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GIAITHEO
CHUYEN DE
• BOI Dl/dNG HQC SINH GIOI
• CHUAN B! THI TU TAI, DAI HOC VA CAO
BOG
Ha
NOI
DANG
NHA XUAT BAN OAI HOC QUOC GIA HA NOI
BAN GIAO V I E N NANG K H I E U TRl/CfNG THI
NGUYEN DLfC D 6 N G {Chu hien)
TUYEN TAP 500
BAITOAN
•
HDIH imm GIAN
C H O N LOG
•
•
•
PHAN LOAI VA PHUdNG PHAP G I A I THEO 2 3 CHUYEN
• B o i difdng hoc s i n h g i o i
• C h u a n b i t h i T i i t a i , D a i hoc v a Cao d a n g
(Tdi ban idn thvt ba, c6 svCa chUa bo
sung)
THir ViEN TiiVH BiKH liik^m
NHA XUAT BAN DAI HOC QUOC GIA H A NOI
NHA XUAT BAN DAI HOC QUOC GIA HA NQI
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Gidm
Bien
Saa
trdch
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nhiem
xuat ban:
doc - Tong bien tap: T S . P H A M T H I T R A M
tap:
THUY
bdi:
THAI
Che ban:
Trinh
HOA
VAN
N h a sach H O N G A N
bay bia:
THAI V A N
SACH LIEN K E T
TUYEN TAP 500 BAI TOAN HJNH KHONG GIAN CHON LOG
Ma so: 1L - 195OH2014
In 1.000 cuon, kho 17 x 24cm tai Cong ti Co phan V3n hoa VSn Lang - TP. Ho Chi IVlinh.
So xuat ban: 664 - 2014/CXB/01-127/OHQGHN ngay 10/03/2014.
Quyet dinh xuat ban so: 198LK - TN/QO - NXBOHQGHN ngay 15/04/2014.
in xong va nop IIAJ chieu quy il nSm 2014.
LCilNOIDAU
Chung t o i x i n g i d i t h i $ u den doc gia bp sdch: Tuyen t a p cdc b ^ i toan d k n h cho
hoc sinh Idp 12, chuan b i t h i vao cac trucrng D a i hoc & Cao d i n g .
Bo sach gom 7 quyen :
.
T U Y E N T A P 546 B A I T O A N T I C H P H A N
.
T U Y E N T A P 540 B A I T O A N K H A O S A T H A M SO
.
T U Y E N T A P 500 B A I T O A N H I N H G I A I T I C H
.
T U Y E N T A P 500 B A I T O A N H I N H K H O N G G I A N
.
T U Y E N T A P 696 B A I T O A N D A I SO
•
T U Y E N T A P 599 B A I T O A N L U O N G G I A C
.
T U Y E N T A P 6 7 0 B A I T O A N RCJI R A C V A C l / C
TRI
NhSm phuc vu cho viec r e n luyen va on t h i vao D a i hoc b k n g phucrng phdp t i m
hieu cac de t h i dai hoc da ra, de tiT n a n g cao va chuan b i k i e n thiJc can t h i e t .
De phuc vu cho cac do'i tUcfng t\i hoc : Cac bai g i a i luon chi t i e t va ddy d u , p h a n
nho tCrng loai toan va dua vao do cac phucfng phap hop l i .
Mac du chiing t o i da co g^ng het siic t r o n g qud t r i n h bien soan, song vSn k h o n g
t r a n h k h o i nhiJng t h i e u sot. Chiing t o i m o n g don n h a n m o i gop y, phe b i n h tii quy
dong nghiep ciing doc gia de Ian xuat ban sau sach ducfc hoan t h i e n hcfn.
Cuoi Cling, chiing toi x i n cam cm N I l A X U A T B A N D A I H O C Q U O C G I A H A N O I da
giiip da chiing t o i m o i m a t d l bo sach dUdc r a dcfi.
NGUYEN
DtfC
DONG
3
•
(i)
B A N G K E CAC K I H I E U V A CHLf V I E T T A T T R O N G
• [ ( A B C ) ; ( E F G ) ] : goc tao bori 2
mp
( A B C ) va ( E F G )
->
• C > : Phep t i n h tien vectcf v
V
• D A : Phep doi xOmg true A
• Do : Phep doi xiiTng true 0
• Q(0; cp) : Phep quay t a m O, goc quay
(p.
• V T ( 0 ; k ) : Phep v i t u t a m 0, t i so k.
• D N : dinh nghla
• D L : dinh ly
• Stp : D i e n t i c h t o a n p h a n
: The t i c h
• C M R : chiJng m i n h r i n g
A
: goc
• B i : budc i
• T H i : t r u d n g hop i
• V T : ve t r a i
xuong dtfcfng thftng (d)
(3r3^
SACH
CAC K I H I E U T O A N HOC v A CAC T l / V I E T T A T
<=> : (i) tUcfng dUcfng
(il
• => : (i) keo theo
• : k h o n g tUdng dilcfng
• d> : k h o n g keo theo
• = : dong n h a t
: k h o n g dong n h a t
• i
• Sv\nc = S ( A B C ) = d t ( A B C ) : d i e n . t i c h
AABC
• V s A H c = V ( S . A B C ) : the t i c h h i n h chop
S.ABC
• H Q : he qua
• Sxq : D i e n t i c h xung quanh
• V
• A ' = ''7(ai A : A ' la h i n h chieu ciia A
xuong m a t p h i n g (a)
• A ' = ''Vfd) A : A ' l a h i n h chieu cua
• d [ M ; (D)l : k h o a n g each tiT d i e m M d e n
ducfng t h i n g (D)
• d [ M ; ( A B C ) I : k h o a n g each tii diem M
den mat phang ( A B C )
• (a; P ) : goc n h i d i e n tao bcfi 2 mfa m a t
phang (a) va ( P )
• ( S ; A B ; D) = ( A B ) : n h i dien c a n h A B
•
tao bdi h a i dUomg t h i n g d
• V P : ve p h a i
• B D T : bat d i n g thijfc
• y c b t : yeu cau b a i toan
• d p c m : dieu p h a i chuCng m i n h
• gt : gia
thiet
• K L : ket luan
• D K : dieu k i e n
• P B : phan ban
va d'
• [ H T C A B C T I : goc tao bdi du&ng t h i n g d
va
• C P B : chiTa p h a n ban
mp(ABC)
4
Chuyen
de 1 :
TONG QUAN V E C A C KHAI NIEM
T R O N G HINH H O C K H O N G G I A N
•
H i n h hoc k h o n g gian la m o t mon hoc ve cac v $ t t h e t r o n g k h o n g g i a n ( h i n h h i n h hoc
t r o n g k h o n g gian) ma cac d i e m h i n h t h a n h nen v a t the do t h u d n g thiTcrng k h o n g ciing n f t m
t r o n g mot m a t phang.
•
N h i f vay ngoai d i e m v a d i i d n g t h d n g k h o n g drfoTc d i n h n g h i a nhiT t r o n g h i n h hoc
phAng; mon h i n h hoc k h o n g g i a n con xay di/ng t h e m mot doi tuong can n g h i e n ciifu nCfa la
k h a i n i # m m g t p h a n g c u n g k h o n g difoTc d i n h n g h i a . K h i noi tori k h a i n i e m nay t a
lien tuang den m o t m a t ban b a n g phang, m o t m a t ho nildc yen l a n g , m o t tb giay dat d i n h
sat t r e n mot m a t da di/gc l a m phang.... No duoc k y hieu b d i cac chCf i n L a T i n h n h a : (P),
(Q), (R), ... hoac cac chCf t h u d n g H y L a p nhU (a), ((5), (y), ....
•
M a t phang k h o n g ducfc d i n h n g h i a qua mot k h a i n i e m k h a c ; n h i f n g thifc te cho thfi'y mSt
ph&ng CO nhutng t i n h chat cu t h e sau, goi la cac t i e n de :
O T I E N D E 1: C o i t n h a t b o n d i e m t r o n g k h o n g g i a n k h o n g t h ^ n g h a n g (nghia la
luon luon c6 i t n h a t 1 d i e m d ngoai m o t m a t p h ^ n g tiiy y).
O T I E N D E 2: N e u m p t dtfdng th&ng v a m p t m a t p h ^ n g c 6 h a i d i e m c h u n g t h i
dUcTng th&ng a y se n S m t r p n v ^ n t r o n g m a t p h a n g n e u t r e n .
O T I E N D E 3: N e u h a i m a t p h & n g c 6 d i e m c h u n g t h i c h t i n g c 6 v 6 so' d i e m c h u n g :
n e n h a i m a t p h S n g do c S t n h a u t h e o m p t d U d n g t h ^ n g d i q u a v 6 so' d i e m
c h u n g a y . Di/cfng t h a n g ay goi la giao tuyen cua h a i m a t ph^ng.
O
T I E N D E 4: C o m p t v a c h i m p t m $ t p h a n g d u y n h a t d i q u a b a d i e m p h a n b i # t
khong th^ng hang.
O T I E N D E 5: T r e n m p t m § t p h a n g t u y y t r o n g k h o n g g i a n c a c d i n h l y h i n h h o c
ph&ng scf c a p (da hoc tCr Idp 6 den Idp 10 va cac d i n h l y n a n g cao) d e u d i i n g .
O T I E N D E 6: M o i d o a n th&ng t r o n g k h o n g g i a n d e u c 6 dp d a i x a c d i n h : t i e n de neu
len sU bao toan ve dp dai, goc va cac t i n h chat lien thuoc da biet t r o n g h i n h hoc p h i n g .
•
TiT do chung t a c6 m o t so each xac d i n h m a t p h 4 n g n h i / sau :
O
H E Q U A 1: C o m p t v a c h i m p t mfit p h S n g d u y n h a t d i q u a m p t d U d n g t h S n g v a
m p t d i e m n S m n g o a i dt^dng t h a n g do.
O
O
H E Q U A 2: C o mpt v a c h i mpt m^t p h d n g duy n h a t d i q u a h a i di^cAig t h ^ n g cSt n h a u .
H E Q U A 3: C o m p t v a c h i m p t m ^ t p h a n g d u y n h a t d i q u a h a i di^c/ng t h d n g
song song.
•
•
Dong t h d i t a phai hieu t h e m r k n g mot m a t phang se r o n g k h o n g bien gidi va dUcmg t h ^ n g c6
do dai v6 t a n mac du t a se bieu dien no mpt each h i n h thiifc hflu h a n va k h i e m t o n nhU sau:
De thuc h i e n dirge phep ve c h i n h xdc m 6 t h i n h h i n h hoc t r o n g k h o n g g i a n ngoai cac dudng
t h a y ve l i e n n e t , t a can p h a i n a m chac di/pc k h a i n i e m di/dng k h u a t ve b k n g net dijft doan:
Mpt dtfdng b i k h u a t t o a n bp h a y c h i k h u a t m p t d o a n c u e bp n a o do k h i v a c h i
k h i t o n t a i i t n h a t m p t m a t p h S n g du'ng p h i a trvC6c h o ^ c p h i a t r e n c h e n o m p t
e a c h t o a n bp h o a c c u e bp ti^cAig uTng.
5
•
Muon xac d i n h n h ^ n h m o t m a t p h ^ n g t r o n g k h o n g gian t a con chon t h u thuat thUc h a n h :
M p t h i n h t a m g i a c , tii" g i a c h o a c d a g i a c ph&ng ( k h o n g g e n h ) , dUcfng i r o n ,
l u d n x a c d i n h m p t m ^ t p h S n g t r o n g k h o n g g i a n . T a gpi c a c m&t p h ^ n g do l a m^it
p h S n g h i n h thvCc v d i c a c k y h i p u ( A B C ) , ( A B C D ) , ( C ) , ... txictng vtng.
M p t dvictng t h d n g n ^ m t r o n g m ^ t p h & n g h i n h thd'c m a m a t do h i k h u a t c u e bp
•
M a t p h d n g h i n h thu^c h i k h u a t n e u c 6 m p t h a y n h i e u m ^ t ph&ng n a o do c h e n o .
•
h a y t o a n bp v a k h i dUcTng t h ^ n g do k h o n g l a b i e n c u a m a t p h d n g b i k h u a t do,
t h i di^dng th&ng do c u n g tii'oTng vlng k h u a t c u e bp h a y t o a n bp.
Noi h a i d i e m m a it n h a t c 6 mpt d i e m k h u a t t h i dUpc mpt dUcfng k h u a t cue bp h a y
•
Mpt d i e m nhm t r o n g m p t m $ t ph&ng h i n h thuTc b i k h u a t t h i goi l a d i e m k h u a t .
•
t o a n bp : n e u h a i diictag do k h o n g l a b i e n c u a c a c m^t phAng h i n h thufc c h e no.
•
C A C H I N H A N H M I N H HQA
\(d)
•
(d) b i (a) che k h u a t cue bo, do (d) c6 1 doan ve
net dijft doan n k m dudi (a).
S
•
(d) b i m a t p h ^ n g (SAC) che k h u a t cue bo, do (d)
CO m p t doan ve duft doan n k m sau (SAC)
(hien
n h i e n (d) cung d sau cac m a t (SAB), (SBC)).
•
C a n h AC b i h a i m a t p h a n g (SBC) v£l (SBC)
che
k h u a t toan bo, do ca doan AC x e m n h u hoan t o a n d
sau dong t h d i h a i m a t p h ^ n g (SAB), (SBC).
-AA.
c./—1—^VFJL^
•
• A ] H b i che t o a n bo do ca doan A ] H n k m sau m a t
p h i n g ( A i A D D i ) , mSc dij no d trU
H a i m a t p h l n g (a), (P) thuf tif chiJa h a i difdng
t h i n g ( d i ) , (da) ma (dj) n (da) = I
=> S I la giao tuyen can t i m .
>
H a i m a t p h l n g (a), (P) thuf t i f chtifa h a i difdng
t h i n g ( d i ) , (da) ma ( d i ) // (da).
S_
D i f n g xSy song song v d i (dj) h a y (da)
=> xSy la giao t u y e n can t i m .
7
m. C A C B A I T O A N C O B A M
Bai 1
Cho tiif giac l o i A B C D c6 cac canh doi k h o n g song song va d i e m S d ngoai (ABCD). T i m
giac tuyen ciia :
a/ (SAC) va (SBD).
hi
(SAB) va (SDC); (SAD) va (SBC).
Giai
a/ Xet h a i m a t p h a n g (SAC) va (SBD), t a c6 :
T r o n g tuT giac l o i A B C D , h a i ducmg cheo A C
•
S la d i e m c h u n g thuf n h a t .
•
(1)
n B D = O : d i e m c h u n g thijf n h i (2).
^
Ti/(1) va (2) suy r a :
(SAC) o (SBD) = SO (ycbt)
hi
Xet hai m a t p h a n g (SAB) va (SDC) cung c6 :
H a i canh ben A B va C D cua t i l giac A B C D
•
S la m o t d i e m chung.
•
theo gia t h i e t k h o n g song song.
^
A B ^ C D = E : la d i e m c h u n g thut h a i .
Do do : (SAB) n (SDC) = SE (ycbt)
Tucfng t i f : (SAD) n (SBC) = SF (ycbt); v d i F = A D ^ BC; do A D / / BC.
Bai 2
Cho t i l d i e n A B C D . Goi G j , Ga la t r p n g t a r n h a i t a m giac B C D va A C D . L a y theo thuT t i i I ,
J , K la t r u n g d i e m ciia B D , A D , C D . T i m cac giac tuyen :
aJ
(G1G2C) o ( A D B )
hi
(G1G2B) n ( A C D )
c/
( A B K ) o (CIJ>.
a/ (G1G2C) n ( A B D ) = I J
(ABK) ^ (CIJ) =
d
(GiGaB) n ( A C D ) = GgK hoSc A K
hi
G,G2
Bai 3
Cho h i n h chop S . A B C D c6 day A B C D la h i n h b i n h h a n h t a m O.
T i m giao t u y e n cua h a i mSt p h i n g (SAB) va (SCD).
hi
T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC).
aJ
c/ T i m giao t u y e n ciia h a i m a t p h ^ n g (SAC) va (SBD).
Giai
aJ
Xet h a i m a t phSng (SAD) va (SBC), t a c6 :
De y A D c ( S A D ) ; BC c (SBC) m a A D // BC.
•
S la d i e m c h u n g thur n h a t .
•
Ta d u n g xSy // A D hoac BC.
[(SAD) = (xSy; AD)
^
|(SBC) = (xSy; BC)
=^ (SAD) n (SBC) = xSy (ycbt).
hi
Tifang t i r , difng uSv // A B hoftc C D
8
=> (SAB) r^ (SCD) = uSv (ycbt)
c/ Goi O = A C n B D , tiTcrng t a b a i 1
=> (SAC) n (SBD) = SO (ycbt).
Bai 4
Cho h i n h chop S . A B C D c6 day la h i n h t h a n g A B C D v d i A B l a day Idtn. Gpi M la m o t d i e m
bat ky t r e n SD va E F l a difang t r u n g b i n h cua h i n h t h a n g .
a/ T i m giao t u y e n ciia h a i mSt p h i n g (SAB) va (SCD).
b/ T i m giao t u y e n cua h a i m a t phSng (SAD) va (SBC),
c/ T i m giao t u y e n cua h a i mSt p h a n g ( M E F ) va ( M A B ) .
Doc gia t u g i a i tUcfng t u n h u cac b a i t r e n .
Bai 5
Cho h i n h chop S . A B C D c6 A B C D l a h i n h b i n h h a n h . Goi G,, G2 l a t r o n g t a m cac t a m giac
SAD; SBC. T i m giao t u y e n cua cac cSp mSt p h a n g :
a/ (SGiG^) va ( A B C D )
b/ (CDGiGz) va (SAB)
UvCdng
0/
(ADG2) va (SBC).
d§Ln
Goi I , J , E, F thur t a Ik t r u n g d i e m cac doan t h i n g A D ,
BC, SA, SB theo thur tvt d6. Thifc h i e n cac l a p l u a n nhtf cac
bai toan t r e n ;
a/ (SG1G2) n ( A B C D ) = I J (ycbt)
b/ (CDGiGa) n (SAB) = E F (ycbt)
c/ (ADG2) ^ (SBC) = xG2y (ycbt)
T r o n g do xGay // A D hoSc BC.
L o a i 2 : T l M G I A O D I £ M C U A D U d N G T H A N G 1fA M A T
L
PHirONG
PHANG
PHAP
Ca sd cua phaang phap t i m giao d i e m O cua dudng t h a n g
(a) va m a t phSng (a) l a xet 2 h a i k h a nSng xay r a :
n
T r i r d n g hop (a) chiJa dudng t h S n g (b) va (b) l a i c&t diicrng
t h d n g (a) t a i O.
T i m O = (a) n (b)
=> O la d i e m can t i m .
n
Trtfdng hap (a) k h o n g chiifa dUcmg t h i n g nao cat (a).
T i m ( P ) ^ ( a ) v a ( a ) n ( P ) = (d)
>
T i m O = (a) o (d)
=> O la d i e m can t i m .
n. CAC
BAI TOAM G O B A N
Bai 6
Cho tuf d i e n A B C D . Goi M , N I a n lugt la t r u n g d i e m cua A C va BC. L a y d i e m K e B D sao
cho K B > K D . T i m giao d i e m ciia h a i dudng t h i n g CD va A D v d i ( M N K ) .
9
•
De y den K B > K D
Do do t r o n g (BCD)
Ma K N c ( M N K )
•
Giai
=> K N k h o n g song song C D
K N o CD = I .
CD
( M N K ) = I (ycbt)
Taong t a xet I M c ( M N K ) , t r o n g ( A D C )
Ta CO :
AD n IM = E
=> A D n ( M N K ) = E (ycbt)
Bai 7
Cho tiJ dien A B C D . L a y d i e m M t r e n A C va h a i d i e m N va K thuf tiT nSm t r o n g cac t a m
giac B C D va A C D . D u n g giao d i e m cua CD va A D \di ( M N K ) .
HtfdTng d i n
Doc gia t u g i a i , x e m h i n h ben.
a/
CD
( M N K ) = P (ycbt)
b/ A D n ( M N K ) = Q (ycbt)
Bai 8
Cho h i n h chop tuf giac S.ABCD. L a y t r e n SA, SB va BC ba d i e m M , N , P theo t h i i t\i sao
cho M P k h o n g t h e c&t A B hay C D . T i m giao d i e m cua SC va A C v d i ( M N P ) .
Giai
ThUdng t h u d n g do ycbt t i m giao d i e m
NP o SC = K
ma NP c (MNP)
Trong mp(SAC)
ma M K c (MNP)
SC n ( M N P ) = K (ycbt)
M K o AC = H1
|
=> A C r> ( M N P ) = H (ycbt)
Bai 9
Cho m o t t a m giac A B C va m o t d i e m S d ngoai m a t p h i n g chila t a m giac. T r e n SA va
SB
ta lay hai d i e m M , N v a t r o n g m a t p h i n g (ABC) ta lay mot d i e m O. D i n h ro giao diem cua
( M N O ) v d i cac dudng t h i n g A B , B C , A C va SC.
Hi^dng d i n
Tuang t u , doc gia t u g i a i (xem h i n h ben)
A B n ( M N O ) = E (ycbt)
BC o ( M N O ) = F (ycbt)
A C n ( M N O ) = G (ycbt)
SC n ( M N O ) = H (ycbt)
10
Loal 3 : Cfll/NG MWfl B A D I £ M T R O N G K H O N G G I A N T H A N G H A N G
I. pmroNG P H A P
Co so cua p h i i o n g phap can p h a i chufng m i n h ba d i e m
trong yeu cau b ^ i t o a n l a d i e m chung cua 2 mSt phSng nao
do (chfing b a n A, B, C nSm t r e n giao t u y e n (d) cua h a i m a t
phSng do nen A, B, C t h a n g hang).
O day k h o n g l o a i triJ k h a n&ng chiJng m i n h difoc difdng
thang A B qua C => A, B, C t h i n g hang.
n. C A C B A I T O A N C O BAM
B a i 10
Xet ba d i e m A, B, C k h o n g thuoc m a t p h i n g (u). Goi D, E, F I a n l u o t l a giao d i e m ciia A B ,
EC, CA va (g). ChCifng m i n h D, E, F t h a n g hang.
Giai
De y t h a y D, E, F viTa a t r o n g m p ( A B C ) vifa d t r o n g mp(a).
Do A, B, C g (a), nen (a) va (ABC) p h a n b i e t nhau.
=> ( a ) n (ABC) = A (A chuTa D , E, F)
D, E, F t h i n g h a n g t r e n A (dpcm).
B a i 11
H a i t a m giac A B C , A B C k h o n g dong p h i n g c6 A B n A B ' = I , A C n A C = J , BC n B C = K
ChiJfng m i n h I , J , K t h i n g h a n g .
Giai
De y I , J , K I a n l u o t d t r e n h a i m a t p h i n g p h a n
(P) ^ (ABC) va (Q) = ( A ' B ' C ) .
N e n no 1^ diim c h u n g cua h a i m a t p h i n g do
I , J , K e (A) = (ABC) n ( A ' B C )
=> 1, J , K t h i n g h a n g (dpcm).
B a i 12
Cho A, B l a h a i d i e m d h a i p h i a khac n h a u doi v d i m a t p h i n g a va A B c i t a t a i O. D i t o g
hai dUdng t h i n g x'Ax, y ' B y song song n h a u theo thuf tiT c i t a t a i M va N . ChuTng m i n h M , N ,
O t h i n g hang.
Kvldng
TifOng
^
t\l:
•
fM, O,
dSn
NG(S)
[8 = (Ax; By) n (a)
=> M , N , O t h i n g h a n g t r e n (8)
11
t o a l 4 : CmiUG MWfl M Q T DtfCiNG T H A N G T R O N G KHONG G I A N
Q U A M O T D I £ M C O DINH
I.
PHirONG
PHAP,
Ca sd cua phucfng phap chuTng m i n h diXcrng t h i n g (d)
qua m o t d i e m co d i n h :
Ta can t i m t r e n (d) h a i d i e m tuy y A ; B va chuTng m i n h
2 d i e m do t h i n g h a n g v d i m o t d i e m I co d i n h c6 sSn t r o n g
khong gian.
=> (d) qua I CO d i n h (dpcm).
IL PHtfONG PHAP,
Co sd cua phiTcfng phap can thuc h i e n ba bifdc ccf ban :
n
B i : T i m dUctng t h i n g a co d i n h d ngoai mSt p h 5 n g co
d i n h (a) ma (a) chila d (liOi dong).
•
B2 : T i m giao d i e m I = a ^ d
=> I l a d i e m co d i n h ma d d i qua
m. C A C B A I T O A N C O B A N
Bai 13
Cho A , B l a h a i d i e m co d i n h t r o n g k h o n g g i a n d ve h a i p h i a khac n h a u cua m&t p h i n g co
d i n h a. X e t d i e m M luu dong t r o n g k h o n g g i a n sao cho M A n a = I va M B n a = J . ChuTng
m i n h difdng t h i n g I J luon d i qua m o t d i e m co d i n h .
Giai
Goi O = A B n (a) => O co d i n h ( v i A , B co d i n h vk
a 2 p h i a cua (a))
T a CO : mp(P) = ( M A ; M B ) n (a) = I J
De y t h a y : O e I J => O, I , J t h i n g hang.
N g h i a l a dacfng t h i n g I J d i qua O co d i n h (dpcm)
Bai 14
Cho h i n h t h a n g A B C D ( A B // C D va A B > CD). X e t d i e m S e ( A B C D ) va m a t p h i n g a luu
dong quanh A C v d i a '-^ SB = M , a n SD = N . ChuTng m i n h difdng t h i n g M N luon luon d i qua
mot d i e m co d i n h .
De t h a y dxiac n g a y M N c (SBD)
va
AC c (SAC) va M N o A C = O t h i O e
B D = (SBD) n
(SAC)
=> M N qua O co d i n h (dpcm).
12
Bai 15
Cho h a i d i f d n g t h f t n g d o n g q u y O x , O y v a h a i d i e m A , B k h o h g n S m t r o n g m a t
phing
(xOy). M o t m a t p h a n g l i f u d o n g ( a ) q u a A B l u o n l u o n c a t O x , O y t a i M , N . C h i i f n g t o M N q u a
mot d i e m co d i n h .
Giai
D e y t h a ' y k h i ( a ) q u a y q u a n h A B co d i n h n h t f n g vAn co :
( a ) n [ ( O x ; O y ) ^ (P)] = A ( q u a M , N )
1 AB CO d i n h
Nhung -j[P
ABo(p) = I e A
CO d i n h
N g h i a la d u d n g t h a n g M N = A l o u dQng nhOng
v a n qua I co d i n h . ( d p c m )
toal S : C H O N G MWfl B A O U d N G T H A « G T R O N G KHONG G I A N D O N G Q U Y
L PHUONG PHAP,
Co so cua p h i f a n g p h a p l a t a c a n c h i i f n g m i n h d U d n g thiif n h a t
qua g i a o d i e m c i i a 2 d i f d n g c o n l a i b a n g 2 budrc co b a n :
•
•
Bi : T i m (d,) o
(d^)
= O
B2 : C h u f n g m i n h (d;j) q u a O .
=> ( d i ) , (d2), (d.i) d o n g q u y t a i O ( d p c m )
Q. P H U O N G
PHAP,
C o s d c u a p h a a n g p h a p l a t a c a n chijfng m i n h
chung
doi m o t cat n h a u v a d o i m o t d t r o n g 3 m a t p h a n g p h a n
b i e t q u a 2 bifdc ca b a n :
d],
•
B i : Xac d i n h <
c: a; d j
0
da = I i
id3
\
da, d;j cz P; da ^ d3 = I2
A
dg, d , e Y; dg n d j = I3
di
\
\
a, p, y p h a n bi§t
•
B2 : K e t l u a n ( d , ) ; (da); (d;,) d o n g q u y t a i 0 = I i = I2 = I3
m. c A c
BAI TOAN C O BAM
Bai 16
C h o t i l d i e n A B C D . G o i E , F , G l a b a d i e m t r e n b a c a n h A B , A C , B D sao c h o E F n B C = I ,
E G o A D = J ( v d i I ^ C wk J
^ B). C h i j f n g m i n h C D , I G v a J F d o n g q u y .
Giai
X e t b a difcfng t h A n g C D ; I G v a J F , t a tha'y :
CD, I G e ( B D C ) va C D
• IG, J F c (EFG)
IG / 0
va I G n J F * 0
J F , C D e ( A C D ) va J F r> C D * 0
Va ba m a t p h a n g ( B C D ) , ( E F G ) , ( A C D ) l u o n p h a n b i e t
( v i I ?t c v a J ?t D ) => C D , I G , J F d o n g q u y t a i O ( d p c m ) .
13
O Cach khac
D o c g i a churng m i n h r S n g J F q u a O = I G n C D => C D ; I G v a J F d o n g quy.
B a i 17
C h o h a i t a m giac A B C , A B C sao cho A B c a t A ' B ' a E , A C cdt A C d F ; B C c a t B C d G .
a/
Chufng m i n h b a d i e m E , F , G t h S n g h a n g .
b/ C h i J n g m i n h difcfng t h a n g A A ' , B B ' , C C d o n g quy.
Gidi
a/
D e y t h a y E , F , G l a b a d i e m chung cua h a i m a t ph^ng p h a n biet
(a) ^ ( A B C ) v a (P) = ( A B ' C ) .
D o do : E , F , G e (A) = ( a ) n (P).
V a y E , F , G th^ng h a n g (dpcm).
b/ N h a n x e t n h u s a u :
: AA', B B ' cr ( E A A ' ) ; A A ' o B B ' # 0
^ B B ' , C C c ( G B B ' ) ; B B ' r^ C C * 0
Ice,
^
AA' c ( F C C ) ; C C n AA' # 0
A A ' , B B ' , C C d o n g quy t a i O (dpcm).
Chuyen
de 2 :
QUAN HE SONG SONG
t o a i 1: C H t J N G MWfl HAI DLfCJNG
THANG
SONG SONG
I. PHirONG PHAP
C o S0 c u a p h a o n g p h a p c a n t h i i c h i e n h a i hxidc CO b a n c h o d i n h n g h i a a // b
j a , b c: (a)
'a^b = 0
•
B i : K i e m t r a h a i difdng t h a n g a c u n g t r o n g m o t m a t
p h a n g h a y hifeu n g a m r a n g h i e n n h i e n d i e u do x a y r a
n e u c h u n g t r o n g 1 h i n h p h a n g n a o do. ( 1 )
•
B 2 : D u n g d i n h ly T h a l e s , t a m giac dong dang, t i n h c h a t bac cau ( t i n h c h a t cung song
s o n g \6i
difdng thiJ b a ) l a h a i c a n h c u a h i n h t h a n g , h a y h a i c a n h doi c u a h i n h b i n h h a n h ,
... de k h a n g d i n h h a i difcfng t h ^ n g do k h o n g c6 d i e m c h u n g . ( 2 )
T i f ( 1 ) v a ( 2 ) => ( y c b t )
n. C A C BAI TOAN CO BAN
B a i 18
C h o h i n h c h o p S . A B C D c6 G j , G 2 , G3, G , I a n lucft l a t r o n g t a m c a c t a m g i a c S A B , S B C ,
S C D , S D A . C h u m g m i n h tiJf g i a c G i G a G g G , l a h i n h b i n h h a n h .
14
Giai
SG,
SE
Theo t i n h chat t r o n g tarn, t a c6 : - i ,
SG3
t
[ SH
SG2
SF
2
SG4
2
SK
3
3
Dinh l y Thales va t i n h chat diTcfng t r u n g b i n h
G,G2// = - E F ; E F 7 / = i A C
'
^
3
2
• G1G2
// = G;jG4
G.G,, // = - H K ; HK// = - AC
^
'
3
2
G1G2G3G4 l a h i n h b i n h h a n h (dpcm).
B a i 19
Cho diem S d ngoai m a t phSng h i n h b i n h h a n h A B C D . X e t m S t p h d n g a qua A D c^t SB
va SC Ian lucft d M va N . Chiirng m i n h A M N D l a h i n h t h a n g .
Giai
S
D6 y thay h a i m S t phSng (a) v a (P) c6 2 d i e m M vfl N 1^ d i ^ m chung.
=> M N = (a) n (SBC)
'(a) 3 AD
ma^(SBC)3BC
iAD//BC
N
va theo each d i m g M N // A D (hoftc BC)
=> A D N M l a h i n h t h a n g day lorn A D . (dpcm)
B a i 20
Cho tuT dien A B C D . Goi M , N I a n li^gt l a t r u n g d i e m cua B C va B D . G g i P l a d i e m t u y y
tren canh A B sao cho P ?t A v a P # B. X e t 1 = P D
A N va J = PC o A M .
ChiJng m i n h r S n g : I J // C D .
Giai
Xet h a i m a t p h a n g ( A M N ) v a (PCD) c6 h a i d i e m chung l a I va J .
IJ = ( A M N )
r-.
(PCD)
'CD c (PCD)
N h i m g < MX CT (AMN)
• va MN // CD
^
I J // M N hoac C D (dpcm).
toai Z : CfltJfJG M W H DiidfiG T H A N G S O N G S O N G TfCl M A T F H A N G
L PHtrOWG P H A P ,
Co so ciia phuong phap m o t l a sii dung d i n h l y phuong giao t u y e n song song.
De chiing m i n h d // a t a can thUc h i e n h a i bade CO b a n chufng m i n h :
•
E l : Chufng m i n h d = y o p m a
•
B2 : K e t l u a n t i f t r e n d // a.
d
y r- a = a
p n a = b.
a//b
15
n . PHOONG PHAP^
Ca sd ciia phifcng phap la stf dung dieu k i e n can va du
chijfng m i n h di/dng t h i n g (d) song song vcJi m a t p h a n g
(a)
b a n g h a i btfdrc :
•
B i : Quan sat va quan l y gia t h i e t t i m dudng t h i n g ou
v i e t (A) cz (a) va chiJng m i n h (d) // (A).
•
B2 : K e t l u a n (d) // (a) theo dieu k i e n can va dii.
m. cAc
BAI T O A N C O BAM
Bai21
T r o n g tuf dien A B C D , chufng m i n h rSng dean no'i h a i t r o n g t a m G i , G2 cua h a i A A B C
va
A A B D t h i song song v6[ ( A C D ) .
Giai
A
Goi A i , A2 l a t r u n g d i e m BC va B D theo thut tiT do, t a c6 :
AG2
3
AA, ' AAg
2
AG)
Theo d i n h l y T h a l e s , t a c6 :
' 0 , 0 2 / / A , A2
B
' m a A,A2 //CD (tinh chat dUcrng trung binh)
Theo t i n h bSc cau
=>
G1G2 // CD c: (ACD)
=j.
G1G2 // (ACD)
(dpcm)
B a i 22
Cho h i n h chop S.ABCD day l a h i n h b i n h h a n h A B C D . G o i M , N l a t r u n g d i e m SA va SB.
Chijfng m i n h : M N // (SCD) v a A B // ( M N C D ) .
Giai
Theo t i n h c h a t dudng t r u n g b i n h t r o n g t a m giac
=> M N // A B , ma A B // CD
=> M N // C D
Theo dieu k i e n can va du
O
cz
(SCD)
=> M N // (SCD)
(ycbt).
Cach khac
De y M N = ( M N C D ) n (SAB) va t r o n g h a i m a t p h a n g do
chiJa theo thijf tiT cac doan t h i n g C D // A B
D
=> M N // (SCD) 3 CD (ycbt)
M N // A B va C D
TifOng tyl :
A B // M N c ( C D M N )
=> A B // ( C D M N ) (dpcm).
B a i 23
Xet h a i h i n h b i n h h a n h A B C D va A B E F k h o n g dong p h l n g . Goi M , N l a h a i d i e m thoa
AM - i AC va BN = - BF . Chufng m i n h r i n g M N // ( D E F ) .
3
3
Giai
De y t h a y M , N l a t r o n g t a m cua b a i t a m giac A B D va
A B E theo thijf t u do.
Keo d a i t h i D M o E N = P : l a t r u n g d i e m A B .
^
PE
PD
PX
PM
1
3
Theo d i n h l y T h a l e s
^
M N // E D c ( E F D C ) ^ ( D E F ) (dpcm)
D
16
Bai 24
H i n h c h o p S . A B C D c6 d a y l a h i n h b i n h h a n h A B C D , t a r n O . G o i M , N I a n \\iqt l a t r u n g
d i e m S A , S B v a x e t h e t h i J c v e c t o : 3 S I - 2 S M = 3 SJ - 2 S N = 0*. ChuTng m i n h r S n g :
a/ I J / / ( S C D )
b/ S C / / ( M N O ) .
Hvfdrng d i n
a/ i
I J // M N , M N // A B ; A B // C D
M N // C D
CD c (SCD)
=> I J / / ( C S D ) ( d p c m )
b/
AM
AO
AS
AC
S C // M O c ( O M N )
S C // ( O M N ) ( d p c m )
Bai 25
C h o A x , B y l a h a i nijfa d i T d n g t h S n g c h e o n h a u . T r e n A x l a y d i e m M , t r e n B y l a y d i e m N
sao c h o A M = B N . C h i j f n g m i n h r S n g dUcfng t h i n g chufa d o a n M N l u o n l u o n s o n g s o n g w6i m a t
p h a n g CO d i n h .
Q u a A d u n g A x ' // B y ; q u a N d i f n g N N ' // B A ; v6i N ' e A x ' . L u c d o tii g i a c A N N B l a h i n h
b i n h h a n h n e n : A N ' = B N => A M = A N '
De y A A M N ' c a n d A n e n t i a p h a n giac n g o a i A t cua STAJT
se s o n g s o n g v6i M N ' v a t i a A t
n a y co d i n h h a y A B v a A t x a c d i n h m a t p h S n g co d i n h ( P ) .
Ta
lMN'//At
CO : <
[
( M N N ' ) // ( P )
N N ' // A B
V a y : M N // ( P ) tiifc l a M N l u o n l u o n s o n g s o n g v6i m&t p h a n g co d i n h
(dpcm).
toal 3 : HAI M A T P H A N G S O N G S O N G
Dang
1 : C H Q N G MINH HAI MAT P H A N G S O N G S O N G
L PmrOHG PHAP
Co
sd cua phuong
phap
chiJng m i n h
hai mat
p h a n g fx v a P s o n g s o n g n h a u t a c a n thiTc h i e n h a i
bUdc CO b a n t r o n g k h i siJf d u n g d i e u k i e n c a n v a d u
nhu sau:
•
B i : Chufng m i n h
" m a t p h a n g ( a ) c h i i a h a i dUcJng
t h a n g a, b d o n g
q u y thijf t i f s o n g
song v d i h a i
dUoing t h a n g a', b ' d o n g q u y t r o n g m a t p h a n g P".
•
B2 : K e t l u a n ( a ) // (P) t h e o d i e u k i e n c a n v a d u .
THL; VJENTifJHglNHTHUAN
17
n . ckc
BAITOAN C OBAM
Bai 26
T r e n b a t i a c u n g c h i e u , s o n g s o n g v a Ichong d o n g p h ^ n g A x , B y , C z M n lifot l a y c a c d i e m
A ' , B ' , C s a o c h o : A A ' = B B ' = C C c 6 do d a i k h a c k h o n g . ChOfng m i n h ( A B C ) // ( A B C ) .
Giai
AA' =3 BB'
D e y : ( A B ' C ) // ( A B C ) ( d p c m )
Bai 27
C h o h i n h b i n h h a n h A B C D . Tir A v a C k e A x c a C y song song cung chieu v a khong n k m
t r o n g m a t p h S n g ( A B C D ) . Chiifng m i n h ( B ; A x ) // ( D ; C y ) .
Gi&i
Tirang t u xet h a i m a t p h i n g ( B ; A x ) v a
( D ; C y ) , thuT t a chuTa c a c c a p d u d n g
thing
d o n g quy.
fAB//CD
IAx//Cy
=> ( B ; A x ) // ( D ; C y ) ( d p c m )
Bai 28
C h o h a i h i n h binh h ^ n h A B C D v a A B E F d trong h a i m a t ph^ng khac nhau. Chilng m i n h
( A D F ) // ( B C E ) .
Giai
H a i m a t p h l i n g ( A D F ) v a ( B C E ) thiif tiT chuTa c a c c a p dirdng
t h d n g d o n g quy.
iAF//BE
AD//BC
/ A;
( A D F ) // ( B C E ) ( d p c m )
Dang 2 : CHUfNG MINH CAC Dl/dNG THANG D6NG
PHANG
LPBirONGPBAP
Ccf s d c u a p h u a n g p h a p chiifng m i n h c a c d u d n g t h i n g d i , d2, dg... d o n g p h i n g l a c a n p h a i
thiTc h i ^ n h a i bi/ d i , d2, d^j, ... dong p h i n g trong (a);
(a) phai
chufa cac giao diem cija d,, da, ds, ....
n. C A C B A I T O A N C O B A N
Bai 29
Cho tiJ dien ABCD c6 AB = AC = AD. Chufng minh rSng ba diTcfng phan giac ngo^i cdc goc
SAC. CAI), I5AB cung nSm trong mot mat phlng.
Giai
Goi A t i , At2, Ata la ba diTdng phan giac
ngoai ciia goc : fiAfc, CXt), I5A6 theo thuT t u do.
Do cac tam gidc can tai dinh A nen cac
phan giac ngoai song song vdi canh day, nen :
At, / / B C c (BCD)
A t a Z / C D e (BCD)
;At3//BDc(BCD)
At,, At2, At3 // (BCD)
=> A t , , At2, Ata dong ph^ng (trong (P) // (BCD) \k (P) qua A) (dpcm).
Bai 30
Cho hinh chop day la luc gidc deu. Chufng minh rang giao tuyen cua mat ben doi nhau thi
dong phlng.
Giai
De y thay :
(SAB) n (SED) = t, // AB, E D
(SBC) o (SEE) = ta // BC, E E
^(SCD) n (SEA) =
=>
// CD, FA
t , , ta, tg//(ABCDEF)
Vay t , , t2, tg dong ph^ng trong (a) // (ABCDEF)
va (a) qua S. (dpcm)
Bai 31
Tren bon tia phan biet Ax, By, Cz va Dt song song cung chieu, lay cac diem A', B', C , D'
sao cho AA' = BB' = CC = DD'. Chutng minh r i n g A B , B'C, CD', D A ' , A C , B'D" cung song
song vdi mat ph^ng ABCD.
Htfdng d i n
Doc gia t u giSi iMng t\l hai bai toan tren.
19
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