ´ng n>n C B!âĽCCf ⼠/ABC70 n>n _Cf ⼠B7 /3l 6 ⼠0. gĂłcw/B7Cf0;/ABC70x =$CC' = mdo
a cĂł = B.h = (ABC7.CCfABC7 l# h8nh vuĂ´ng n>n (ABC7 = a2
CC'V vuĂ´ng n>n CCf = _C.tanmdo
=
a &
2
=3a &
2
T% &U <# Cho h8nh hN ch nht ABC7 AfBfCf7f cĂł AAf = 2a ; $Qt NhKng /AfBC0 hN vi 3 /ABC70 $t gĂłc mdo v# AfC hN vi 3 /ABC70 $t gĂłc 6do .+nh thZ t+ch Lh%i hN ch nht.
2=
(3Z
(KZ
DI
CIGI
AI
D
C
G
A
a cĂł AAf /ABC70⼠â AC l# h8nh chiu cSa AfC t<>n /ABC70 .
gĂłcwAfC/ABC70x =$ o
AfCA 6d=BC ⼠AB â BC ⼠AfB /3l 6 ⼠0 .
gĂłcw/AfBC0/ABC70x = $ oAfBA md=
AfAC âV AC = AAf.cot6do = 2a 6
AfAB âV AB = AAf.cotmdo =2a 6
62 2 a m
ABC BC AC AB6
â = â =V
= AB.BC.AAf =61ma 2
6
$. D*+ $: K'Q R*+ U [\*
T% &U 1# Cho lkng t< 9i>n ta$ gic ABC AfBfCf cĂł 3 ABC l# ta$ gic 34u c5nh a !it c5nh !>n l# a 6 v# hN vi
3 ABC $t gĂłc mdo .+nh thZ t+ch lkng t<.
H
(KZ
=
G
AC
C
G
A
jJi giWi: a cĂł CfH /ABC0 CH⼠â l# h8nh chiu cSa CCf t<>n /ABC0
$ o
gĂłcwCCf /ABC0x CfCH md= =d 6a
CHCf CfH CCf.sinmd2
â = =V
(ABC =2 6a
= . = (ABC.CfH =
66a 6q
T% &U !# Cho lkng t< 9i>n ta$ gic ABC AfBfCf cĂł 3 ABC l# ta$ gic 34u c5nh a . H8nh chiu cSa Af 9u%ng /ABC0 l#t$ _ 3IJng t<En ngo5i tiN ta$ gic ABC !it AAf hN vi 3 ABC $t gĂłc md .
10 Ch*ng $inh <ng BBfCfC l# h8nh ch nht.20 +nh thZ t+ch lkng t< .
*guy+n H,i H- /0$$!2$/11
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
H
O
(KZ
CI
A
=
GI
AI
C
G
jJi giWi: 10 a cĂł Af_ /ABC0 _A⼠â l# h8nh chiu cSa AAf t<>n /ABC0
$ ogócwAA f/ABC0x _AA f md= = a có BBfCCf l# h8nh !8nh h#nh / v8 $Qt !>n cSa lkng t<0 A_ BC⼠t5i t<ung 3iZ$ H cSa BC n>n BC AfH⼠/3l 6 ⼠0
BC /AAfH0 BC AAfâ ⼠â ⼠$# AAf))BBf n>n BC BB fâĽ
. BBfCCf l# h8nh ch nht.20 ABCV 34u n>n
2 2 a 6 a 6A_ AH6 6 2 6
= = =
oA_Af Af_ A_t anmd aâ = =V
= (ABC.Af_ =6a 6
T% &U $# Cho h8nh hN ABC7.AFBFCF7F cĂł 3 l# h8nh ch nht vi AB = 6 A7 = .Hai $Qt !>n /ABBFAF0 v#
/A77FAF0 lYn lIt t5o vi 3 nhng gĂłc d v# mdd. . +nh thZ t+ch Lh%i hN nu !it c5nh !>n !ng 1.
*+
!'
C'
B'
A'
!
C
B
A
jJi giWi:X| AFH 0/ ABCD⼠HM AD HN AB âĽâĽ
AD N A AB M A âĽâĽâ ff /3l 6 ⼠0
$ $o oA'* #- ,A'+* &â = =ÄQt AFH = 9 . Xhi 3Ăł
AFG = 9 : sin mdd =6
2 x
AG = HM x N A AA =â=â6
6ff222
M# HM = 9.cot d = 9
Ggh}a l# 9 =
6
6
6 2
=ââ
x x
ABC7.AFBFCF7F = AB.A7.9
=6
6. . 6
=
LOáş I 2: TH+ TĂCH ,HáťI CH'"
1. D*+ 1: K'Q ,'W; ,W ,*' \* >B]*+ +W, >4 0-Y
T% &U 1# Cho h8nh chĂłN (ABC cĂł (B = (C = BC = CA = a . Hai $Qt /ABC0 v# /A(C0 cRng vuĂ´ng gĂłc vi /(BC0. +nhthZ t+ch h8nh chĂłN .
*guy+n H,i H- /0$$!2$/12
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
^
_
=
G
S
C
A jJi giWi: a cĂł
/ABC0 /(BC0
/A(C0 /(BC0
âĽâĽ
AC /(BC0â âĽ
7o 3Ăł2 6
(BC
1 1 a 6 a 6 ( .AC a6 6 12
= = =
T% &U !# Cho h8nh chĂłN (ABC cĂł 3 ABC l# ta$ gic vuĂ´ng cn t5i B vi AC = a !it (A vuĂ´ng gĂłc vi 3 ABC v#(B hN vi 3 $t gĂłc mdo.
10 Ch*ng $inh cc $Qt !>n l# ta$ gic vuĂ´ng .20+nh thZ t+ch h8nh chĂłN .
=
(Z
S
C
A
jJi giWi:10 SA (ABC) SA AB . SA AC⼠â ⼠⼠$# BC AB BC SB⼠â ⼠/ 3l 6 ⼠0. cc $Qt !>n chĂłN l# ta$ gic vuĂ´ng.
20 a cĂł SA (ABC) AB⼠â l# h8nh chiu cSa (B t<>n /ABC0. gĂłcw(B/ABC0x = $ oSAB &= .
ABCV vuĂ´ng cn n>n BA = BC =a
2
(ABC =21 a
BA%BC2 #
=
o a &SAB SA AB%tan&
2â = =V
2 3
ABC
1 1a a & a &
V S %SA3 3 # 2 2#= = =
T% &U $# Cho h8nh chĂłN (ABC cĂł 3 ABC l# ta$ gic 34u c5nh a !it (A vuĂ´ng gĂłc vi 3 ABC v# /(BC0 hN vi3 /ABC0 $t gĂłc mdo. +nh thZ t+ch h8nh chĂłN .
=
(KZ
M
C
G
A
SjJi giWi: Ml# t<ung 3iZ$ cSa BCv8 ta$ gic ABC 34u n>n AM ⼠BCâ (A ⼠BC /3l6 ⼠0 .
gĂłcw/(BC0;/ABC0x = $ oSA &= .
a cĂł = ABC1 1B%h S %SA3 3=
o 3aSA SA Atan&
2â = =V
=3
ABC
1 1 a 3B%h S %SA
3 3 = =
T% &U <# Cho h8nh chĂłN (ABC7 cĂł 3 ABC7 l# h8nh vuĂ´ng cĂł c5nh a v# (A vuĂ´ng gĂłc 3 ABC7 v# $Qt !>n/(C70 hN vi 3 $t gĂłc mdo.
10 +nh thZ t+ch h8nh chĂłN (ABC7.20 +nh LhoWng cch t^ A 3n $Qt NhKng /(C70.
*guy+n H,i H- /0$$!2$/16
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
H
=
D
CG
A
S
(KZ
jJi giWi: 10a cĂł SA (ABC)⼠v# C! A! C! S!⼠â ⼠/ 3l 6
⼠0./10
gĂłcw/(C70/ABC70x = $S!A = mdo .
SA!V vuĂ´ng n>n (A = A7.tanmdo = a 3
2
3
ABC! a
1 1 a 3V S %SA a 3
3 3 3= = =
20 a ,~ng AH S!⼠v8 C7 ⼠/(A70 /,o /10 0 n>n C7 ⼠AHâA* (SC!)⼠AH l# LhoWng cch t^ A 3n /(C70.
2 2 2 2 2 2
1 1 1 1 1 #SA!
A* SA A! 3a a 3aâ = + = + =V
AH = a 3
2
2. D*+ 2: K'Q ,'W; ,W ` \* >B]*+ +W, >4 0-Y
T% &U 1# Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh vuĂ´ng cĂł c5nh a. MQt !>n (AB l# ta$ gic 34u n$ t<ong $Qt NhKng vuĂ´ng gĂłc vi 3ABC7
10 Ch*ng $inh <ng chn 3IJng cao Lh%i chĂłN t<Rng vi t<ung 3iZ$ c5nh AB.20 +nh thZ t+ch Lh%i chĂłN (ABC7.
=
H
D
C
G
A
S
jJi giWi:10 eOi H l# t<ung 3iZ$ cSa AB.
SABV 34u S* ABâ âĽ$# (SAB) (ABC!) S* (ABC!)⼠â ⼠H l# chn 3IJng cao cSa Lh%i chĂłN.
20 a cĂł ta$ gic (AB 34u n>n (A = a 32
su <a3
ABC!
1 a 3V S %S*
3 &= =
T% &U !# Cho t* ,i-n ABC7 có ABC l# ta$ gic 34u BC7 l# ta$ gic vuông cn t5i 7 /ABC0 ⼠/BC70 v# A7 hN vi/BC70 $t góc mdo . +nh thZ t+ch t* ,i-n ABC7.
(Z
=
H D
C
AjJi giWi: eOi H l# t<ung 3iZ$ cSa BC.
a cĂł ta$ gic ABC 34u n>n AH ⼠/BC70 $# /ABC0 ⼠/BC70 â AH(BC!)⼠.
a cĂł AH ⼠H7â AH = A7.tanmdo = a 3
H7 = A7.cotmdo = a 3
3
BC!âV BC = 2H7 = 2a 3
3su <a
=3
BC!
1 1 1 a 3S %A* % BC%*!%A*
3 3 2 "= =
T% &U $# Cho h8nh chĂłN (.ABC cĂł 3 ABC l# ta$ gic vuĂ´ng cn t5i B cĂł BC = a. MQt !>n (AC vuĂ´ng gĂłc vi 3cc $Qt !>n cEn l5i 34u t5o vi $Qt 3 $t gĂłc d.
*guy+n H,i H- /0$$!2$/1
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
a0 Ch*ng $inh <ng chn 3IJng cao Lh%i chĂłN t<Rng vi t<ung 3iZ$ c5nh AC. !0 +nh thZ t+ch Lh%i chĂłN (ABC .
45
IJ
H
A
C
B
S jJi giWi: a0 X] (H ⼠BC v8 $N/(AC0 ⼠$N/ABC0 n>n (H ⼠$N/ABC0.
eOi r ⢠l# h8nh chiu cSa H t<>n AB v# BC â (r ⼠AB (⢠⼠BC tho giW
thit $ $ o
SI* S/* #-= =
a cĂł: HJ HI SHJ SHI =ââ=â n>n BH l# 3IJng Nhn gic cSa
ABCV ^ 3Ăł su <a H l# t<ung 3iZ$ cSa AC.
!0 Hr = H⢠= (H =2
aâ (ABC=
12.
6
1 6aSH S ABC =
3. D*+ 3: K'Q ,'W; 0XB
T% &U 1# Cho chĂłN ta$ gic 34u (ABC c5nh 3 !ng a v# c5nh !>n !ng 2a. Ch*ng $inh <ng chn 3IJng cao L| t^( cSa h8nh chĂłN l# t$ cSa ta$ gic 34u ABC. +nh thZ t+ch chĂłN 34u (ABC .
=
2=
HO
C
G
A
S
jJi giWi: 7~ng (_ ⼠/ABC0 a có (A = (B = (C su <a _A = _B = _C _ l# t$ cSa ta$ gic 34u ABC.a có ta$ gic ABC 34u n>n
A_ = 2 2a 3 a 3A*
3 3 2 3= =
22 2 2 11a
SA S SA A3
â = â =V
a 11S
3â = .
3
ABC
1 a 11V S %S
3 12= =
Va bU 2:Cho Lh%i chĂłN t* gic (ABC7 cĂł t[t cW cc c5nh cĂł 3 ,#i !ng a .10 Ch*ng $inh <ng (ABC7 l# chĂłN t* gic 34u.
20 +nh thZ t+ch Lh%i chĂłN (ABC7.
=
O
DC
A
S
jJi giWi:
7~ng (_ ⼠/ABC70 a cĂł (A = (B = (C = (7 n>n_A = _B = _C = _7 â ABC7 l# h8nh thoi cĂł 3IJng t<En gno5itiN n>n ABC7 l# h8nh vuĂ´ng . a cĂł (A2 & (B2 = AB2 &BC2 = AC2 n>n ASC V vuĂ´ng t5i (
2
2
aOS â =
â 6
21 1 2 2.
6 6 2 m ABCD
a aV S SO a= = =
T% &U $# Cho Lh%i t* ,i-n 34u ABC7 c5nh !ng a M l# t<ung 3iZ$ 7C.a0 +nh thZ t+ch Lh%i t* ,i-n 34u ABC7.
*guy+n H,i H- /0$$!2$/1
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
!0+nh LhoWng cch t^ M 3n $N/ABC0.(u <a thZ t+ch h8nh chĂłN MABC.
=I
HO
M
C
A
D
jJi giWi:a0 eOi _ l# t$ cSa ABC â / 0 DO ABC â âĽ
1
.6
ABC V S DO=
2 6
ABC
a
S =
2 6
6 6
a
OC CI = =
2 2Ă´ Ăł : DOC vu ng c DO DC OC â = â m
6
a=
2 6
1 6 m 2.
6 6 12
a a aV â = =
!0 X| MH)) 7_ LhoWng cch t^ M 3n $N/ABC0 l# MH
1 m
2 m
a MH DO= =
2 61 1 6 m 2. .
6 6 m 2 MABC ABC
a a aV S MH â = = =
$. D*+ $: K'Q ,'W; c PP d @Q ' a,'
T% &U 1# Cho h8nh chĂłN (.ABC cĂł ta$ gic ABC vuĂ´ng cn áť B 2 AC a= (A vuĂ´ng gĂłc vi 3 ABC SA a=10 +nh thZ t+ch cSa Lh%i chĂłN (.ABC.20 eOi e l# t<Ong t$ ta$ gic ABC $Qt NhKng /Îą 0 \ua Ae v# song song vi BC c@t (C (B lYn lIt t5i M G. +nh thZt+ch cSa Lh%i chĂłN (.AMG
G
M
N
I
C
A
SjJi giWi:a0a cĂł:
.
1.
6S ABC ABC
V S SA= v# SA a=
& i Ăł : 2 ABC c n c AC a AB aâ = â =
21
2 ABC S aâ = :
621 1
. .6 2 m
SABC
aV a a= =
!0 eOi r l# t<ung 3iZ$ BC.
e l# t<Ong t$ta cĂł :2
6
SG
SI
=
Îą )) BC â MG)) BC2
6
SM SN SG
SB SC SI â = = =
.
SAMN
SABC
V SM SN
V SB SC â = =
:6 2
2SAMN SABC
aV V = =
T% &U ! : Cho ta$ gic ABC vuĂ´ng cn áť A v# AB a= . <>n 3IJng thKng \ua C v# vuĂ´ng gĂłc vi $Qt NhKng /ABC0
l[ 3iZ$ 7 sao cho CD a= . MQt NhKng \ua C vuông góc vi B7 c@t B7 t5i ⏠v# c@t A7 t5i .
a0 +nh thZ t+ch Lh%i t* ,i-n ABC7. !0 Ch*ng $inh / 0CE ABDâĽc0 +nh thZ t+ch Lh%i t* ,i-n C7⏠.
*guy+n H,i H- /0$$!2$/1m
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
=
=
e
f
A
C
D
jJi giWi:
a0+nh ABCDV :3
ABC! ABC
1 aV S %C!
3 &= =
!0acĂł: AB AC AB CD⼠⼠/ 0 AB ACDâ ⼠AB EC â âĽ
a cĂł: DB EC ⼠/ 0 EC ABDâ âĽ
c0 +nh â˘âŹ DC V :a cĂł: . /"0 DCEF
DABC
V DE DF V DA DB
=
M# 2. DE DA DC = chia cho 2 DA
2 2
2 2
1
2 2
DE DC a
DA DA aâ = = =
Ibng t~:2 2
2 2 2
1
6
DF DC a
DB DB DC CB= = =
+
^/"01
m
DCEF
DABC
V
V
â = .61
m 6m DCEF ABCD
aV V = =
T% &U $# Cho Lh%i chĂłN t* gic 34u (ABC7. Mt $Qt NhKng 0/Îą \ua A B v# t<ung 3iZ$ M cSa (C . +nh t s% thZ t+chcSa hai NhYn Lh%i chĂłN !áť Nhn chia !áťi $Qt NhKng 3Ăł.
+
S
)
,
B
!
C
A
jJi giWi: X| MG )) C7 /G 0SDâ th8 h8nh thang ABMG l# thit ,i-n cSa Lh%ichĂłN Lhi c@t !áťi $Qt NhKng /ABM0.
& SABCDSADBSANB
SADB
SAND V V V SD
SN
V
V
1
2
1
2
1==â==
SABCDSBCDSBMN
SBCD
SBMN V V V SD
SN
SC
SM
V
V
q
1
1
1
2
1.
2
1. ==â=== M#
(ABMG = (AGB & (BMG = SABCDV q
6.
(u <a ABMG.ABC7 = SABCDV q
7o 3Ăł :
6
.
= ABCD ABMN
SABMN
V
V
T% &U < : Cho h8nh chĂłN t* gic 34u (.ABC7 3 l# h8nh vuĂ´ng c5nh a c5nh !>n t5o vi 3 gĂłc mdÎż . eOi M l# t<ung
3iZ$ (C. MQt NhKng 3i \ua AM v# song song vi B7 c@t (B t5i v# c@t (7 t5i âŹ.a0 HV 9c 3áťnh $N/AMâŹ0
!0 +nh thZ t+ch Lh%i chĂłN (.ABC7c0 +nh thZ t+ch Lh%i chĂłN (.AMâŹ
*guy+n H,i H- /0$$!2$/1
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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A
S
I
O
D
B
C
C'
D'
B'
I
O
A
B C
D
S
E
F
M
ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
jJi giWi:a0 eOi I SO AM = ⊠. a cĂł /AMâŹ0 ))B7 â ⏠)) B7
!0. 7 7
1.
6S ABC ABC V S SO= vi
2
7 ABC S a=
& SAV cĂł :m
.tanmd2
aSO AO Îż
= =
:6
. 7
m
mS ABC
aV =
c0 Phn chia chĂłN t* gic ta cĂł
. â˘MâŹS AV = (AM⏠& (AM =2(AMâŹ
.S ABCDV = 2(AC7 = 2 (ABC
â?t Lh%i chĂłN (.AM⏠v# (.AC7
a cĂł :1
2
SM
SC â =
SAC â cĂł t<Ong t$ r ⏠)) B7 n>n:
2
6
SI SF
SO SDâ = =
7
1.
6
SAMF
SAC
V SM SF
V SC SDâ = =
6
7 7
1 1 m
6 m 6mSAMF SAC SAC
aV V V â = = =
6 6
. â˘M âŹ
m m2
6m 1qS A
a aV â = =
T% &U : Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh vuĂ´ng c5nh a (A vuĂ´ng gĂłc 3 2SA a= . eOi BF 7F l#h8nh chiu cSa A lYn lIt l>n (B (7. MQt NhKng /ABF7F0 c@t (C t5i CF.
a0 +nh thZ t+ch Lh%i chĂłN (.ABC7. !0 Ch*ng $inh / f f0SC AB DâĽ
c0 +nh thZ t+ch Lh%i chĂłN (.ABFCF7F
jJi giWi:
a0 a cĂł:6
.
1 2.
6 6S ABCD ABCD
aV S SA= =
!0 a cĂł / 0 f BC SAB BC AB⼠â âĽ
fSB AB⼠(u <a: f / 0 AB SBC âĽ
n>n ABf ⼠(C .Ibng t~ A7f ⼠(C. (C ⼠/ABf7f0
c0 +nh . f f fS A B C DV
&+nh . f fS AB C V : a cĂł:f f
f f. /"0SAB C
SABC
V SB SC
V SB SC =
SAC â vuĂ´ng cn n>nf 1
2
SC
SC =
a cĂł:2 2 2
2 2 2 2
f 2 2 2
6 6
SB SA a a
SB SB SA AB a= = = =
+ *guy+n H,i H- /0$$!2$/
1q
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
^ f f 1
/"06
SAB C
SABC
V
V â =
6 6
f f
1 2 2.
6 6 SAB C
a aV â = =
&6
. f f f . f f
2 2
2 S A B C D S A B C
a
V V = =
Jg D*+ J : !* h; i'Q ,'W; >? R*+ U
T% &U 1# Cho h8nh chĂłN (.ABC7 cĂł ABC7 l# h8nh vuĂ´ng c5nh 2a (A vuĂ´ng
gĂłc 3. eĂłc gia (C v# 3 !ng mdÎż v# M l# t<ung 3iZ$ cSa (B.
10 +nh thZ t+ch cSa Lh%i chĂłN (.ABC7.20 +nh thZ t+ch cSa Lh%i chĂłN MBC7.
.2=
(KZ
H
DC
GA
S
jJi giWi:
a0a cĂł 1 .6
ABCDV S SA=
& 2 2/2 0 ABCDS a a= =
& Ăł : tan 2 mSAC c SA AC C aâ = =6
21 q m .2 m
6 6
aV a aâ = =
!0 X| ) ) / 0 MH SA MH DBC â âĽ
a cĂł: 12
MH SA= 12
BCD ABCDS S =
6
7
1 2 m
6 MBC
aV V â = =
T% &U !# Cho h8nh chĂłN ta$ gic (.ABC cĂł AB = a BC = ma CA = a. Cc $Qt !>n (AB (BC (CA t5o vi 3 $t gĂłc mdo .+nh thZ t+ch Lh%i chĂłN.
jJi giWi:H5 (H 0/ ABC ⼠L] H ⼠AB H⏠⼠BC H⢠⼠AC su <a ( ⼠AB (⏠⼠BC (⢠⼠AC . a cĂł$ $ $ )S0* S* S/* &= = = â
SJH SFH SAH â=â=â n>n H =H⏠= H⢠= < / < l# !n L+nh 3IJng t<En ngOai tiN ABC â 0
a cĂł (ABC = 00/0// c pb pa p p âââ
vi N = acba
2
=++
G>n (ABC = 22.6.. a
MQt Lhc (ABC = N.<6
m2 a
p
S r ==â
a$ gic vuĂ´ng (H:
*guy+n H,i H- /0$$!2$/1
7/17/2019 Chuyen de Hinh Khong Gian Co Dien Full
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
60
AC
B
H
S
FE
J
(H = <.tan mdd = aa
226.6
m2=
(ABC = 62 6q22.mm6
1aaa = .
T% &U $: Cho h8nh hN ch nht ABC7.AFBFCF7F cĂł 6 AB a= A7 = a
AAF = a _ l# giao 3iZ$ cSa AC v# B7.a0 +nh thZ t+ch Lh%i hN ch nht Lh%i chĂłN _AFBFCF7F
!0 +nh thZ t+ch Lh%i _BBFCF.c0 +nh 3 ,#i 3IJng cao 3nh CF cSa t* ,i-n _BBFCF.
MO
DC
A
DC
A
jJi giWi:a0 eOi thZ t+ch Lh%i hN ch nht l# .
a cĂł : . 7.AA fV AB A= 2 66. 6a a a= =
2 2
Ăł : 2 ABD c DB AB AD aâ = + =
" Xh%i _AFBFCF7F cĂł 3 v# 3IJng cao gi%ng Lh%i
hN n>n:6
f f f f
1 6
6 6OA B C D
aV V â = =
!0 M l# t<ung 3iZ$ BC / f f0OM BB C â âĽ
2 6
f f f f
1 1 6 6. . .
6 6 2 2 12O BB C BB C
a a aV S OM â = = =
c0 eOi CFH l# 3IJng cao 3nh CF cSa t*
,i-n _BBFCF. a cĂł : f f
f
6f OBB C
OBB
V C H S =
2 2Ăł : 2 ABD c DB AB AD aâ = + =
2
f
1
2OBBS aâ = f 2a 6C H â =
T% &U <: Cho h8nh lN NhIbng ABC7.AFBFCF7FcĂł c5nh !ng a.+nh thZ t+ch Lh%i t* ,i-n ACBF7F.
jJi giWi:H8nh lN NhIbng 3Ic chia th#nh: Lh%i ACBF7F v# !%n Lh%i CBF7FCF BBFAC 7FAC7 ABFAF7F. &Cc Lh%i CBF7FCF BBFAC 7FAC7 ABFAF7F
*guy+n H,i H- /0$$!2$/2d
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
=
DI
CI
GIAI
D C
GA
cĂł ,i-n t+ch 3 v# chi4u cao !ng nhau n>n cĂłcRng thZ t+ch.
Xh%i CBF7FCF cĂł2 6
1
1 1 1. .
6 2 mV a a a= =
&Xh%i lN NhIbng cĂł thZ t+ch:6
2V a=
â 6 6 6f f
1 1. m 6
ACB DV a a a= â =
T% &U : Cho h8nh lkng t< 3*ng ta$ gic cĂł cc c5nh !ng a.a0 +nh thZ t+ch Lh%i t* ,i-n AFBF BC.
!0 l# t<ung 3iZ$ c5nh AC $N/AFBF0 c@t BC t5i âŹ. +nh thZ t+ch Lh%i CAFBFâŹ.
j
I e
f
C
GA
C
GA
jJi giWi:a0 Xh%i AFBF BC:eOi r l# t<ung 3iZ$ AB
f f f f
1.
6 A B BC A B BV S CI =
2 61 6 6.
6 2 2 12
a a a= =
!0Xh%i CAFBFâŹ: Nhn <a hai Lh%i CâŹAF v# CâŹAFBF.&Xh%i AFCâŹcĂł 3 l# C⏠3IJng cao AFA n>n
f â˘âŹ â˘âŹ
1. f
6 A C C V S A A=
2
â˘âŹ
1 6
1mC ABC
aS S = =
6
f â˘âŹ
6
q A C
aV â =
&eOi ⢠l# t<ung 3iZ$ BFCF. a có Lh%i AFBFC⏠có 3 l#
CâŹBF 3IJng cao â˘AF n>nf f ⏠âŹBf
1. f
6 A B C C V S A J =
2
âŹBf f
1
2 C CBB
aS S = =
2 6
f f âŹ
1 6 6
6 2 2 A B C
a a aV â = =
& :6
AfBfâŹâ˘
6
1mC
aV =
*guy+n H,i H- /0$$!2$/21
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
ĂI TkPCHlmN Än HoNH KH!NG GIAN
? 1: Cho h8nh chĂłN (.ABC 3 ABC l# ta$ gic vuĂ´ng t5i B cĂł AB = a BC = a 3 (A vuĂ´ng gĂłc vi $Qt NhKng /ABC0 (A = 2a. eOi M G lYn lIt l# h8nh chiu vuĂ´ng gĂłc cSa3iZ$ A t<>n cc c5nh (B v# (C. +nh thZ t+ch cSa Lh%i chĂłN A.BCGM.
? 2: Cho h8nh chóN (.ABC7 có 3 ABC7 l# h8nh ch nht; (A ⼠/ABC70; AB = (A= 1; AD 2= . eOi M G lYn lIt l# t<ung 3iZ$ cSa A7 v# (C; r l# giao 3iZ$ cSa BM v# AC.+nh thZ t+ch Lh%i t* ,i-n AGrB.
? 3: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# $t h8nh vuĂ´ng t$ _. Cc $Qt !>n/(AB0 v# /(A70 vuĂ´ng gĂłc vi 3 /ABC70. Cho AB = a (A = a 2 . eOi H X lYn lIt l# h8nhchiu cSa A t<>n (B (7. +nh thZ t+ch Lh%i chĂłN _.AHX.
? $: Cho lkng t< 3*ng ABC.A1B1C1 có AB = a AC = 2a AA1 a2 -= v# ¡ oBAC 12= .
eOi M l# t<ung 3iZ$ cSa c5nh CC1. Ch*ng $inh MB ⼠MA1 v# t+nh LhoWng cch , t^ 3iZ$ Ati $Qt NhKng /A1BM0.
? J: Cho h8nh chĂłN t* gic 34u (.ABC7 cĂł c5nh !>n !ng a $Qt !>n hN vi 3 gĂłcÎą . 8$ Îą 3Z thZ t+ch cSa Lh%i chĂłN 35t gi t<áť ln nh[t.
? : Cho h8nh chĂłN (.ABC7 cĂł 3 l# h8nh ch nht AB =2a BC= a cc c5nh !>ncSa h8nh chĂłN !ng nhau v# !ng 2a . eOi M G tIbng *ng l# t<ung 3iZ$ cSa cc c5nh AB
C7; X l# 3iZ$ t<>n c5nh A7 sao cho6
a AK = . HV t+nh LhoWng cch gia hai 3IJng thKng MG
v# (X tho a.? : Cho h8nh chĂłN (.ABC cĂł gĂłc gia hai $Qt NhKng /(BC0 v# /ACB0 !ng mdd ABC
v# (BC l# cc ta$ gic 34u c5nh a. +nh LhoWng cch t^ B 3n $N/(AC0.? p: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thoi vi Âľ d12d= A B7 = a Ćd. C5nh
!>n (A vuĂ´ng gĂłc vi 3. eĂłc gia $Qt NhKng /(BC0 v# 3 !ng mdd. Mt $Qt NhKng /â0 3i\ua B7 v# vuĂ´ng gĂłc vi c5nh (C. +nh t s% thZ t+ch gia hai NhYn cSa h8nh chĂłN ,o $Qt
NhKng /â0 t5o <a Lhi c@t h8nh chĂłN.
? q: Cho h8nh hN 3*ng ABC7.AFBFCF7F cĂł cc c5nh AB=A7 = a AAF = 6
2
a v# gĂłc
BA7 = mdd . eOi M v# G lYn lIt l# t<ung 3iZ$ cSa cc c5nh AF7F v# AFBF. Ch*ng $inh <ng
ACF vuĂ´ng gĂłc vi $Qt NhKng /B7MG0. +nh thZ t+ch Lh%i chĂłN A.B7MG.? 1Z: Cho lkng t< ta$ gic ABC.A1B1C1 cĂł t[t cW cc c5nh !ng a gĂłc t5o !áťi c5nh !>n v# $Qt NhKng 3 !ng 6dd. H8nh chiu H cSa 3iZ$ A t<>n $Qt NhKng /A1B1C10 thuc3IJng thKng B1C1. +nh LhoWng cch gia hai 3IJng thKng AA1 v# B1C1 tho a.
? 11: Cho h8nh lkng t< 3*ng ABC.AFBFCF cĂł âABC l# ta$ gic vuĂ´ng t5i B v# AB =a BC = b AAF = c / 2 2 2⼠+c a b 0. +nh ,i-n t+ch thit ,i-n cSa h8nh lkng t< !áť c@t !áťi $Qt NhKng/P0 3i \ua A v# vuĂ´ng gĂłc vi CAâ˛.
? 12: Cho Lh%i chĂłN (.ABC cĂł (A ⼠/ABC0 âABC vuĂ´ng cn 3nh C v# (C = a . +nh
gĂłcĎ
gia 2 $Qt NhKng /(CB0 v# /ABC0 3Z thZ t+ch Lh%i chóN ln nh[t.? 13: Cho h8nh lN NhIbng ABC7.AfBfCf7f c5nh !ng a v# 3iZ$ M t<>n c5nh AB saocho AM = 9 /d ⌠9 ⌠a0. MQt NhKng /MAfCf0 c@t BC t5i G. +nh 9 tho a 3Z thZ t+ch Lh%i 3a
*guy+n H,i H- /0$$!2$/22
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
,i-n MBGCfAfBf !ng1
6thZ t+ch Lh%i lN NhIbng ABC7.AfBfCf7f.
? 1$: <>n c5nh A7 cSa h8nh vuĂ´ng ABC7 cĂł 3 ,#i l# a l[ 3iZ$ M sao cho AM = 9/d ⤠$ ⤠a0. <>n nza 3IJng thKng A9 vuĂ´ng gĂłc vi $Qt NhKng /ABC70 t5i 3iZ$ A l[ 3iZ$( sao cho (A = / Ć d0. +nh thZ t+ch Lh%i chĂłN (.ABCM tho a v# 9. 8$ gi t<áť ln nh[tcSa thZ t+ch Lh%i chĂłN (.ABCM !it <ng 92 & 2 = a2
? 1J: Cho h8nh nón 3nh ( 3IJng t<En 3 có t$ _ v# 3IJng L+nh l# AB = 2U. eOiM l# 3iZ$ thuc 3IJng t<En 3 v# ¡ 2ι = ASB ¡ 2β = ASM . +nh thZ t+ch Lh%i t* ,i-n (A_M tho
U ι v# β .? 1: Cho h8nh chóN ta$ gic 34u (.ABC có 3 ,#i c5nh !>n !ng 1. Cc $Qt !>n hN
vi $Qt NhKng 3 $t gĂłc â. +nh thZ t+ch h8nh cYu ni tiN h8nh chĂłN (.ABC.? 1: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh vuĂ´ng c5nh a. (A ⼠/ABC70 v#
(A = a. eOi M G lYn lIt l# t<ung 3iZ$ A7 (C. +nh thZ t+ch t* ,i-n B7MG v# LhoWng ccht^ 7 3n $N/BMG0.
? 1p: Cho h8nh chóN S.ABC có AB = AC = a. BC =2a . 6=SA a ¡ ¡ d6d= =SAB SAC +nh thZ
t+ch Lh%i chĂłN S.ABC .? 1q: Cho h8nh lkng t< ABC . AF BFC F cĂł 3 l# ta$ gic 34u c5nh a h8nh chiu vuĂ´ng
gĂłc cSa AF l>n $Qt NhKng / ABC 0 t<Rng vi t$ O cSa ta$ gic ABC . Mt $Qt NhKng / P 0 ch*a
BC v# vuĂ´ng gĂłc vi AAF c@t lkng t< tho $t thit ,i-n cĂł ,i-n t+ch !ng2 6
q
a . +nh thZ t+ch
Lh%i lkng t< ABC . AF BFC F.? 2Z: âc 3áťnh váť t<+ t$ v# 3 ,#i !n L+nh cSa $Qt cYu ngo5i tiN t* ,i-n ABCD cĂł
2 6 1 1d 16= = = = = = AB AC AD CD DB BC .? 21: +nh thZ t+ch cSa h8nh chĂłN (.ABC !it 3 ABC l# $t ta$ gic 34u c5nh a
$Qt !>n /(AB0 vuĂ´ng gĂłc vi 3 hai $Qt !>n cEn l5i cRng t5o vi 3 gĂłc â.? 22: Cho h8nh chĂłN S . ABC cĂł 3 l# â ABC vuĂ´ng cn t5i A AB AC a. MQt !>n
\ua c5nh hu4n BC vuĂ´ng gĂłc vi $Qt 3 hai $Qt !>n cEn l5i 34u hN vi $Qt 3 cc gĂłcmdd. +nh thZ t+ch cSa Lh%i chĂłN S . ABC .
? 23: Cho h8nh chĂłN t* gic (.ABC7 cĂł 3 ABC7 l# h8nh thang vuĂ´ng tai A v# 7.Bit A7 = AB = a C7 = 2a c5nh !>n (7 vuĂ´ng gĂłc vi $Qt NhKng 3 v# (7 = a. +nh thZ t*
,i-n A(BC tho a.? 2$: Cho h8nh chóN lc gic 34u (.ABC7⏠vi (A = a AB = !. +nh thZ t+ch cSa
h8nh chóN 3ó v# LhoWng cch gia cc 3IJng thKng (A B.? 2J: Cho h8nh chóN (.ABC7 có 3 ABC7 l# h8nh thoi c5nh a ¡ dmd= BAD (A vuông
góc $Qt NhKng /ABC70 (A = a. eOi CⲠl# t<ung 3iZ$ cSa (C. MQt NhKng /P0 3i \ua ACⲠv#
song vi B7 c@t cc c5nh (B (7 cSa h8nh chĂłN lYn lIt t5i BⲠ7â˛. +nh thZ t+ch cSa Lh%i chĂłN
(.ABâ˛Câ˛7â˛.? 2: +nh thZ t+ch h8nh chĂłN (.ABC !it (A = a (B = ! (C = c ¡ dmd= ASB
¡ ¡d dd 12d= = BSC CSA .
*guy+n H,i H- /0$$!2$/26
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
? 2: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thang AB = a BC = a 2 dd BAD = c5nh 2SA a= v# (A vuĂ´ng gĂłc vi 3 ta$ gic (C7 vuĂ´ng t5i C. eOi H l# h8nh chiu cSa At<>n (B. +nh thZ t+ch cSa t* ,i-n (BC7 v# LhoWng cch t^ 3iZ$ H 3n $Qt NhKng /(C70.
? 2p: Cho lkng t< 3*ng ABCA1B1C1 có AB = a AC = 2a AA1 2 = a v# ¡ 12d= ! BAC .eOi M l# t<ung 3iZ$ cSa c5nh CC1. +nh LhoWng cch , t^ 3iZ$ A ti $Qt NhKng /A1BM0.
? 2q: Cho h8nh hN ABC7.Aâ˛Bâ˛Câ˛7ⲠcĂł 3 ABC7 l# h8nh vuĂ´ng AB = AAⲠ= 2a.H8nh chiu vuĂ´ng gĂłc cSa AⲠl>n $Qt NhKng 3 t<Rng vi t$ cSa 3. M l# t<ung 3iZ$ cSa
BC. +nh thZ t+ch h8nh hN v# cosin cSa gĂłc gia hai 3IJng thKng AM v# Aâ˛C? 3Z: Cho h8nh chĂłN (.ABC cĂł $Qt !>n (BC l# ta$ gic 34u c5nh a c5nh !>n (A
vuĂ´ng gĂłc vi $Qt NhKng 3. Bit gĂłc BAC = 12dd t+nh thZ t+ch cSa Lh%i chĂłN (.ABC tho a.? 31: +nh thZ t+ch cSa h8nh chĂłN (.ABC !it 3 ABC l# $t ta$ gic 34u c5nh a
$Qt !>n /(AB0 vuĂ´ng gĂłc vi 3 hai $Qt !>n cEn l5i cRng t5o vi 3 gĂłc a.? 32: Cho h8nh lkng t< 3*ng ABC7.AFBFCF7F cĂł 3 ABC7 l# $t h8nh thoi c5nh a
góc¡
BAD = mdd
. eOi M l# t<ung 3iZ$ AAⲠv# G l# t<ung 3iZ$ cSa CCâ˛. Ch*ng $inh <ng !%n3iZ$ BⲠM G 7 3ng NhKng. HV t+nh 3 ,#i c5nh AAⲠtho a 3Z t* gic Bâ˛M7G l# h8nhvuĂ´ng.
? 33: Cho lkng t< ABC.AfBfCf cĂł Aâ˛.ABC l# h8nh chĂłN ta$ gic 34u c5nh 3 AB = a
c5nh !>n AAⲠ= !. eOi Îą l# gĂłc gia hai $Qt NhKng /ABC0 v# /Aâ˛BC0. +nh tanÎą v# thZ t+ch
cSa Lh%i chĂłN Aâ˛.BBâ˛Câ˛C.? 3$: Cho h8nh chĂłN t* gic 34u (.ABC7 cĂł 3 ,#i c5nh 3 !ng a cc $Qt !>n t5o
vi $Qt 3 gĂłc mdo. MQt NhKng /P0 ch*a AB v# 3i \ua t<Ong t$ cSa ta$ gic (AC c@t (C (7
lYn lIt t5i M G. +nh thZ t+ch Lh%i chĂłN (.ABMG tho a.? 3J: Cho h8nh lN NhIbng ABC7.Aâ˛Bâ˛Câ˛7Ⲡc5nh a. eOi M G lYn lIt l# t<ung 3iZ$
cc c5nh C7 Aâ˛7â˛. ÄiZ$ P thuc c5nh 77F sao cho P7Ⲡ= 2P7. Ch*ng t /MGP0 vuĂ´ng gĂłc
vi /Aâ˛AM0 v# t+nh thZ t+ch cSa Lh%i t* ,i-n Aâ˛AMP.? 3: Cho Lh%i chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh ch nht vi AB = 2A7 = 2a
s5nh (A vuĂ´ng gĂłc vi $Qt NhKng /ABC70 c5nh (C t5o vi $Qt 3 /ABC70 $t gĂłc #- .
eOi e l# t<Ong t$ cSa ta$ gic (AB $Qt NhKng /eC70 c@t (A (B lYn lIt t5i P v# . +nhthZ t+ch Lh%i chĂłN (.PC7 tho a.
? 3: Cho h8nh chĂłN 34u (.ABC7 cĂł c5nh 3 !ng a c5nh !>n hN vi 3 gĂłc & .eOi M l# 3iZ$ 3%i 9*ng vi C \ua 7 G l# t<ung 3iZ$ cSa (C. MQt NhKng /BMG0 chia Lh%ichĂłN th#nh hai NhYn. +nh t s% thZ t+ch cSa hai NhYn 3Ăł.
? 3p: Cho h8nh lN NhIbng ABC7.Aâ˛Bâ˛Câ˛7Ⲡc5nh a. eOi X l# t<ung 3iZ$ cSa c5nh BC
v# r l# t$ cSa $Qt !>n CCâ˛7â˛7. +nh thZ t+ch cSa cc h8nh 3a ,i-n ,o $Qt NhKng /AXr0 chiah8nh lN NhIbng.
? 3q: Cho 3IJng t<En /C0 3IJng L+nh AB = 2U. <>n nza 3IJng thKng A9 vuĂ´ng gĂłcvi $Qt NhKng ch*a /C0 l[ 3iZ$ ( sao cho (A = h. eOi M l# 3iZ$ ch+nh gia cung AB. MQt
NhKng /P0 3i \ua A v# vuĂ´ng gĂłc vi (B c@t (B (M lYn lIt t5i H v# X.. +nh thZ t+ch cSaLh%i chĂłN (.AHX tho U v# h.
*guy+n H,i H- /0$$!2$/2
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
? $Z: Cho h8nh lN NhIbng ABC7.Aâ˛Bâ˛Câ˛7Ⲡc5nh !ng a. eOi M G lYn lIt l# t<ung
3iZ$ cSa AB v# Câ˛7â˛. +nh thZ t+ch Lh%i chĂłN Bâ˛.Aâ˛MCG v# cosin cSa gĂłc t5o !áťi hai $Qt
NhKng /Aâ˛MCG0 v# /ABC70.? $1: +nh thZ t+ch cSa Lh%i chĂłN (.ABC !it 3 ABC l# $t ta$ gic 34u c5nh a
$Qt !>n /(AB0 vuĂ´ng gĂłc vi 3 hai $Qt !>n cEn l5i cRng t5o vi 3 gĂłc Îą.
? $2: +nh thZ t+ch Lh%i chóN (.ABC !it (A = a (B = b (C = c¡
ASB &= ¡BSC "= ¡CSA 12= .
? $3: Cho h8nh chóN (.ABC7 có 3 ABC7 l# h8nh thoi c5nh a ¡ ABC &= chi4u cao
(_ cSa h8nh chĂłN !ng a 3
2 t<ong 3Ăł _ l# giao 3iZ$ cSa hai 3IJng ch?o AC v# B7. eOi M l#
t<ung 3iZ$ cSa A7 $Qt NhKng /P0 ch*a BM v# song song vi (A c@t (C t5i X. +nh thZ t+chLh%i chĂłN X.BC7M.
? $$: Cho h8nh lkng t< ta$ gic ABC.Aâ˛Bâ˛Câ˛cĂł 3 l# ta$ gic 34u c5nh !ng a Aâ˛M
⼠/ABC0 Aâ˛M = a 3
2 /M l# t<ung 3iZ$ c5nh BC0. +nh thZ t+ch Lh%i 3a ,i-n ABAâ˛Bâ˛C.
? $J: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thang vuĂ´ng t5i A v# 7; AB = A7= 2a C7 = a; gĂłc gia hai $Qt NhKng /(BC0 v# /ABC70 !ng
& . eOi r l# t<ung 3iZ$ cSa A7.Hai $Qt NhKng /(Br0 v# /(Cr0 cRng vuĂ´ng gĂłc vi $Qt NhKng /ABC70. +nh thZ t+ch Lh%i chĂłN(.ABC7 tho a.
? $: Cho h8nh chĂłN t* gic 34u (.ABC7 cĂł t[t cW cc c5nh 34u !ng a. +nh tho a thZ t+ch Lh%i chĂłN (.ABC7 v# t+nh !n L+nh $Qt cYu tiN 9Tc vi t[t cW cc $Qt cSa h8nh chĂłN
3Ăł.? $: Cho Lh%i t* ,i-n ABCD. <>n cc c5nh BC BD AC lYn lIt l[ cc 3iZ$ M N
P sao cho =BC BM# =BD BN2 v# = AC AP3 . MQt NhKng / MNP 0 chia Lh%i t* ,i-n ABCD l#$hai NhYn. +nh t s% thZ t+ch gia hai NhYn 3Ăł.
? $p: Cho lkng t< ta$ gic 34u . f f f ABC A B C cĂł c5nh 3 l# a v# LhoWng cch t^ A 3n
$Qt NhKng / A"BC 0 !ng2
a. +nh tho a thZ t+ch Lh%i lkng t< . f f f ABC A B C .
? $q: Cho h8nh chĂłN t* gic 34u (.ABC7 cĂł 3 ,#i c5nh 3 !ng a $Qt !>n t5o vi$Qt 3 gĂłc mdd. MQt NhKng /P0 ch*a AB v# 3i \ua t<Ong t$ ta$ gic (AC c@t (C (7 lYn lItt5i M G. +nh thZ t+ch h8nh chĂłN (.ABMG tho a.
? JZ: Cho h8nh chĂłN ta$ gic 34u (.ABC cĂł cc c5nh !>n cĂł 3 ,#i !ng a v# cc $Qt !>n hN vi $Qt 3 gĂłc d . +nh thZ t+ch cSa h8nh chĂłN 3Ăł tho a.
? J1: Cho h8nh hN 3*ng ABCD. A#B#C#D# cĂł cc c5nh AB = AD = a AAf = 6
2
a v# gĂłc
BAD = mdd. eOi M v# N lYn lIt l# t<ung 3iZ$ cSa cc c5nh Af Df v# Af Bf. Ch*ng $inh AC fvuĂ´ng gĂłc vi $Qt NhKng / BDMN 0. +nh thZ t+ch Lh%i chĂłN A. BDMN .
? J2: Cho h8nh chĂłN (.ABC7 cĂł (A = x v# t[t cW cc c5nh cEn l5i cĂł 3 ,#i !ng a.
Ch*ng $inh <ng 3IJng thKng B7 vuĂ´ng gĂłc vi $Qt NhKng /(AC0. 8$ x tho a 3Z thZ t+ch
cSa Lh%i chĂłN (.ABC7 !ngm
26
a .
*guy+n H,i H- /0$$!2$/2
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
? J3: Cho lkng t< 3*ng ABC.Aâ˛Bâ˛CⲠcĂł 3 ABC l# ta$ gic vuĂ´ng vi AB = BC = a
c5nh !>n AAⲠ= a 2 . M l# 3iZ$ t<>n AAⲠsao cho AM AA1
'3
=uuur uuur
. +nh thZ t+ch cSa Lh%i t* ,i-n
MAâ˛BCâ˛.? J$: Cho ta$ gic vuĂ´ng cn ABC cĂł c5nh hu4n AB = 2a. <>n 3IJng thKng $ 3i
\ua A v# vuĂ´ng gĂłc $Qt NhKng /ABC0 l[ 3iZ$ ( sao cho $N/(BC0 t5o vi $N/ABC0 $t gĂłc !ng mdd. +nh ,i-n t+ch $Qt cYu ngo5i tiN t* ,i-n (ABC.? JJ: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh ch nht vi AB = a A7 = 2a .
C5nh (A vuĂ´ng gĂłc vi $Qt NhKng 3 c5nh !>n (B t5o vi $Qt Nh@ng 3 $t gĂłc & . <>n
c5nh (A l[ 3iZ$ M sao cho AM = a 3
3 $Qt NhKng /BCM0 c@t c5nh (7 t5i G. +nh thZ t+ch
Lh%i chĂłN (.BCGM.? J: Cho Lh%i lkng t< 3*ng ABC A B C% ' ' ' cĂł 3 ABC l# ta$ gic vuĂ´ng t5i A $Qt
NhKng ABC( ')
t5o vi 3 $t gĂłc
& LhoWng cch t^ 3iZ$ C 3n $Qt NhKng ABC( ')
!nga
v# LhoWng cch t^ 3iZ$ A3n $Qt NhKng BCC B( ' ') !ng a. +nh tho a thZ t+ch Lh%i lkng t< ABC A B C% ' ' ' .
? J: Cho lkng t< ABCAB CⲠⲠâ˛cĂł 3 l# ta$ gic ABC vuĂ´ng cn t5i A BC = 2a AAâ˛
vuông góc vi $Qt NhKng /ABC0. eóc gia AB C( )Ⲡv# BB C( )Ⲡ!ng & . +nh thZ t+ch lkng t< ABCAB CⲠⲠⲠ.
? Jp: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thang vuĂ´ng t5i A v# B AB BC a AD a 2= = = . C5nh !>n (A vuĂ´ng gĂłc vi 3 ABC7 v# (A= a. eOi l# t<ung 3iZ$ cSaA7. +nh thZ t+ch Lh%i chĂłN (.C7 v# t8$ t$ !n L+nh $Qt cYu ngo5i tiN Lh%i chĂłN (.C7.
? Jq: Cho h8nh lkng t< 3*ng ABC.AFBFCF có ¡ AC a BC a ACB , 2 , 12= = = v# 3IJng
thKng A C' t5o vi $Qt NhKng ABB A( ' ') gĂłc 3 . +nh thZ t+ch Lh%i lkng t< 3V cho v# LhoWngcch gia hai 3IJng thKng A B CC' , ' tho a.
? Z: Cho Lh%i chĂłN S.ABCD cĂł 3 ABCD l# h8nh ch nht !it AB %a & AD a .
<>n c5nh AB l[ 3iZ$ M sao choa
AM2
= c5nh AC c@t MD t5i H . Bit SH vuĂ´ng gĂłc vi $Qt
NhKng 'ABCD( v# SH a . +nh thZ t+ch Lh%i chĂłN S. HCD v# t+nh LhoWng cch gia hai3IJng thKng SD v# AC tho a.
? 1: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thoi c5nh a h8nh chiu vuĂ´ng gĂłccSa 3nh ( t<>n /ABC70 l# t<ung 3iZ$ H cSa AB 3IJng t<ung tun AM cSa âAC7 cĂł 3 ,#i
a 3
2 gĂłc gia /(C70 v# /ABC70 !ng 6dd. +nh thZ t+ch Lh%i chĂłN (.ABC7 v# t+nh ,i-n t+ch
$Qt cYu ngo5i tiN h8nh chĂłN (.ABC.? 2: Cho h8nh lkng t< ta$ gic 34u ABC A B C% ' ' ' cĂł AB CC m m1, ' ( )%= = > 8$ m
!it <ng gĂłc gia hai 3IJng thKng AB' v# BC ' !ng & .
? 3: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh thoi hai 3IJng ch?o AC = a2 3 B7 = 2a c@t nhau t5i _. Hai $Qt NhKng /(AC0 v# /(B70 cRng vuĂ´ng gĂłc vi $Qt NhKng
*guy+n H,i H- /0$$!2$/2m
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
/ABC70. Bit LhoWng cch t^ 3iZ$ _ 3n $Qt NhKng /(AB0 !ng a 3
# t+nh thZ t+ch Lh%i chĂłN
(.ABC7 tho a.? $: Cho $t h8nh t< t<En 9oa v# h8nh vuĂ´ng ABC7 c5nh a cĂł hai 3nh li>n tiN A
B n$ t<>n 3IJng t<En 3 th* nh[t cSa h8nh t< hai 3nh cEn l5i n$ t<>n 3IJng t<En 3 th*hai cSa h8nh t<. MQt NhKng /ABC70 t5o vi 3 h8nh t< $t gĂłc d. +nh ,i-n t+ch 9ung
\uanh v# thZ t+ch cSa h8nh t<.? J: Cho h8nh chĂłN S.ABCD cĂł 3 ABCD l# h8nh ch nht vi AB = 2a BC = a. Cc
c5nh !>n cSa h8nh chĂłN !ng nhau v# !ng a 2 .10 +nh thZ t+ch Lh%i chĂłN S.ABCD tho a.20 eOi M N E F lYn lIt l# t<ung 3iZ$ cSa cc c5nh AB CD SC SD. Ch*ng $inh
3IJng thKng SN vuĂ´ng gĂłc vi $Qt NhKng / MEF 0.? : Cho h8nh chĂłN (.ABC7 cĂł 3 l# h8nh thang cn 3 ln AB !ng lYn 3
nh C7 chi4u cao cSa 3 !ng a /a Ć d0. B%n 3IJng cao cSa !%n $Qt !>n *ng vi 3nh ( cĂł
3 ,#i !ng nhau v# !ng a. +nh thZ t+ch cSa Lh%i chĂłN tho a.? : Cho h8nh chĂłN (.ABC cĂł 3 ABC l# ta$ gic vuĂ´ng cn 3nh A AB a 2= . eOi
r l# t<ung 3iZ$ cSa c5nh BC. H8nh chiu vuĂ´ng gĂłc H cSa ( l>n $Qt NhKng /ABC0 tha $Vn
IA IH2= â . eĂłc gia (C v# $Qt 3 /ABC0 !ng & . HV t+nh thZ t+ch Lh%i chĂłN (.ABC v#LhoWng cch t^ t<ung 3iZ$ X cSa (B 3n $Qt NhKng /(AH0.
? p: Cho h8nh chĂłN (.ABC cĂł (A= 6a /vi a Ć d0; (A t5o vi 3 /ABC0 $t gĂłc !ng mdd. a$ gic ABC vuĂ´ng t5i B ¡ ACB 3= . e l# t<Ong t$ cSa ta$ gic ABC. Hai $Qt NhKng /(eB0 v# /(eC0 cRng vuĂ´ng gĂłc vi $Qt NhKng /ABC0. +nh thZ t+ch cSa h8nh chĂłN
(.ABC tho a.? q: Cho h8nh lkng t< . f f f ABC A B C cĂł 3 ABC l# ta$ gic cn AB = BC = 6a
2 AC a= . Cc $Qt NhKng / f 0 / f 0 / f 0 B AB B AC B BC cRng t5o vi $Qt NhKng / 0 ABC gĂłc dmd . +nhthZ t+ch Lh%i lkng t< . f f f ABC A B C .
? Z: Cho t* ,i-n ABC7 !it AB = C7 = a A7 = BC = b AC = B7 = c. +nh thZ t+chcSa t* ,i-n ABC7.
? 1: Cho h8nh lQng t< ta$ gic 34u ABC.AFBFCF cĂł c5nh 3 !ng a. Bit LhoWng
cch gia hai 3IJng thKng AB v# AFC !ng a 1-
-
. +nh thZ t+ch cSa Lh%i lkng t<.
? 2: Cho t* ,i-n 34u ABC7 cĂł c5nh !ng 1. eOi M G l# cc 3iZ$ lYn lIt ,i 3ngt<>n cc c5nh AB AC sao cho DMN ABC( ) ( )⼠. ÄQt AM = x AG = ). +nh thZ t+ch t* ,i-n 7AMGtho x v# ). Ch*ng $inh <ng: x y xy 3 %+ =
? 3: Cho h8nh chĂłN S ABCD% cĂł 3 ABCD l# h8nh thoi t$ O hai $Qt NhKng / SAC 0
v# / SBD 0 cRng vuĂ´ng gĂłc vi $Qt NhKng / ABCD0. Bit AC a2 3= BD a2= LhoWng cch t^
3iZ$ O 3n $Qt NhKng / SAB0 !ng a 3
#. +nh thZ t+ch Lh%i chĂłN S ABCD% .
? $: Cho h8nh chĂłN S.ABCD cĂł 3 l# h8nh thang vuĂ´ng t5i A v# D. Bit AB %a& ADa& DC a 'a * +( v# SA ⼠/ ABCD0. eĂłc t5o !áťi gia $Qt NhKng 'SBC( vi 3 !ng
#- .+nh thZ t+ch Lh%i chĂłN S.ABCD v# LhoWng cch t^ B ti $Qt NhKng 'SCD( tho a.
*guy+n H,i H- /0$$!2$/2
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ChuyĂŞn Äáť hĂŹnh khĂ´ng gian cáť Äiáťn
? J: Cho h8nh lkng t< 3*ng ABC.AFBFCF có ¡ AC a BC a ACB , 2 , 12= = = v# 3IJng thKng A C' t5o vi $Qt NhKng ABB A( ' ') góc
3 . +nh thZ t+ch Lh%i lkng t< 3V cho v# LhoWng cch giahai 3IJng thKng A B CC' , '.
? : Cho t* ,i-n ABCD cĂł !a c5nh AB BC& CD 3Ă´i $t vuĂ´ng gĂłc vi nhau v# AB BC CD a= = = . eOi C F v# DF lYn lIt l# h8nh chiu cSa 3iZ$ B t<>n AC v# AD. +nh thZ t+ch
t+ch t* ,i-n ABC"DF.? : Cho h8nh chóN S.ABC có SA a SB b SC c, ,= = = ;¡ ¡ ASB BSC &= = ° v# ¡CSA "= . +nhthZ t+ch Lh%i chóN S.ABC .
? p: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh ch nht AB = a A7 = 2a c5nh(A vuĂ´ng gĂłc vi 3 c5nh (B t5o vi 3 gĂłc & . <>n c5nh (A l[ 3iZ$ M sao cho
a AM
3
3= . MQt NhKng /BCM0 c@t (7 t5i G. +nh thZ t+ch Lh%i chĂłN (.BCMG.
? q: Cho h8nh chĂłN (.ABC7 cĂł 3 ABC7 l# h8nh ch nht AB a AD a, 2 2= = . H8nh
chiu vuĂ´ng gĂłc cSa 3iZ$ ( t<>n $Qt NhKng /ABC70 t<Rng vi t<Ong t$ ta$ gic BC7.ÄIJng thKng (A t5o vi $Qt NhKng /ABC70 $t gĂłc d . +nh thZ t+ch cSa Lh%i chĂłN (.ABC7v# LhoWng cch gia hai 3IJng thKng AC v# (7 tho a
? pZ: Cho h8nh lkng t< ABC . AF BFC F cĂł 3 l# ta$ gic 34u c5nh a 3nh AF cch 34ucc 3iZ$ A B C . MQt NhKng / P 0 ch*a BC v# vuĂ´ng gĂłc vi AAF c@t lkng t< tho $t thit
,i-n cĂł ,i-n t+ch !ng a2 3
. +nh thZ t+ch Lh%i lkng t< ABC . AF BFC F .