Chuyên đề trọng điểm bồi dưỡng học sinh giỏi hình học không gian phần 1

Hinh chop tu- giac c6 day la hinh binh hanh .1 n,>-> QDa .'JBA ,i:r:>/, Cac ban phai nho ky: Giao tuyen la duong thang chung cua hai mat phJing, CO nghla la giao tuyen la duong th5ng v

Danh cho hoc sinh khoi 12 chifong trinh chuan va nang cao

On tap va nang cao kl nang lam bai

Bien scan theo noi dung va cau true de thi cua Bo GD&OT

DU0nc3 Higc sin

Danh cho hpc sinh khd'i 12 chaong trlnh chuan va nang cao

On tap va nang cao kl nang l^m b^i

Bien soan theo noi dung cau true de thi cua Bp GD&OT

Cuon sach H I N H H Q C K H 6 N G G I A N , du(?c bien soan n h ^ mvc dfch

H I N H K H 6 N G G I A N K H 6 N G C 6 N L A M6'I L O L A N G C H O C A C B A N

H O C S I N H P H O T H O N G

Cuon H I N H K H 6 N G G I A N du<?c h | thong hoa loan hg kien thuc tu- co; ^ J

ban den nang cao H I N H K H 6 N G G I A N L 6 P 11, H I N H K H 6 N G G I A N

L O P 12 va dac bi?t L U Y $ N T H I D A I H O C

Noi dung cuon sach gom :

Chuong I : Dai cuong hinh hpc khong gian

Chuong I I : Quan h^ song song trong khong gian

Chuang I I I : Quan h$ vuong goc trong khong gian

Chuong I V : The tich khol tru, the tich khoi chop

Chuong V : Matcau, mat tru, mat non ' ^ •

Chuong V I : Bai tap tong hgp lop 12 hoc ky I „ , ' ,

Chuong V I I : Bai tap tong hop luyen thi Dai Hoc

Cuon sach duoc trinh bay ngan gon, ro rang, voi mot luong bai tap rat ?

Ion va rat day du, cac bai tap duoc giai chi tiet, chat che, de hieu Nham

giiip cac em c6 mot djnh huong de giai quyet van de ciia bai tap do

Cuon sach ditgc phan bo 6 chuang, moi chuang dugc tom tat ly thuye't day

du. Bai tap trong moi chuang duoc phan dang ro r^ng, moi dang c6 tom tat

phuong phap giai bai tap Bai tap phan bo trong moi chuang hoac moi

phan la t u de den kho, trong moi dang deu c6 long vao nhung bai tap ciia

dang thi Dai Hoc Nhiing bai tap de nham muc dich giiip cac ban nam ro ly

thuye't va phuong phap de chung minh hoac giai quyet mot van de cu the,

va t u do cac ban c6 ky nang de giai quyet nhung bai tap kho hon Tac gia

rat hy vpng quyen sach nay la nguoi ban dong hanh tot cho cac em trong

qua trinh hpc va trong nhung ky thi i

Tuy c6' gang nhieu trong qua trinh bien soan, nhung cuon sach khong

the tranh khoi nhirng thieu sot Tac gia rat mong nhan dugc y kien dong

gop chan thanh qui bau tu phia ban doc, cac em hoc sinh va cac ban dong

nghiep gan xa de nhOng tai ban Ian sau sach se hoan thien hon

Xin chan thanh cam on !

Tac gia NGUYEN QUANG SON

Cni/ONG I D A I C U O N G n l N H H Q C K H O N G G I A N

Buoc dau tien lam quen voi Hmh hpc khong gian, cac b^n cac b^n phki nho

ky cac khai ni^m va nhung tinh chat sau sau:

L K H A I N I ^ M M d D A U ^

1 MStphing:

M|t bang, m^t ban, mat nuoc ho yen l?uig, m§t san nha, cho ta hiiJti knh mpt phan cixa m$t phang M$t ph5ng khong c6 be day va khong c6 gioi h^n

De bieu dien m^t phJing ta thuong diing hinh binh hanh hay mpt mien goc

va ghi ten cua m^t phJing do vao mpt goc ciia hinh bieu dien (nhu hinh 1)

De ki hi?u mat phang , ta thuong dung chu cai in hoa hole chu cai H i L^p

dat trong dau ( ) Vi d\i: mat phSng (P), mat phSng (Q), m|t phSng ( a ) , m^t

phang (p) hoSc viet tat la mp(P), mp(Q), m p ( a ) , m p ( p ) , ho|c (P), (Q),

{«)'(P)-Hit ,p,rmrt;r

Hlnhl

Cho diem A va mat phing ( a )

Khi diem A thupc m^t phSng ( a ) , ta

noi A nam tren (a) hay m|it phSng (a) chua A, hay mat phSng (a) di qua

diem A va ki hi^u A G ( a ) , dupe bieu

dien 6 hinh 2

Khi diem A khong thupc mat phSng (a) ta noi diem A nam ngoai m^t phSng (a) hay m|t phSng (a) khong chiia diem A va ki hi?u la A g ( a ) , dupe bieu dien d hinh 3

II CAC TINH CHAT DLTOCTHl/A N H A N Tinh chat 1: Co mpt va chi mpt duong thang di qua hai diem phan bi?t

Hinh 2

Hinh 3

.A

• D

T i n h c h a t 2: C o mot v a chi mot mat

phang di qua ba d i e m idiong thang hang

T i n h c h a t 3: Ne'u mot d u o n g thang c6

hai d i e m p h a n bi^t thuoc mpt mat

thang c h u n g d o d u p e goi la giao tuyen

ciia hai mat p h a n g

V i d u : D u o n g t h a n g c h u n g d c u a hai mat p h a n g p h a n biet ( a ) v a ( p ) d u p e

gpi la G I A O T U Y E N cua hai mat phang ( a ) va v a ki hieu la d = ( a ) n ( p )

T i n h chat 6: 7 ren moi mat phang, cac ket q u a d a biet trong h i n h hpc p h a n g

Mat phSng d u p e hoan toan xac djnh khi biet no chiia hai d u o n g thSng c3t nhau:

d u p e n tam giac S A j A j , S A j A - , , , S A , ^ A , H i n h g o m d a giac

A j A 2 A 3 A „ v a n tam giac S A j A j , S A 2 A 3 S A ^ A , d u p e gpi la hinh

chop, ki h i e u la S A , A 2 A 3 A , , T a gpi S la dinh cm hinh chop, con d a giac

A , A 2 A 3 A | , la mat day cm hinh chop, cac tam giac S A j A j , S A 2 A 3 ,

S A j ^ A j d u p c gpi la cac mat hen cm hinh chop, cac d o a n thang

S A , , S A 2 , S A 3 , , S A , , d u p e gpi la cac canh ben ciia hinh chop

T a gpi h i n h c h o p c 6 d a y la tam giac, t u giac, ngij giac, , l a n l u p t la hinh chop

tam i^idc, hinh chop ti'f giac, hinh chop ngu giiic,

C h o b o n d i e m A , B , C , D k h o n g d o n g p h S n g H i n h g o m b o n tam giac

A B C , A C D , A B D va B C D gpi la hinh tie dien (hay ngMn gpn gpi la tu dien) v a

d u p e ki hi^u la A B C D C a c d i e m A , B, C , D gpi la cac dinh cm tie dicn C a c

d o a n thSng A B , B C , C D , D A , C A , B D gpi la c a c canh cm tu dien H a i canh

k h o n g di q u a mpt d i n h gpi la hai canh doi dien cua tie dien C a c t a m giac A B C ,

A C D , A B D , B C D gpi la cac mat ciia tie dien D i n h k h o n g n S m tren m a t gpi la

dinh doi dien cm mat do

%

,;V •• '

5

H i n h tu di#n c6 bon m3t la cdc tarn gidc deu gpi la hinh tii diftt deu

A S

Hinh chop tir giac

s

Hinh chop tu- giac c6 day la hinh binh hanh

.1 n,>-> QDa 'JBA ,i:r:>/,

Cac ban phai nho ky: Giao tuyen la duong thang chung cua hai mat phJing,

CO nghla la giao tuyen la duong th5ng vua thupc mat phiing nay vua thuQC

mat phang kia

Dgng toan t i m giao tuyen, thuong giao tuyen ciia nhOng cau hoi dau hay

dupe su dyng de t i m giao diem de lam bai t?ip 0 nhung cau sau Ta xet cy

thenhij'ng bai toan sau:

Cho t u giac ABCD sao cho cac canh doi khong song song v o i nhau Lay mpt diem S khong thupc mat phang (ABCD) Xac dinh giao tuyen cua :

a) M p (SAC) va m p (SBD)

b) M p (SAB) va m p (SCD) (Sj tj

c) M p (SAD) va m p (SBC)

L O I G I A I a) N h i n hinh ta de dang tha'y S la

diem chung t h u nhat, noi A C va

BD lai chung cat nhau tai O, thi

O la diem chung t h u hai, A C va

BD cat nhau la v i chung ciing thupc mat phSng day (ABCD)

Cach trinh bay giao tuyen ciia (SAC) va (SBD) n h u sau:

Ta CO S e ( S A C ) n ( S B D ) (1)

Trong mp(ABCD) gpi O = A C n B D

(4)

f O e A C , A C c : ( S A C ) , ^ ,

V\\ / = ^ O e { S A C ) n ( S B D ) (2)

O e B D , B D c z ( S B D ) V / V ; v ^

Tir (1) va (2) suy ra (SAC) n ( S B D ) = SO

b) Cau b cung t u o n g t u (SAB) va (SCD) c6 diem chung t h i i nhat la S, diem

chung t h i i hai la E, v o i E la giao diem ciia AB va C D v i hai d u a n g thSng nay

cung thuQC mat phang (ABCD) va chung khong song song v o i nhau Cach

trinh bay cung n h u cau a):

c) Cau nay cung vay S la diem chung t h u nhat, diem chung thiV hai la giao

diem ciia A D va BC, v i hai d u o n g thSng nay cung thuoc m p (ABCD) va

chiing khong song song, ta trinh bay n h u sau:

Cho t u dien A B C D Goi I , J Ian lugt la trung diem cac canh A D , BC

a) T i m giao tuyen ciia 2 m p (IBC) va m p (JAD) , <

b) Lay diem M thuoc canh AB, N thuoc canh A C sao cho M , N khong la

trung diem T i m giao tuyen ciia m p (IBC) va m p ( D M N )

b) M p ( M N I ) va m p (ABD)

c) M p ( M N I ) va m p (ACD)

L O I G I A I a) M p ( M N I ) va m p (BCD)

Goi H = M N n B C ( M N , B C c ( A B C ) ) Taco: I e ( l M N ) n ( B C D ) ( l ) //

Cho h i n h chop S.ABCD c6 day A B C D la hinh thang c6 A B song song C D

Gpi I la giao d i e m cua A D va BC Lay M thupc c^nh SC T i m giao tuyen

Cho h i n h chop S.ABCD day la h i n h binh hanh tam O Gpi M , N , P Ian l u p t la

t r u n g d i e m cac canh BC, CD, SA T i m giao tuyen cua : a) M p ( M N P ) va m p (SAB) b) M p ( M N P ) va m p (SAD) c) M p ( M N P ) va m p (SBC) • d) M p ( M N P ) va m ^ (SCD)

K e SB,SB c (SBC) " ^ ^ ^ ^ ^ ^ ^-^^^9

Vac6<| ' ' , \>M6(MNP)n(SBC)

M e B C B C c ( S B C ) ^ / v / (6) Tir (5) va (6) suy ra ( M N P ) n (SBC) - M K

d) M p ( M N P ) va m p (SCD)

Gpi H - P E n SD ( P E , SD c ( S A D ) ) , c6:

11

Cho t u dien S.ABC Lay M e S B , N e AC, l e S C sao ciio M I khong song

song vai BC, N I khong song song voi SA Tim giao tuyen ciia mat phang

(MNI) voi cac mat (ABC) va (SAB)

Cho tu dien ABCD, M la mot diem ben trong tam giac ABD, N la mot

diem ben trong tam giac ACD Tim giao tuyen cua cac cap mp sau :

a) ( A M N ) va (BCD) " M'Ti;^f7 • '

-b) ( D M N ) va ( A B C )

LOI GIAI

Trong (ABD ) goi E = A M n BD, c6:

a) Tim giao tuyen cua ( A M N ) va (BCD)

b Tim giao tuyen cua ( D M N ) va (ABC)

Trong (ABD ) goi P = D M n AB , c6:

Cho t i i dien ABCD Lay I e AB, ] la diem trong tam giac BCD, K la diem

trong tam giac ACD Tim giao tuyen cua mat phang (IJK) voi cac mat cua t u dien

DANG 2: Tim giao diem ciia duong thang va m|t phing

Phuang phap: Muon tim giao diem cua duong thiing d va m|t phling (P),

CO hai each lam nhu sau:

Cach 1: Nhung bai don gian, c6 san mot mat phaiTg (Q) chua duong thSng d

va mpt duong thSng a thuQC mat ph3ng (P) > s

Giao diem ciia hai duong thang khong

song song d va a chinh la giao diem ciia

d va mgt phang (P)

Cach 2: Tim myt mat phang (Q)

chua duong thang d, sao cho de

dang tim giao tuye'n voi mat phang

(P) Giao diem ciia duong thang d

va mat phang (P) chinh la giao diem

cua duong thang d va giao tuyen a

vua tim

Cho tu dien A B C D Goi M, N Ian lugt la trung diem cua A C va B C K la

dien-tren BP sao cho K D < KB Tim giao diem cua C D va A D voi mp (MNK)

Vi

LOI G I A I

Tim giio diem ciia C D voi mp (MNK)

Cac ban Je y N K C D cung thuoc mat ph^ng (BCD) va chiing khong song

song ne 1 hai duong thang nay se cat nhau tai mot diem I, nhung N K lai

thuoc mat phang (MNK) suy ra I thuoc mat phSng (MNK)

Vay I chinh la giao diem cua C D va (MNK)

Ta ccS the trinh bay nhu sau :

Trong mp (BCD) goi I = C D n N K

l e C D

l e N K , N K c ( M N K )

=>I = C D n ( M N K )

-Tim giao diem ciia A D voi mp (MNK)

Chon mat phang ( A C D ) chiia A D

Sau do tim giao tuye'n ciia (ACD) va (MNK), ta trinh bay nhu sau :

Cho tu dien A B C D Tren A B , A C , B D Ian luot lay 3 diem M , N , P sao cho

M N khong song song voi B C , MP khong song song voi A D Xac djnh giao diem cua cac duong thing B C , A D , C D voi mp (MNP)

Tim j^iao diem cua CD vii iiip(MNP) • • •

l e A D , A D c : ( A C D )

N e A C , A C c = ( A C D ) Goi J = N I n C D (Voi N I , C D c ( A C D ) )

J e C D

Ta CO

J e N I , N I c ( M N P ) •J = C D n ( M N P ) Cho tu di^n A B C D Tren A C va A D lay hai diem M, N sao cho M N khong song song voi C D Goi O la diem ben trong tam giac B C D

a) Tim giao tuye'n cua (OMN ) va (BCD ) b) Tim giao diem cua BC voi (OMN) c) Tim giao diem cua BD voi (OMN)

LOI GIAI a) Tim giao tuye'n ciia (OMN ) va (BCD ):

T a c 6 : O e ( O M N ) n ( B C D ) (l) Trong ( A C D ) , goi I = M N n C D

b) Tim giao diem cua BC voi (OMN):

Gpi P = BCnOl(BC,OIc:(BCD))

Vay : P = BC o ( OMN )

c) Tim giao diem cua BD voi (OMN);

Trong (BCD), goi Q = BD n OI

Vay : Q = BD n (OMN)

Cho tu di^n ABCD, lay M thupc AB, N thupc AC, I la diem thupc mien trong

cua tam giac BCD Xac dinh giao diem cua cac duong thMng BC, BD, AD,

Tir (1) \i (2): H1 = (M NI) n (ACD)

Tim giao diem ctia BD va mp(MNl)

Cho tu dien ABCD Gpi M, N Ian iupt la trung diem cac canh AC, BC

Tren canh BD lay diem P sao cho BP = 2PD Lay Q thupc AB sao cho QM cat BC Tim:

a) Giao diem ciia CD va mp (MNP)

b) Giao diem ciia AD va mp (MNP)

c) Giao tuyen ciia mp (MPQ) va mp (BCD)

d) Giao diem ciia CD va mp (MPQ)

d) Giao diem ciia AD va mp (MPQ)

Gpi E = C D n NP(CD,NP c mp(BCD)) <

E e C D Co: , ^ =>E = C D n ( M N ] [ E e N P , N P c ( M N P ) ^ b) Giao diem ciia AD va mp (MNP)

Tim giao tuyen ciia m p ( A C D ) va mp(MNP)

d) Tim giao diem ciia CD v a mii ^ T ^ V J H N T i M H O I N f i Tl lUAN

1 7

Gpi L = K P O C D ( K P , C D C ( B C D ) ) , C 6 :

L e C D , ^

e) Tim giao diem ciia A D va mp ( M P Q )

Dau tien tim giao tuyen ciia m p ( A C D ) va m p ( M P Q )

Co M , L la hai diem chung ciia ( A C D ) va ( M N P )

Suy ra M L = ( M P Q ) n ( A C D )

Goi T = A D n M L ( A D , M L c ( A C D ) ) T = A D n ( M P Q ) ( v i M L C ( M P Q ) )

ifc *.<fi, ; 'A

11 ,

Cho hinh chop S.ABCD c6 AB va CD khong song song Gpi M la mot diem

thupc mien trong ciia tam giac SCD

a) Tim giao diem N ciia duong th^ng CD va mp (SBM)

b) Tim giao tuyen ciia 2 mat phSng (SBM) va (SAC)

c) Tim giao diem I ciia duong th^ng BM va mat phSng (SAC)

d) Tim giao diem P ciia SC va mp(ABM), t u do suy ra giao tuyen ciia hai

b) Tim giao tuyen ciia 2 mp (SBM) va (SAC)

Cac ban de y mat ph^ng ( S B N ) cung

1 8

l £ S O , S O c ( S A C ) ^

d) Tim giao diem P ciia SC va mp (ABM), suy ra giao tuyen ciia hai m p (SCD)

va (ABM)

Tim giao tuyen ciia hai mat phang (AMB) va (SAC) ,

Ta CO A va I la hai diem chung ciia hai mat phSng (ABM) va (SAC)

Vay: ( A B M ) n ( S A C ) = A I Gpi P la giao diem ciia A I va SC, thi P la giao diem ciia SC va m p ( A B M ) Hai m p (SCD) va (ABM) c6 M va P la hai diem chung, vay giao tuyen ciia chiing la PM

Cho t i i giac ABCD va mot diem S khong thupc mp (ABCD ) Tren doan

AB lay mpt diem M ,tren doan SC lay mot diem N ( M , N khong trung voi cac dau m i i t )

a) T i m giao diem ciia duong thSng A N voi mat phSng (SBD) f b) Tim giao diem ciia duong thang M N voi mat phang (SBD)

LOI GIAI a) Tim giao diem ciia duong thang A N voi mat phang (SBD)

• Chpn mp phu (SAC) =3 AN J ' f i l l ;

Tim giao tuyen ciia ( SAC) va (SBD) Trong (ABCD) gpi P = AC n BD / / / ! \ Vay S va P la hai diem chung cua hai / '' ' A ' ^ ^

Cho hinh chop S.ABCD Gpi O la giao diem ciia AC va BD M , N , ? Ian lupt

la cac diem tren SA , SB , SD

a) Tim giao diem I ciia SO voi mat phang ( MNP ) b) Tim giao diem Q ciia SC voi mat ph^ng ( M N P ) ^ _ _ _ _

1 9

b) Tim giao diem Q ciia SC voi mp (MNP )

• Chon mp phu (SAC) 3 SC ^

• Tim giao tuyen ciia ( SAC ) va (MNP)

Cho t u dien ABCD Goi M , N la hai diem tren AC va A D O la diem

trong tam giac BCD Tim giao diem cua :

a) M N v a ( A B O )

b) AO va (BMN )

ben

LOI GIAI a) Tim giao diem ciia M N va (ABO ):

• Chon mp phu (ACD) =) M N Tim giao tuyen ciia (ACD ) va (ABO)

Ta c6: A la diem chung cua (ACD ) va (ABO)

Trong (BCD) goi P = BO n DC

P e B O , B O c ( A B O )

P e C D , C D c ( A C D ) /

=^ P G (ABO) n (ACD) (2) t

Tir (1) va (2) ^ (ACD) n (ABO) = AP

- Trong (ACD), goi Q = AP n M N , c6:

Tim giao tuyen cua (ABP ) va (BMMj

Ta c6:B la diem chung cua (ABP ) va (BMN) (3)

b) SD va (IJK )

c) SCva (IJK)

LOI GIAI a) Tim giao diem cua IK va (SBD)

• Chon mp phu (SAK) chua IK Tim giao tuyen ciia (SAK ) va (SBD) Taco: SG(SAK)r^,(SBD) ( l ) Trong (ABCD) goi P = AK n BD

b) Tim giao diem cua SD va (IJK ) :

• Chon mat phang phu (SBD) chua SD Tim giao tuyen ciia (SBD ) va ( I J K )

Theo cau a) ta c6: Q e (SBD) r^ ( I J K ) (3)

Trong (ABCD), goi M = J K n BD M la diem chung cua ( I J K ) va (SBD) (4)

T u ( 3 ) v a ( 4 ) = v ( I J K ) n (SBD) = Q M • ' ,

• Trong (SBD) gQi N = Q M m SD

c) T i m giao diem ciia SC va ( I J K ) : ' ' "'* '

• Chpn m p phu ( S A C ) Z3 SC T i m giao tuyen cua ( S A C ) va (IJK) !,

, [ l € S A , S A c : ( S A C ) ^ / / ^ , =, I 1 u T

C(?i E = A C n J K ( v i A C J K c ( A B C D ) ) Vay E e ( I J K ) n ( S A C ) (6) > ,

T u (5) va (6) ( I J K ) n ( S A C ) = I E >; 0 / = " ! '

• Trong (SAC), gpi F = I E n S C

Cho t u d i ^ n S.ABC Goi I , H Ian lupt la trung diem ciia SA, AB Tren canh

SC lay diem K sao cho CK = 3 S K

FB a) T i m giao diem F cua BC voi m p (IHK) Tinh ti so

FC b) G Q I M la trung diem cua doan I H T i m giao diem cua K M va m p ( A B C )

LOI G I A I a) T i m giao diem ciia BC voi m p (IHK)

Ta t i m giao tu)'e'n ciia (ABC) va (IHK) trudc

AE NF

FB FB 1 i/iae)zvCc Ket luan: FB

P A N G 3: T i m thiet dign ciia hinh (H) khi cit bai mat phang (P) ^ ' ' Thie't di^n la phan chung ciia mat phang (P) va hinh ( H )

Xac dinh thiei di^n la xac dinh giao tuyen ciia mp (P) voi cac mat ciia hinh (H)

Thuong ta t i m giao tuyen dau tien ciia mat phSng (P) v a i mpt mat phang (a) nao do thupc hinh (H), giao tuyen nay de t i m duoc Sau do keo dai giao tuyen nay c l t cac canh khac ciia hinh (H), t u do ta t i m dupe cac giao tuyen tiep theo Da giac gioi han boi cac doan giao tuyen nay khep kin thanh mpt thiet dien can t i m

Thong qua cvi the nhung bai tap sau thi cac b^n se hieu ro hom

Cho hinh chop S.ABCD Gpi M la mpt diem trong tam giac SCD

a) T i m giao tuyen ciia hai m p (SBM) va (SAC) * '-^'f'

b) T i m giao diem ciia duong thSng B M va m p (SAC) ""^'^^^

c) Xac djnh thie't dien ciia hinh chop khi cat bai m p (ABM)

rr'i'

LOI G I A I a) T i m giao tuyen ciia hai m p (SBM) va (SAC).'5V''' *

Ket luan : thiet dien can tim la tu giac ABIJ

Cho tu di^n A B C D Tren AB, A C lay 2 diem M, N sao cho M N khong

song song B C Goi O la 1 diem trong tarn giac B C D

a) Tim giao tuyen (OMN) va (BCD)

b) Tim giao diem cua D C , BD voi (OMN)

c) Tim thiet dien ciia mp(OMN) voi hinh chop

LOI GIAI a) Tim giao tuyen (OMN) va (BCD)

[ H e M N , M N c ( M N O ) Trong mp (ABC), gpi H = M N n B C , c6: ^ '

[J € H O , H O c ( M N O ) H

r:^J = C D n ( M N O ) c) Tim thiet dien cua mp(OMN) voi hinh ch Theo each dung diem o cau a) va b) thl : ( A B C ) n ( M N O ) = M N , ( A B D ) n ( M N O ) ( A C D ) r ^ ( M N O ) = NJ, ( B C D ) n ( M N O ) = IJ

Vay thiet dien can tim la tu giac MNJI

Cho tu dien S A B C Goi M € A S , N e ( S B C ) , P e ( A B C ) , khong c6 duong thang nao song song

a) Tim giao diem cua M N voi (ABC), suy ra giao tuyen cua ( M N P ) va (ABC) b) Tim giao diem cua AB vol ( M N I ' )

c) Tim giao diem cua NI^ voi (SAB)

d) Tim thiet dien ciia hinh chop cat boi mat phang ( M N P )

LOI GIAI a) Tim giao diem cua M N voi (ABC) 5 Chpn mat phang phu (SAH) chua M N / Tim giao tuyen cua

mp(SAH) va mp(ABC)

A e ( A B C ) n ( S A H ) ( l ) Trong mp(SBC), goi: /

a) Tim giao diem ciia IJ voi mp (ABC)

b) Tim giao tuyen ciia mp (IJK) voi cac mat ciia hinh chop Tu do suy

ra thiet dien ciia mp(IJK) cat boi hinh chop ' •

LOI GIAI a) Tim giao diem ciia IJ voi mp (ABC) ^ ''

E = HK n A C ( H K , A C c ( A B C ) ) Tim giao tuyen ciia mp(IJK) va mp(SBC)

fFeDJ,DJc(lJK) , , ,

] \€ IJK n SAB

FeSB,SBc(SAB) v / v i

Tu (3) va (4): FI = (IJK) n (SAB) Tim giao tuyen ciia mp(IJK) va mp(SAC)

Tu (5) va (6): LE = ( I J K ) n (SAC) Ket luan: Tu each tim giao diem cua mat phSng (IJK) voi cac canh ciia hinh

chop S.ABC, suy ra thiet di?n ciia mp(IJK) ck boi hinh chop la tii giac DFLE

(4)

•l! (XI i I J- ' -

L G ( l J K ) n ( S A C ) (5)

Ee(lJK)n(SAC) (6)

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh tarn O Goi M , N, I

Ian lugt nam tren ba canh AD, CD, SO Tim thiet di§n ciia hinh chop voi mat

phang (MNI)

LOI GIAI Trong (ABCD), goi: ] = BD n M N ,

K = M N n A B , H = M N n B C '

Trong (SBD), goi Q = IJ n SB

Trong (SAB), goi K = KQ n SA

Trong (SBC), goi P = QH n SC

Vay : thiet dien la ngu giac MNPQR

Cho hinh chiSp S.ABCi") Goi M, N , I ' Ian liiot la tnmg diem lay tren AB

AD V a SC

Tim thiet dien ciia hinh chop voi mat phang (MNP)

LOI GIAI Trong (ABCD), goi:

E = MN n DC,

F = MN BC

Trong (SCD), goi Q = EI' n SD

Trong (SBC), goi R = FP,",SB

Vay : thiet diC-n la ngu giac MNPQR

DANG 4: Chung minh ba diem thing hang, ba duang thing dong qui,

chung minh mpt diem thupc mgt duang thing co dinh^

Phuang phap: * ^'

Muon cliiing minh ba diem A, B, C thang

hang, ta chirng minh ba diem do Ian krot

thupc hai mat phang phan biet (a) va (p),

thi suy ra ba diem A , B, C nam tren giao

tuye'n ciia (a) va ((i), nen chung thang hang

PHl/ONG PHAP CHLfNG MINH BA D l / O N G THANG DONG QUY:

Ta tim giao diem cua hai duong thang trong ba duong thang da cho, roi chung minh giao diem do nam tren duong thang thii ba Cu the nhu sau: Cach chung minh 3 duang thing a, b, c dong quy tai mot diem, f Chon mot mat phing (P) chua duong thang (a)va (b).Goi I = ( a ) n ( b ) Tim mot mat phing (Q) chua duong thang (a), tim mot mat phang (R) chua duong thing (b),saocho (c) = ( Q ) n ( R ) :

Vay: 3 duong thing (a),(b),(c) dong quy tai diem I (a),(b)c:mp(P)

( a ) n ( b ) = I

m p ( P ) n m p ( Q ) = (a) ^ ( a ) n ( b ) n ( c ) = I

m p ( P ) n m p ( R ) - ( b )

m p ( Q ) n m p ( R ) = : ( c )

Cho tu dien S.ABC Tren SA, SB, va SC Ian luot lay cac diem D, E, F sao cho

DE cit AB tai I, EF cit BC tai j , FD cit CA tai K Chiing minh ba diem I , J, K thang hang^

LOT^GIAI

I = A B n D E ( AB,DE c ( S A B ) ) Co: AB,ABe(ABC)

l 6 D E , D E c ( D E F ) Ie(ABC)n(DEF) (l)

K - A C n D F ( A C , D F C ( S A C ) )

K e A C , A C C ( A B C ) KeDF,DFc:(DEF)

K 6 ( A B C ) n ( D E F ) (2)

J = BCnEF(BC,EFc(SBC)) ] € B C , B C c ( A B C ) =^Je(ABC)n(DEF) JeEF,EFc(DEF)

Co

Tu (1) (2) va (3) suy ra 3 diem I , J, K thing hang

29

Cho tii di#n ABCD c6 G la trpng tam tam giac BCD Gpi M , N, P km iugt

la trung diem ciia AB, BC, CD

a) Tim giao tuyen ciia (AND) va (ABP) n v t i b i

-b) Goi I = AG n MP, ] = CM n A N Chung minh D, I , J thing hang

Ket luan vay ba diem D , I , J thang hang '

Cho hinh binh hanh ABCD S la diem khong thuQc (ABCD), M va N Ian

lugt la trung diem cua doan AB va SC

a) Xac djnh giao diem I = A N n (SBD) v

b) Xac djnh giao diem J = M N n (SBD)

c) Chung minh I, ], B thcing hang

it]

LOI GIAl a) Xac dinh giao diem I = A N n (SBD ) ' '

• Chon mp phu (SAC) chua AN ' , y\

Tim giao tuyen cua (SAC ) va (SBD):

Trong mat phang (ABCD) gpi O la giao diem ciia AC va BD Hai mat phing

(SAC) va (SBD) c6 hai diem chung la S va O -AA } 'o iXtpii \

• Trong (SAC) goi I = A N n SO Vay: I = A N n ( SBD) '[

b) Xac djnh giao diem J = M N o (SBD)

• Chpn mp phu (SMC) chua M N Tim giao tuyen ciia (SMC ) va (SBD):

Trong (ABCD) goi K = MC n BD

Hai mat phiing (SMC) va (SBD) c6 hai diem chung la S va K

Vay: (SMC) n (SBD) = SK

• Trong (SMC), goi J = M N n SK

Vay J = M N n ( SBD) ^ l;:J*ai/

c) Chung minh I , J , B thing hang g

Ta CO : B la diem chung cua (ANB) va ( SBD) l€SO,SOc(SBD)

V , = 5 l 6 ( S B D ) n ( A B N )

l e A N , A N c ( A B N ) ^ ' ^ ^^ ' ^

Je(SBD)n(ABN)

j6SE,SEc(SBD) ' J e M N , M N c ( A B N ) '

Tu (]) (2) (3) suy ra ba diem B , I , J thang hang

if')

LOI GIAI a) Tim giao diem K = IJ n (SAC)

• Chon mp phu (SIB) chua IJ

• Tim giao tuyen cua (SIB ) va (SAC) Se(SBl)n(SAC) ( l )

Trong (ABCD) gpi E = AC n B I , c6:

E e A C c ( S A C )

E € BI c (SBI)

Tu (1) \'a (2) (SBI) n (SAC) = SE Ql^Z)

Trong (SIB) gpi K = IJ n SE

T u ( 3 ) , ( 4 ) , ( 5 ) , ( 6 ) siiy ra bcVn diem A, K, L, M cimg thuoc giao tuye'n ciia

hai mat phang (SAC) va (AJO)

Vay: A, K, M thang hang

• Chon m p p h u (SBD) chua B N

• T i m giao tuyen cua (SBD ) va (SAC)

T r o n g ( A B C D ) goi O = A C n BD

/ H a i mat phang (SAC) va (SBD) c6 hai diem chung la S va O V^y giao tuye'n

cua chiing la SO U = , t , l

• Trong (SBD), gpi I = B N n SO

V ^ y giao tuye'n cua c h i i n g la SK

hai mat phang ( B C N ) va (SAC) => Ba diem C, I , J ciing thupc giao tuyen ciia hai mat phcing (BCN) va (SAC) Ket luan C, I , J thiing hang

Cho hinh chop S.ABCD Goi M , N, P Ian l u o t la t r u n g d i e m ciia SA, SB,

SC Goi !-: = A B n C D , K = A D n B C a) T i m giao tuye'n cua ( S A C ) n ( S B D ) , ( M N P ) n ( S B D ) b) T i m giao d i e m Q cua d u o n g thang SD v o i mat phang ( M N P ) c) Goi H = N M n PQ C h u n g m i n h 3 diem S, H , E t h ^ n g hang • - ^ d) C h u n g m i n h 3 d u o n g thang SK, Q M , N P dong q u i

T i m giao tuyen ciia ( M N P ) n ( S B D )

T r o n g m p ( S A C ) goi F = M P n S O , co: , i riiffrt

T u (*),(**),(***) suy ra ba d i e m S , H , E thupc giao tuyen ciia hai m | t

phiing (SAB) va (SCD) nen ba diem S, H , E t h i n g hang , /

Cho h i n h chop S.ABCD c6 day la h i n h binh hanh G p i M la trung d i e m

cua canh SD, I la d i e m tren canh SA sao cho A I = 215

Gpi giao d i e m K cua I M v o i mat p h i n g (ABCD) Tinh t i so KD/KA Gpi N

la trung diem BC T i m thiet di?n cua hinh chop S.ABCD c i t boi mat ( A M N )

Tir d o s u y ra thiet d i ^ n ciia

h i n h chop S.ABCD b j c i t b o i mSt ( A M N ) la t i i giac A M P N

Cho t i i d i ^ n A B C D G p i P va Q Ian l u p t la nhCrng d i e m tren hai doan thang BC va B D , M la d i e m tren doan A C Gia s i i k h o n g t o n tai song song trong hinh ve ciia bai toan -.|,/. ,, i\,,,,

a) T i m giao d i e m ciia d u o n g thang A B va m a t phang ( M P Q ) Suy ra giao

d i e m N ciia d u o n g thang A D va mat phang (MPQ) b) PQ cat C D tai I T i m giao tuyen ciia hai mat phang ( M P Q ) v o i mat phang (ACD) N h a n xet g i ve vj t r i ciia M , N , I?

c) DP va CQ c i t nhau tai E, M Q va N P cJt nhau tai F Chiinp to A, E, F thang hang

L O I G I A I a) T r o n g m p ( A B C ) gpi H = A B n M P , c6:

H Q H Q c i t A D tai diem N , t h i N la giao d i e m ciia A D va (MPQ)

b) M va I la hai diem chung cua hai mat phSng (MPQ) va (ACD)

Vay giao tuye'n cua (ACD) va (MPQ) la duong thing MI

Vi N 6 (MQP) o (ACD) =^ N € M I Vay ba diem M , N , I th^ng hang

c) Vi ba diem A , E , F la ba diem chung cua hai mat ph^ng (ADP) va (ACQ)

•" Nen chiing thupc giao tuye'n cua (ADP) va (ACQ)

Ket lu^n ba diem A , E, F th3ng hang "-^

BAI TAP TONG HOP CHl/ONG I

Cho hinh chop S.ABCD vdi ABCD la hinh binh hanh Gpi M la diem bat

ky thupc SB, N thupc mien trong tam giac SCD ^

a) Tim giao diem cua M N va mp (ABCD) ,_

b) T i m S C n ( A M N ) , S D n ( A M N )

c) r i m S A n ( C M N )

LOIGIAI \

a) Tim giao diem ciia M N va mp (ABCD)

Gpi I = SN n CD(vi SN,CD c (SCD)) Chpn mat phSng (SBI) chua M N

s

Ta CO B va I la hai diem chung cua hai

mat ph3ng (SBI) va (ABCD)

Dau tien ta tim giao tuye'n cua mat phc^ng (SAC) va (SBI)

Gpi O = AC n BI (vi AC, BI c (ABCD))

Ta c6: S va O la hai diem chung ciia hai mat phling (SAC) va (SBI)

Vay: (SBl)n(SAC) = SO „ ^ , , ,

Gpi E = SO n M N (vi SO, M N c (SBl))

Chpn mat ph^ng (SAC) chua SC

Tim giao tuyen cua (SAC) va (AMN)

Ta CO K va N la hai diem chung cvia hai mat phSng (AMN) va (SCD)

' 4 , •

V^y ( A M N ) n ( S C D ) = KN Gpi P la giao diem ciia K N va SD Suy ra P cung la giao diem ciia SD va mp(AMN)

c) Tim giao diem cua SA va mat phang (CMN)

Chpn mat ph^ng (SAC) chua SA Tim (SAC)n(CMN) Taco C e ( S A C ) n ( C M N ) (3)

Theo cau b) E = SO n M N (SO, M N c (SBl)), c6:

Cho hinh chop S A B C D c6 day A B C D la hinh thang voi A B song song C D O

la giao diem ciia 2 duong cheo, M thupc S B ' M ' i a) Xac dinh giao tuyen ciia ( S A C ) n ( S B D ) ; ( S A D ) n ( S B C ) ' '

b) Tim giao diem ciia S A n ( M D C ) ; S O n ( M D C ) ' •

L O I G I A I

a) Xac djnh giao tuyen ciia ( S A C ) n ( S B D ) ' ' '

Ta CO S la diem chung thu nha't va O la diem chung thu hai ciia 2 mat phang

( S A C ) va (SBD) (0 ci

37

-Cho hin! chop S.ABCD c6 ABCD la hinh binh hanh tarn O Gpi M , N Ian

lug-t la trung diem cua AB, SC

a) T i m l = A N n ( S B D )

b) TimK = M N n ( S B D )

c) Tmh ti so

K N d) Chung minh B, I , K thang hang Tinh ti so

IK LOIGIAI a) Tim giao tuyen ciia mp(SAC) va mp(SBD) i

' l€ A N , A N c ( A B N ) ;

= > l G( A B N ) n ( S B D ) (3)

B e ( A B N ) n ( S B D ) (4) Tir (3) va (4) BI = (ABN) n (SBD);

K = B I n M N :, fuwi^ A\

rr^, U • - K M c) Tmhtiso — GQI Q trung diem cua A I

Ta CO AQ = QI = I N (Vi I la trong tarn cua tam giac SAC)

Ta c6: MQ la duong trung binh cua tam giac AABI

=^ MQ // Bl

Ta co: IK la duong trung binh ciia tam giac ANMQ

V|y K la trung diem ciia M N

K M , Suy ra = 1 ' K N d) Theo each tim giao tuyen ciia cau b) thi 3 diem B, K, I thiing hang

b) Mp (KMN) cat AB tai L Tinh ti so , LA

LB LOI GIAI a) Tim thiet di?n ciia hinh chop voi mp (KMN)

Trong mat phSng (SAC), gpi I la giao diem ciia K M va AC Trong mat phSng (ABC), gpi L la giao diem cua I N va AB Ket luan thiet dien can tim la MNLK

3 9

b) Mp (KMN) cit AB tai L Tinh ri so

Trong mat phiing ( S A C ) , ke A E // K M vai E thupc SC

Ta CO K M la duang trung binh ciia tarn

giac SAE nen M trung diem cua SE

Doan SC dugc chia lam 5 phan, S M

chiem 2 phan, M C chiem 3 phan suy

Tu (1) va (2) c6:

i A B = I AL c> 2AB = SAL 2(LA + LB) = SLA « 2LB = SLA LA 2

LB ~ 3 Cho tu di?n ABCD Goi I , J la trung diem cua AC, BC Lay K thupc canh

BD sao cho BK = 2KD

a) Tim E = CD n (IJK) Chung minh DE = DC

b) Tim giao diem F = AD (IJK) Chung minh FA = 2FD

c) Tim thiet dien cua tu di?n ABCD voi mp (IJK) Xac djnh hinh tinh ciia

Trong ABCE ke DP // EJ

Trong ABDP c6 JK // PD nen:

40

LOIGIAI

BJ_BK = 2 =>BJ = 2JP =j>CI = 2JP

JP KD Tir do suy ra DP la duong trung binh cua tam giac CEJ

Suy ra D trung diem ciia CE M*-'' b) Tim giao diem F = AD n (IJK) Chung minh FA = 2FD

Vi IE, A D c ( A C D ) G 9 i F = I E n A D B

Ma I E c ( l J K ) ^ F = A D n ( l J K ) Xet trong tam giac ACE c6 F la giao /

diem ciia 2 duong trung tuyen AD va P

EI Suy ra F la trpng tam ciia AACE

Vay FA - 2FD c) Tim thiet di^n ciia tir di#n ABCD voi mp (IJK) Xac djnh hinh tinh ciia thiet dien

( I J K ) n ( A B C ) = IJ; ( I J K ) n ( B C D ) = JK ( I J K ) n ( A B D ) = KF; ( I J K ) n ( A C D ) = FI

Thiet di^n can tim la tu giac IJKF

Tir (1) va (2) suy ra tir giac IJKF la hinh thang

(2)-Cho tu di?n S.ABC Tren SB, SC Ian lugt lay 2 diem l, J sao cho IJ khong

song song voi BC Trong tam giac ABC lay mpt diem K

a) Xac djnh giao tuyen ciia 2 mp (ABC) va (IJK)

b) Xac djnh giao diem ciia AB, AC va (IJK)

c) Tim giao tuyen ciia (SAB) va (IJK) ' rinifl oilD

d) Tim giao diem cua BC, IJ voi mp (SAK) '" nnxb J y c l Urui n'ki 88

e) Xac djnh thiet dien ciia mp (IJK) voi tit dien S.ABC

LOI GIAI a) Xac djnh giao tuyen ciia 2 mp (ABC) va (IJK)

D 6 l J , I J c ( l J K )

Gpi D = IJnBC(vi IJ, BC c (SBC)), c6:

r ^ D e ( l J K ) n ( A B C ) (l)

K € ( l J K ) n ( A B C ) (2) Tir(l)va(2) (lJK)n(ABC) = D K

D e B C , B C c ( A B C ) il A H,

41

b) Xac djnh giao diem cua A B , A C va (IJK)

Goi E va F Ian lupt la giao diem ciia

AB, AC voi DK ( vi AB , A C , DK ciing

t h u Q C mat ph^ng (ABC )) Ngoai ra DK

lai t h u Q C mat phJng (IJK):

Vay: AB n mp(lJK) = E; A C n mp(lJK) = F

c) Tim giao tuyen ciia (SAB) va (IJK) A 4,

Ta CO I va E la hai diem chung ciia hai

mat phiing (SAB) va (IJK) nen:

e) Xac dinh thie't dien ciia mp (IJK) voi tu dien S.ABC

Theo each dung diem 6 nhii-ng cau tren ta c6:

(IJK) n (ABC) = EF; (IJK) n (SAC) = FJ

(IJK)n(SAB) = IE; (IJK)n(SBC) = JI

Vay: thie't di^n can tim la tu giac IJFE

'in I

•L = IJn(SAK)

Cho hinh chop S.ABCD c6 ABCD la hinh thang, day Ion la AB Tren SA,

SB Ian lugt lay 2 diem M, N sao cho M N khong song song voi AB Gpi

O = AC n DB

a) Tim giao diem ciia duong thang AB voi mp (MNO)

b) Tim giao tuyen ciia mp (MNO) voi cac mat (SBC) va (SAD)

c) Xac djnh thie't di^n cua (MNO) voi hinh chop S.ABCD

d) Goi K la giao diem ciia hai giao tuyen 6 cau b, E = A D n BC Chung minh

3 diem S, K, E thgng hang

LOIGIAI a) Tim giao diem ciia duong thSng AB voi mp (MNO)

Gpi H - A B n M N ( v i AB, M N c (SAB)) (s|i) (£)'>ivr^r)w

Vliy: H e Ap

H e M N , M N c (MNO)

=>H = A B n ( M N O ) b) Tim giao tuyen ciia mp (MNO) vol cac

m i t phSng (SBC) va (SAD)

G(?i F = B C n H O ( B C , H O e ( A B C D ) ) , c 6 : fFeBC,BCc(SBC) , ^ , , , ,

^ ' , = ^ F 6 M N O n(SBC ( l

F 6 H O, H O c ( M N O ) ^ ' ^ ' ^ '

N € ( M N O ) n ( S B C ) (2) Tir ( 1 ) va (2) (MNO) n (SBC) = FN Trong mp(ABCD), gpi G = AD n HO ,c6:

f G € A D , A D c ( S A D )

G e H O , H O c ( M N O ) ^ ' ^ ' ^ ' Co: M e ( M N O ) n (SAD) (4)

Tu (3) va (4) (MNO) n (SAD) = MG c) Xac djnh thie't dien ciia (MNO) voi hinh chop S.ABCD

Theo each dyng diem 6 nhirng cau tren, ta c6:

( M N O ) n ( A B C D ) = GF; ( M N O ) n ( S B C ) = FN

( M N O ) n ( S A B ) = N M ; ( M N O ) n ( S A D ) = MG Vay: thie't dien can tim la t i i giac MNFG

d) Chung minh 3 diem S, K, E thang hang

E = A D n B C , A D C ( S A D ) , BCc(SBC) ;

= > E 6( S A D ) n ( S B C ) (»)

K = GM n FN, GM c (SAD), FN c (SBC) '^^^A ^ J O S A :m -f

=> K 6 (SAD)n(SBC) (* *) • S€(SAD)n(SBC) :

Tu (*)(* *)(* * *) suy ra ba diem E, K, S thupc giao tuyen ciia hai mat phang (SAD) va (SBC) nen ba diem E, K, S thang hang

b) T i m giao diem H cua M K va mp (BCD) Chung minh K la trpng tarn cua

tam giac A B H

c) Tren B C lay diem N, tim giao diem P, Q ciia C D va A D vai mp (MNK)

d) Chung minh 3 duang thang M Q , N P , B D dong qui

^fe* L O I G I A I a) Xac dinh giao tuyen ciia (AKM) va ( B C D )

Mat phang ( A B G ) cung chinh la mat phang (AKM)

Tim giao tuyen cua (ABG) va ( B C D )

b) Tim giao diem H ciia M K va mp ( B C D )

Trong mp(ABG) gpi H = MK n BG , c6:

H e M K ,

^ ^ / ^ , ^ H = M K n ( B C D )

H € B G , B G c ( B C D ) ^ '

C h u n g minh K la trpng tarn ciia tam giac A B H

Vi K la trpng tam cua tam giac A C D nen

K chia doan A G thanh ba phan bang nhau

Goi L la diem do'i xung ciia K qua G

Vay K la trung diem cua A L

' Trong A A B L , M K la duong trung binh

cua tam giac

Ta c6: ABGL = A H G K (g.c.g) => BG = H G

Vay K la trong tam cua tam giac A B H

c) Tim giao diem P, Q ciia C D va A D voi mp (MNK)

Trong mp(ABC) goi E = M N n A C

Trong m p ( A C D ) duong thSng E K cSt C D va A D Ian lupt tai P va Q , thi P va

Q chinh la giao diem ciia C D va A D voi m p ( M N K ) a i,,^'u a)

d) Chung minh 3 duong thSng M Q , NP , BD dong qui j j i u u<) } I

a) Tim giao diem N ciia M G va mat phSng (ABCD)

b) Chung minh ba diem C, D, N thang hang, va D trung diem ciia C N

LOI GIAI Trong mat phang (SBE) chua M G , gpi N la giao diem ciia M G va BE

Vi BE thupc mat phSng (ABCD), nen N thupc mat phSng ( A B C D ) Trong mat phSng (SBN) ke E F // M N ( F thupc S B )

Trong tam giac S E F c6 M G // E F nen c6:

SM S G

= 2 =>SM = 2MF « B M = 2MF

MF G E Vgy F trung diem ciia BM Trong A B M N c6 E F // M N nen c6:

BF BE , „ ^

= = 1 => BE = E N ,

FM E N Vay E la trung diem ciia BN

De dang chiing minh AAEB = ADEN(c.g.c) ABE = E N D Hai goc nay bang nhau theo truong hop so le trong nen

AB // D N , ma A B // C D Vay ba diem C , D , N th^ng hang

E D la duong trung binh ciia tam giac N B C suy ra D la trung diem ciia C N Cho hinh chop S A B C D , day la hinh binh hanh A B C D c6 tam la O Goi M

la trung diem cua SC

a) Xac djnh giao tuyen ciia (ABM) va (SCD)

b) Gpi N la trung diem ciia BO Hay xac djnh giao diem I ciia (AMN) voi

SD Chiing minh rang | ^ ^ ' ^'^'^^ "^'^^ "^^''^ '^'"'^ chop S A B C D cat boi mat phang (AMN)

LOI G I A I Xac djnh giao tuyen ciia (ABM) va (SCD)

45

(K la trong tam ASAC)

T u (1) va (2) suy ra — = - ^

^ ' ^ I D 3

Thiet dien cua hinh chop S.ABCD cat boi mat phJing ( A M N ) u i u i„i

Goi L la giao diem cua A N va BC Ke't luan thiet dien can tim la t u giac A L M I

Cho t u dien ABCD Tren A D lay N sao cho A N = 2 N D , M la trung diem ciia

AC, tren BC lay Q sao cho BQ = 1 BC

.;.! JinurlD 03 •

IC a) T i m giao diem I cua M N voi (BCD) Tinh ti so ^ •

b) T i m giao diem J cua BD voi ( M N Q ) Tinh ti so — , —

T u D k e D G / / I M ( G e AC) »

A M A N

T r o n g A A G D c o : — = — 2

A M = 2 M G Vay: G trung diem ciia C M suy ra D G

la d u o n g trung binh ciia tam giac C M I , suy ra D la trung diem ciia C I

IC Ke't luan: — = 2

I D b) T i m giao diem J ciia BD v o i ( M N Q )

Ta ccS QJ la d u o n g trung binh cun tam giac BEF => BJ = JF

N Ian lugt trung diem cua SC va SA; E trpng tam ciia tam giac ABC

a) Tim giao diem I cua SD va mat phSng (AME) Chung minh EI // SB

b) Tim giao diem H ciia SD va mat phang (MNE)

c) Tim thie't dien ciia hinh chop c^t boi mat phang (MNE)

LOIGIAI

Ta CO SO = (SAC) n (SBD) Goi J = A M n SO (AM, SO e (SACj)

a) Tim giao diem ciia I cua SD va mat phang (AME)

Chpn mat phSng (SBD) chiia SD

Tim giao tuyen ciia hai mat phang (AME) va (SBD)

Co E la diem chung thii nhat, J la giao

diem ciia A M va SO Ma AM, SO Ian

luot thuoc trong hai mat phang (AME)

va(SBD)

Vay giao tuyen cua chiing la EJ •

Keo dai EJ cat SD tai mot diem thi do la

diem I can tim

Chung minh EI // SB:

Vi J la giao diem cua AM, SO la hai_

duong trung tuye'n ciia tam giac SAC

Nen J la trong tam ciia tam giac SAC /,'

OJ _ OE \

~3

Ta CO EJ // SB (theo djnh ly dao Talet), hay EI // SB

OS OB

b) Tim giao diem H ciia SD va mat ph^ng ( M N E )

Chon mat ph3ng (SBD) chita SD / ]

Tim giao tuyen ciia hai mat phang ( N M E ) va (SBD)

c) Tim thiet di^n ciia hinh chop di hai mat phSng (MNE)

Dau tien tim giao tuyen ciia hai mat phang (MNE) va (ABCD)

Ta c6: E la diem chung thu nhat, c6 M N // AC, ma hai duong thang M N ,

AC Ian lugt thuoc hai mat phSng (MNE) va (ABCD)

Vay giao tuyen ciia chiing qua E va // voi AC cat AB, BC Ian lugt tai K va L Ke't luan thie't dien can tim la da giac KLMHN

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh Gpi M , N Ian lugt la trung diem ciia cac canh AB va SC

a) Tim giao diem K ciia duong thang M N voi mat phang (SBD) Tinh ti so K M

KN b) Goi E trung diem ciia SA Tim giao diem F ciia SD va mat phSng (EMN) Chung minh tii giac MEFN la hinh thang

c) Tim thiet dign ciia hinh chop vol mat phang (EMN)

LOI GIAI a) Tim giao diem K ciia duong thSng M N voi mat phSng (SBD)

Chon mat phang (SMC) chua M N Tim giao tuyen ciia (SMC) va (SBD) Taco Se(SMC)n(SBD) (l)

KN

KN IG

4 9

b) T i m giao diem F ciia SD va mat phang (EMN)

Chon mat phang (SBD) chiia SD, t i m giao tuye'n ciia hai m|t phSng (SBD)

C h u n g m i n h t u giac M E F N la hinh thang

I V i E N la d u o n g t r u n g binh ciia tarn giac SAC ''• '"•''^•'^

De dang chirng m i n h J trung diem cua EN '

Trong tarn giac M N E , KJ la d u o n g trung binh ciia tarn giac nen:

Vay: t u giac EMKF la hinh binh hanh T,

Ket luan t u giac M N F E la hinh thang vi CO EF//MN

c) T i m thiet dien ciia hinh chop voi mat phang (EMN) '4>">OTT

Tim giao tuye'n ciia (MNE) va (ABCD) c6 : , ,

SE 1 b) Goi E la diem tren SC sao cho —• = - , ggi H la giao diem ciia KE va

H E mp(SAD) Tinh ti so — ? ~ c) Goi I la diem tren canh SD ( D I > SI), P la giao diem ciia A K va (SDC), Q

la giao diem C I va (SAB) C h u n g m i n h P, Q , S thgng hang

LOI G I A l a) T i m giao diem F cua DK \ mp(SMC) Tinh ti so Goi J trung diem cua A M

Ta CO M W la d u o n g trung binh cua ADJO

=> W la trung diem cua O D Trong mat phang (SBD) d u n g

d u o n g thang d // SW d cat DK

va SB Ian luot tai G va N WF la

d u o n g trung binh ciia A D G O

=> F trung diem ciia D G

51

Ve lai mat phing (SBC) nhu hinh 2

Gpi L trung diem cua EC, KE duong

trung binh cua tam giac SBL nen

K E = 1 B L (1)

Ta c6: Sj = Cj (so le trong); ^ * ^ ' ^ ^ C

^ ^ ^ ^ ^ _ H i n h 2

Ej = Ej (dol dinh ), Ej = Lj (dong vj ) => Ej = '

Tir do suy ra: ASEH = ACLB(g.c.g) r:> BL = HE (2) ('>

Gpi P = A K n S y ( A K , Syc(SAB)):^P = A K n ( S C D ) ; fti

Goi Q = CI n Sy (CI, Sy c (SCD)) :^ Q = CI n (SAB);

Vi ba diem P, S, Q cung nam tren giao tuyen Sy nen chiing thang hang

;i7

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh Goi M , N Ian lupt

la trung diem cua AB va SC

lA

IN

KM

a) Tim giao diem I cua A N va (SBD) Tinh

b) Tim giao diem K ciia M N va (SBD) Tinh

IB c) Chung to B, I , K thang hang Tinh — Gpi E la trung diem ciia SA Tim

IK thiet di?n ciia (MNE) va hinh chpp

LOI GIAI a) Tim giao tuyen cua mp(SAC) va mp(SBD)

b) Tim giao diem K ciia N M va mp(SBD)

Chpn mp(ABN) chira M N Tim giao tuyen cua mp (ABN) va mp (SBD)

1 6 SO , SO c (SBD) ' l e A N , A N c : ( A B N )

Vay: K la trung diem cua MN Suy ra: KM

KN = 1

c) Theo each tim giao tuyen ciia cau b) thi 3 diem B, K, I thiing hang

Trong ANMQ, ta c6: IK = - Q M

Va trong AABI: Q M = ^ B I ^ IB = 4 I K o ^ = 4

Hai mat phSng (MNE) va (ABCD) c6 M la diem chung va c6 NE // AC, nen:

giao tuyen d cua chung qua M va d / / A C / / N E '

Gpi F = d n C D ( d , C D c ( A B C D ) ) ; gpi H - F N n S D ( F N , S D c ( S C D ) )

Ket luan thie't di^n cua mat ph^ng (MNE) cat hinh chop S.ABCD la da giac EMLNH i,-4i^rtt,iiii j.nf,f'

5 3

Trong khong gian, qua mpt diem

khong nMm tren duong th3ng cho

truoc, CO mpt va chi mpt duong th5ng

song song voi duong thing da cho

Dinh li 2:

Neu ba mat phang phan bi^t doi mpt cix nhau theo ba giao tuyen phan bi?t

thi ba giao tuyen ay hoac dong quy ho§c doi mpt song song vol nhau

M

He qua:

Neu hai mat phiing phan bi?t Ian lug-t chua hai duong thSng song song thi

giao tuyen ciia chiing (neu c6) cung song song voi hai duong thSng do ho|c

trung voi mpt trong hai duong thang do

Dinh li 3:

Hai duong th^ng phan bi^t cung song

song voi duong thSng thu ba thi song

song voi nhau

Dl/OfNG T H A N G V A M A T P H A N G SONG SONG

BinhJiJ:

Neu duong thiing d khong nam trong

m i t phang (a) va d song song voi duong thiing d' nam trong (a) thi d song song voi (a)

pinh li 2:

Cho duong thang a song song voi mat phing (a) Ne'u mat phang (p) chua

a va cat (a) theo giao tuyen b thi b

song song voi a

He qua:

Ne'u hai mat phang phan bi^t ciing song song voi mgt duong thang thi giao tuyen ciia chiing ( ne'u c6 ) cung song song voi duong thang do

Dinh li 3:

Cho hai duong thang cheo nhau Co duy nha't mpt mat phang chua duong thang nay va song song voi duong thing kia ,^

DANG 1: Chung minh duong thing song song voi ducmg thing, duong th^g song song voi mpt mat phang " ;

Chung minh hai duong thing song song thi dya vao hinh hpc phang: Djnh

ly Talet dao, duong trung binh

Muon chung minh duong thing d song song voi mat phing (P), ta phai chung minh duong thing d song song voi mpt duong thing thuoc mp (P)

Tim giao tuye'n each 2: Tim mpt diem chung cua hai mat phing, tim trong

hai mat phSng Ian luot c6 hai dirong thang song song vai nhau Giao tuyen

can tim di qua diem chung va song song vai hai duong thang song song

Cho hinh chop S.ABCD voi day A B C D la hinh thang voi canh day A B va

C D (AB >CD) Gpi M, N Ian lupt la trung diem cac canh SA, SB

a) Chung minh: M N / / C D

b) T i m P = S C n ( A D N )

c) Keo dai A N va D P c^t nhau tai I Chung minh: SI // A B // C D T u giac

SABI la hinh gi?

L O I GIAI a) Chung minh: M N / / C D :

Trong tam giac SAB, ta c6 M N // AB (vi M N la duang trung binh)

Ma A B // C D ( A B C D la hinh thang) V$y: M N // C D n,*wj yub

b) T i m P = S C n (ADN):

• Chpn mp phu (SBC) chua SC Tim giao tuyen cua (SBC ) va (ADN)

Ta c6: N la diem chung cua (SBC ) va (ADN) (1)

1 Trong (ABCD), gpi E = A D n BC ^ ^ — i

V l y M N la duong trung binh cua tam giac Suy ra: SI = 2 M N

SI // A B SI // A B

Ta co: => • [SI = 2 M N , A B = 2MN [SI - A B Vay tu giac SABI la hinh binh hanh

Cho hinh chop S.ABCD c6 day A B C D la hinh thang (day Ion AB) Gpi I, J Ian lupt la trung diem A D va BC, K la diem tren canh SB sao cho SK =

isB

a) Tim giao tuyen cua (SAB) va (IJK)

b) Tim thiet di?n cua (IJK) voi hinh chop S.ABCD Tim dieu kien de thiet dien la hinh binh hanh

L O I G I A I a) Tim giao tuyen cua (SAB) va (IJK):

^ J(IJK) n (ABCD) = IJ , ( I J K ) n (SBC) = JK

'^^^°"|(lJK)n(SAB) = K L , (lJK)n(SAD) = LI ' ^ 5

Thiet di?n can tim la hinh thang IJKL (Vi IJ // LK // AB ) ^ • ' -A

Do IJ la duong trung binh ciia hinh thang ABCD => IJ = - (AB + CD)

XetASABco: = = _ =>LK = - A B

AB SB 3 3

De IJKL la hinh binh hanh » IJ = KL - ' ^ ' A P ' ^ % i -^u^ (€) t i t

<^ -^(AB + CD) = |.AB « AB = 3.CD '"^

Vay: thiet di^n IJKL la hinh binh hanh o AB = 3CD

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh Ggi M,N,P, Q Ian

lirgt la cac diem nam tren cac canh BC, SC, SD,AD sao cho M N // BS, NP // CD,

MQ // CD

a) Chung minh: PQ // SA

b) GQI K = M N n PQ Chung minh diem K nam tren duong thang co djnh

khi M di dpng tren canh BC

LOIGIAI a) Chung minh: PQ//SA

DP CN

I Xet ASCD co: NP // CD ^ — = — (1)

DS CS ' CN CM

Xet ACBS co: M N // SB = > ^ = ^ (2)

Cho hinh chop tu giac S.ABCD Goi M, N, E, F Ian lugt la trung diem ciia

SA, SB, SC, SD Chung minh rang: ' ' v\ ' a) ME // AC, NF // BD r tiiM U mil nk, n{^ih ;

-b) Ba duong thang ME, NF, SO (O la giao diem cua AC va BD, dong qui)

c) Bon diem M, N, E, F dong phang

LOI GIAI a) Chung minh ME // AC, NF // BD

ME la duong trung binh cua tam giac SAC

=> ME // AC

NF la duong trung binh ciia tam giac SBD

NF // BD ^ b) Ba duong th^ng ME, NF, SO

(O la giao diem cua AC va BD) dong qui

Trong tam giac SAC, goi K = ME n A C Suy ra K la trung diem ciia SO ^ Trong tam giac SDO co FK la duong trung binh cua tam giac

=> FK // DO « FK // BD (l) Trong tam giac SBD co FN la duong trung binh ciia tam giac

=>FN//BD (2) •—

Tu (1) va (2) thi K thupc NF , , 1 ^ j j n n Ket luan: ba duong thang ME, NF, SO dong quy tai diem K , ^ c) Tu chiing minh 6 cau b) thi ME va NF cat nhau tai K ^^^^

Suy ra bon diem M, N, E, F dong phang

Cho tu dien ABCD, goi I va J Ian lugt la trung diem ciia BC va BD, E la mgt diem thugc canh AD

a) Xac djnh thiet dien ciia tu di^n khi cit boi mp (IJE )

b) Tim vj tri ciia E tren AD de thiet di^n la hinh binh hanh

c) Tim dIeu ki^n ciia tii dien ABCD va vj tri ciia diem E tren A D de thiet di^n la hinh thoi

LOI GIAI

(lJE)n(ACD) = Ex

a) Xac djnh thiet di?n ciia tu di?n khi cat boi mp (IJE )

Ta CO IJ la dvrong trung binh cua ABCD nen IJ // C D

J ( l J E ) n ( A C D ) = E

' I J C : ( I J E ) , C D C ( A C D )

Voi Ex // CD // IJ Gpi F = E x n A C

Vay: thiet dien can tim la hinh thang IJEF

b) De IJEF la hinh binh hanh thi IJ = EF

Vay E phai la trung diem cua AD

c) Khi IJEF la hinh binh hanh thi EJ la duong

trung binh cua A DAB, suy ra EJ = — AB

,^ Vay: de IJEF la hinh thoi thi IJ = EJ

« - C D = 1 A B « A B = C D

2 2

Ket luan: de thiet di?n IJEF la hinh thoi thi

E la trung diem ciia AD va AB = CD

Cho hinh chop S.ABCD c6 day ABCD la t i i giac loi Goi M , N Ian lupt la

trpng tam cua tam giac SAB va SAD, E la trung diem cua CB

a) Chung minh M N // BD

b) Xac djnh thiet dien ciia hinh chop khi cat boi mp (MNE )

c) G(?i H va L Ian lugt la cac giao diem aia mp (MNE) voi cac canh SB va

SD Chung minh LH // BD

LOI GIAI a) Chung minh M N // BD

Gpi K trung diem cua SA

Theo tinh chat trpng tam, c6:

K M

KB 1^ = 1 ^ M N I I B D

••i>i

•it

b) Xac djnh thiet di^n cua hinh chop

khi cat boi mp (MNE ) "'^

E la diem chung ciia mp (MNE) va ( A B C D ) , nen giao tuyen ciia chiing qua E

va song song voi M N va song song voi B D Giao tuyen nay cM A B va C D

Ian lupt t^i F va G

• •0

.roi rong mp ( S A B ) duong thang F M cat S A va SB Ian luat tai P va H Con

" trong mp ( S A D ) duong thang PN cit S D tai L Tu do suy ra thiet di^n can tim la ngu giac EHPLG A'( * c) Chung minh LH // B D ^ ^ \ :^^^\i,A,

HL = ( M N E ) n ( S B D )

Co: \N // B D

M N c : ( M N E ) , B D C ( S B D )

HL // M N // B D 1 qoii-

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh Gpi M, N Ian lupt

la trung diem cac canh AB va CD ' a) Chung minh M N // (SBC), M N // (SAD), ^ , ^ ^ b) Gpi P la trung diem canh SA Chung minh SB va SC deu song song voi (MNP)

c) Gpi G ] , G2 Ian lupt la trong tam ciia A ABC va ASBC. vi^jut

Chung minh GjGj // (SAB)

Xet A SCD, ta co: QN // SC (QN la duong trung binh ciia tam giac SCD)

SC <z (PMN), SC / /QN

QN c (PMN)

SB // PM, PM c (PMN) ^ SB // mp(PMN)

c) Chung minh GjGj // (SAB):

Cho hinh chop S.ABCD c6 day (ABCD) la hinh thang AD la day Ion va

AD = 2BC Goi O la giao diem cua AC va BD, G trong tam cua tam giac SCD

a) Gpi O va O' Ian lupt la tam ciia ABCD va ABEF Chung minh rang OO' song song voi (ADF) va (BCE)

b) Gpi M va N Ian lupt la trpng tam ciia AABD va AABE Chung minh ring: M N // (CEF)

)

LOI GIAI Chung minh r^ng OO' song song voi (ADF) va (BCE)

Ta co: OO' // DF (OO' la duong trung binh ABDF )

Ma DF c (ADF) OO' // (ADF)

Ta co: OO' // CE (OO' la duong trung binh AACE )

MaCEc(BCE) => OO'//(BCE)

Chung minh rang: M N // (CEF)

Gpi H la trung diem ciia AB,

Cho hinh chop S.ABCD c6 day A B C D la hinh binh hanh Gpi M , N Ian

lugt la trong tam cua hai tam giac SAB va SAD

a) C h u n g m i n h M N // mp(ABCD)

b) G o i E la trung diem cua BC Xac dinh thiet di^n cua hinh chop S.ABCD

khi cat boi m a t phang (MNE)

L O I G I A I a) C h u n g m i n h M N / / m p ( A B C D ) '

Gpi I , J Ian l u g t la trung diem ciia cua AB va A D

TU u u - ' SM SN 2

Theo t m h chat tron^ tam, co; = = —

• ^ SI SJ 3

=> M N // IJ (theo tinh chat Taiet dao)

M a IJ thuoc mat phSng (ABCD)

^ M N //(ABCD),

b) Trong mat phang day, qua E ke d u o n g thang song song IJ cat A C tgi F, cat

CD tai G EG la giao tuyen cua (MNE) va day (ABCD)

Ket luan: thiet dien can t i m la da giac OPEGQ

Cho t u dien A B C D Goi G la trong tam cua t u di^n A B C D

a) C h u n g m i n h rang d u o n g thang d d i qua G va m p t dinh cua t u d i ^ n se d i

qua trong tam ciia mat doi dien voi dinh ay

b) Ggi A ' la trgng tam cua mat phang BCD C h u n g m i n h G A = 3GA'

L O I G I A I Gpi M , N Ian l u g t la trung diem ciia A B va C D

Goi G trung d i e m ciia M N

Suy ra G la trong tam ciia t i i di^n ' " •

a) Buoc 1: Tim giao diem cm AG va mat pMng (BCD)

Bugcli Chi'mg minh A' la trong tam cua tam giac BCD

Trong mat phang ( A B N ) ke M I song song v o i A A ' (Voi I thugc BN)

Xet A A B A ' c6 M I la d u o n g trung binh ciia t a m giac, nen I trung diem ciia

B A ' S u y r a BI = I A ' (1) r -.,^.:},V,.iV

Xet A I M N CO G A ' la d u o n g trung binh ciia tam giac, nen A ' trung diem ciia

i N S u y r a A ' N = I A ' (2) - w ^ Ngoai ra trong ABCD c6 B N la d u o n g trung tuyen, ket h g p v o l (1) va (2) la

Cho t i i dien A B C D Goi M , N Ian lugt la trung diem ciia cac canh AB, C D va

G la trung diem ciia doan N M a) T i m giao diem A ' ciia d u o n g th^ng A G va mat p h l n g (BCD)

b) Qua M ke d u o n g thang M x song song v o i A A ' va M x cat m p (BCD) t^i

M ' C h i i n g m i n h B, M ' , A ' thang hang va B M ' = M ' A ' = A ' N c) C h i i n g m i n h GA=3GA'

L O I G I A I a) Chgn m p ( A B M ) chiia A G

Hai mat phang ( A B N ) va (BCD) c6 hai diem chung la B va N Suy ra giao tuyen ciia chiing la B N , B N cSt A G tai A', t h i A ' = A G n ( B C D )

b) V i M x // A A ' , ma A A ' C ( A B N ) va M € ( A B N y

= > M X C ( A B N ) '•

Goi M ' = M x n B N => M ' = M x n ( B C D ) 'k> f.':.) ;CiMV/

\r,v ;jnrv; 'y{\&> JH cv 'I^^A 'I \

Tif do suy ra ba diem B, M', A' thang hang A

CoMM'duongtrungbinhcua ABAA'=>BM' = M ' A ' (1)

' Va GA' la duong turng binh ciia A N M M ' M ' A ' = A ' N (2) ^.^

Tu (1) va (2) suy ra BM'= M'A'= A'N

c) Tu chung minh cau b) c6:

G A ' = i M M ' v a M M ' = - A A ' : ^ G A ' = | A A ' = > A G = 3 G A '

2 2 4

hJ \,j ' v y <\'- •: S;'iO'I.;,'.i hi •/'.ti

D A N G 2: Thiet di?n ciiaa hinh chop bi cSt b6i mat phang (a) va song song

v6i mpt dutmg thang cho trudc Tinh di?n tich thiet di?n

Dang toan nay cac ban phai nho ky tinh chat: £i •^Wci

| M e ( a ) n ( P ) ^ , ( p ) ^ (^x//d)

[ ( a ) / / d , d c ( P ) ^ ^ ^ ^ ^ '

Cho hinh chop S.ABCD c6 day ABCD la hinh thang c6 AD day Ion Gpi M

trung diem cua CD, (a) la m|it phSng qua M va song song voi SA va BC

a) Hay xac dmh thie't di?n cua hinh chop S.ABCD voi mat phang ( a )

b) Tim giao tuyen ciia hai mat phlng (a) va (SAC) Chung minh giao tuyen

vua tim duyc song song voi mat phang (SAD)

LOI GIAI a) Xac djnh thiet di?n ciia hinh chop S.ABCD voi mat p h i n g ( a ) , i

V M la diem chung ciia hai mat phSng

(a) va (ABCD), c6 (a) // BC nen giao

tuyen cua chiing qua M va song song

voi BC, giao tuyen nay cat AB tai E

E la diem chung ciia hai mat phang

(a)va (SAB), c6 (a)// SA nen giao

tuyen ciia chiing qua E va song song

voi SA, giao tuyen nay cat SB tai F

F la diem chung ciia hai mat ph5ng

(a)va (SBC), c6 (a)// BC nen giao

tuyen ciia chiing qua F va song song

voi BC, giao tuyen nay cat SC tai G

Ket luan mat phSng (a) cit hinh chop S.ABCD theo mpt thiet di?n la hinh

thang MEFG, vi c6 ME va FG cimg song song voi BC

b) Gpi H la giao diem ciia ME va AC, ta c6 H va G la hai diem chung ciia hai m|it phing (a) va m|t phSng (SAC) Vay ( a ) n ( S A C ) = HG Vi (a)// SA nen giao tuyen HG // SA, ma SA thuQc m l t phing (SAD) nen giao tuyen HG//mp(SAD)

Cho tu dien ABCD , lay diem M la mpt diem thupc mien trong ciia tarn giac BCD Gpi (a) la mat phang qua M va song song voi AC va BD Hay xac djnh thiet di?n ciia mat phJing (a) voi h i di?n ABCD Thiet di^n

la hinh gi? ^

LOI GIAI

M la diem chung ciia hai mat ph5ng (a)va (BCD), c6 (a)// BD nen giao tuyen ciia chiing qua M va song song voi BD, giao tuyen nay cSt BC tai E va cat CD tai F

E la diem chung ciia hai mat phing (a) va (ABC), c6 (a) // AC nen giao tuyen ciia chiing qua E va song song voi AC, giao tuyen nay cat AB tai H

H la diem chung ciia hai mat phlng (a) va (ABD), c6 (a) // BD nen giao

tuyen ciia chiing qua H va song song voi BD, giao tuyen nay cit AD tai G

G va F la hai diem chung ciia hai mat phSng (a) va mat phang (ACD)

Vay giao tuyen ciia chiing la FG v" 7 - ' d w VM rtnim •

Vi mat phiing (a) // AC, nen giao tuyen FG // AC

Ket luan: thie't di?n can tim la hinh binh hanh EFGH, vi c6 EF // HG // BD va HE//FG//AC

Cho hinh chop S.ABCD c6 day ABCD la hinh binh hanh Lay mpt diem M

di dpng tren canh SC Gpi (a) la mat phing chiia A M va song song voi BD a) Chung minh r3ng mat phSng (a) luon di qua mpt duong thMng co dinh

k h i M t h a y d o i j- ^ ; y r > <

b) Mat phSng (a) cat SB va SD tai E va F Hay neu each dyng E va F

c) Gpi I la giao diem ciia ME va CB, J la giao diem ciia MF va CD Chung

minh ba diem I , J, A thAng hang

LOI GIAI

a) A la m g t d i e m chung cua hai mat phang

( a ) va (ABCD), c6 ( a ) // BD, nen giao tuyen

cua chung qua A va song song v o i BD

Vay: ( a ) n ( A B C D ) = A x ( A x // B D )

V i A x la d u o n g thang co d j n h k h i M thay doi

Ket l u | n : m p ( a ) luon d i qua d u o n g

thang CO d|nh A x <> , ^

b) Gpi 0 = A C n B D

-Ta c6: SO la giao tuyeh cua

hai mat ph5ng (SAC) va (SBD)

Gpi G = A M n S O ( A M , S 0 cz ( S A C ) )

Ta c6: G la diem chung ciia mat p h i n g ( a ) va mat phSng (SBD), c6 ( a ) // BD

nen giao tuyen cua chung qua G va song song v o i BD, giao tuyen nay cat SB

va SD Ian lup-t tai E va F

c) I va F la hai diem chung cua mat phang ( a ) va mat phang day (ABCD),

nen I va F phai thupc giao tuyen Ax ciia hai mat phang. , i„,j:>

Vay ba d i e m I , J , A thang hang

Cho hinh binh hanh ABCD va diem S khong nam trong mat phang chi'ra ABCD

a) T i m giao tuyen cua cac cap mat ph^ng sau (SAC) va (SBD), (SAB) va (SCD)

b) M o t mat phang ( a ) qua BC, cat SA tai N va cat SD tai M

BC c ( a ) , A D c (SAD)

^ M N // A D // BC c) Gpi I = B N n C M ( B N , C M c ( a ) )

l € B N , B N c ( S B D )

V i : I e (SAC) n (SBD)

I e C M , C M c ( S A C ) Suy ra I thupc giao tuyen SO co djnh ciia hai mat phang (SAC) va (SBD)

d) Gpi E va F Ian lupt la trung diem ciia SA va A D •, -, „ - , i

V i K chia doan A C thanh ba phan bang nhau va A K chiem 1 phan, de dang chung m i n h K la trpng tam ciia tam giac A B D ; ,, ^

Theo tinh chat trong tam co: — = — = - ^ G K // EF ' ' " ( 3 )

• ^ BE BF 3

M a EF la d u o n g t r u n g b i n h ciia tam giac A D S => EF // S D (4)

T u ( 3 ) va (4) CO GK // S D c ( S C D ) ^ GK // ( S C D )

Cho t u di^n A B C D , gpi M , N Ian lupt la trung diem ciia BC va BD

a) T i m giao tuyen ciia hai mat phang ( A M N ) va (ACD)

b) M o t mat phSng (P) qua C D va cMt A M , A N Ian l u p t tai F va E.Tu giac CDEF la hinh gi?

c) CF va DE cat nhau tai K C h u n g to A , B , K thang hang "

d) C h u n g to giao diem I ciia CE va DF luon nam tren m o t d u o n g thang co

Vi K = C F n D E , m a

(p)r^(AMN) = EF

b) • MN // CD => EF // MN // CD

M N c ( A M N ) , C D c ( P ) /< 0 « )

Vi CD // MN suy ra CDEF la hinh thang

c) Ta CO AB la giao tuyen ciia hai m^t

ph^ng (ABC) va (ABD) „; ; ,'

K e C F , C F c ( A B C )

K e D E , D E c ( A B D )

Vi K la diem chung ciia hai mat ph5ng (ABC) va (ABD) nen K thupc giao

tuyen Vgy ba diem A, B, K th^nghang " ' l/i J v l ' 1 ' 3 r m 1 t.,? j {i

d) Trong m|it phling (BCD) gpi O la giao diem cua CN va DM ,

Ta CO A va O la hai diem chung cua hai mat phiing (ANC) va (AMD), nen

giao tuyen ciia chiing la AO

Gpi I la giao diem cua C E va DF '

p'^-^): :

l € C E , C E c ( A N C ) , ,

r Ta c6: \ , => I e ( A N C ) n ( A M D )

I e D F , D F c ( A M D ) V 7 V ) : (AMD)

Suy ra I thupc giao tuyen AO cua hai mSt phSng

Vi hai diem A va O co dinh nen diem I thupc doan AO co djnh

Cho hinh chop S.ABCD M, N la hai diem tren AB, CD Mat phing (a) qua

MN va song song v6i SA

a) Tim cac giao tuyen cua (a) voi (SAB) va (SAC)

b) Xac djnh thiet di^n ciia hinh chop vol (a) ''^ ' *^ '

c) Tim dieu kifn ciia MN de thiet di^n la hinh thang

=> (a) n (SAB) = MP (voi MP // SA, P G SB)

Tim cac giao tuyen ciia (a) voi (SAC)

Gpi R = M N n A C ( M N , A C c ( A B C D ) )

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b) Xac djnh thiet dien ciia hinh chop voi (a):

Theo cau a) thiet di^n la tu giac M P Q N

c) Tim dieu ki#n cua M N de thiet di^n la hinh thang: "

M N / / P Q (2)

Ta CO : MPQN la hinh thang =>

, fSA//MP (1), ta CO <^ ^ „ SA// Q N Xet (1)

Do do: Xet (2), ta CO

'.OS r> r A? W>v Sf/'Mr;

gi-MBc=(a),BCc(SBC) Vay de thiet dign la hinh thang thi MN // BC

M N / / P Q ^' 1

Cho tu di^n ABCD Tren canh AD lay trung diem M, tren cgnh BC lay diem N bat ky Gpi (a) la mat phSng chira duong thiing MN va song son§ voi CD

a) Hay xac djnh thiet di^n ciia mat phMng (a) voi tu di?n ABCD " ' b) Xac djnh vj tri ciia N tren BC sao cho thiet dien la hinh binh hanh

LOIGIAI a) Xac djnh thiet dif n ciia m|t phing (a ) y/,£^

voi tu dif n ABCD /

(a) // CD \ Taco: | c D c ( A C D )

Va ( a ) o ( A B D ) = M Q ( s ) ;

( a ) n ( A B C ) = PN (4)

T u (1) va (2), ta dugc : M P // N Q Vay thiet dien la hinh thang M P N Q

b) Xac d i n h vj t r i ciia N tren BC sao cho thiet dien la hinh binh hanh

Vay: N la t r u n g diem BC thi M P N Q la hinh binh hanh

Cho hinh thang A B C D c6 day Ion AB va S la mot diem o ngoai mat ph^ng

ciia h i n h thang Gpi M la mpt diem ciia CD; (a) la mat phMng qua M va

T u (1) va (2), ta dugc M N // PQ Vay thiet dien la hinh thang M N P Q

b T i m giao tuyen ciia (a) voi mat ph^ng (SAD). ^-f''

Trong (ABCD), ggi I = A D r-i M N => I la diem chung cua (a) va (SAD)

f < " ' " ^ ^ - ^ ^ = < 5 ' ^ ' " ^ ( a , o ( S A D ) = , , V 6 i I t / / S A )

Co:

Ta c6:

I e ( a ) n ( S A D )

Trong mat phSng (a) cho tarn giac ABC vuong tai A, ABC = 60°, AB = a

Ggi O la t r u n g diem ciia BC Lay diem S 6 ngoai mat phang (a) sao cho SB =

a va SB 1 O A Ggi M la mgt diem tren canh AB, mat phJing (P) qua M song song v o i SB va O A , c i t BC, SC, SA Ian lugt tai N , P, Q Dat x = B M ( 0 < x < a)

a) C h u n g m i n h M N P Q la hinh thang vuong

b) Tinh di?n tich o i a hinh thang theo a va x Tinh x de di^n tich nay Ion nha't

L O I G I A I ,,,, , a) Chung m i n h M N P Q la hinh thang vuong:

Xet tarn giac ABC, ta c6: cosB =

3

Cho h i n h v u o n g A B C D canh a, tam O GQI S la mot diem 6 ngoai mat p h i n g

(ABCD) sao cho SB = SD G g i M la diem tiay y tren A O v o i A M = x M a t

phang (a) qua M song song v o i SA va BD cat S O , SB , A B tai N , P , Q

a) T u giac M N P Q la h i n h gi?

b) Cho SA = a Tinh di?n tich M N P Q theo a va x Tinh x de dien tich Ion nhat

L O I G I A I a) T u giac M N P Q la hinh gi?

Tir (4) va (5), suy ra M N // PQ // SA (6) , (QAe) \\

-Tir (3), (6) va (*), suy ra M N P Q la hinh c h u nhat f iSm yomii ^(tCuu

b) T i n h dien tich M N P Q theo a va x: i/\hAfi )f,m £.u;> tmui ofiij< m i i

T a c o : S M N P Q = M Q M N V - ^' ' • • j ^ ^(<:mAJvr^.6^^•i >f.-vi r

Tinh M O : Xet tam giac A Q M : ' '''' ^''^ "

Ta c6: A = 45°, Q = 45", M = 90" A A Q M can tai M Vay M Q = A M = x

T i n h M Q : Xet tam giac SAO: • - o rfmh V

Vay: x = ^ - j - t h i S^^,VJ,,Q dat gia trj Ion nhat

Cho h i n h chop SABCD c6 day la hinh vuong canh a Tren canh A B lay mot

d i e m M v o i A M = x Goi (a) la mat phang qua M va song song v o i m a t

p h i n g (SAD) cat SB, SC, va C D Ian l u o t tai N , P, Q

a) T i m thiet d i f n cua (a) vdi mat phang hinh chop Thiet dien la h i n h gi?

75

b) T i m qui tich giao diem I ciia M N va PQ khi M di dpng tren do^in AB

c) Cho S A D = 90" va SA = a Tinh di?n tich ciia thiet dien theo a va x

Tim X de dien tich ciia thiet di^n bang 3a'

L O I G I A I a) Tim thiet dien cua (a) voi mat phang hinh chop:

Vi mp ( a ) // ( S A D ) mp ( a ) // vol

mpi duong thuoc mat phang (SAD)

• T i m giao tuyen ciia mat phang (a)

va mat ph3ng (ABCD)

Co M la diem chung cua hai mat phang

mp(a) va mp ( A B C D ) , vi mp(a)//AD,

nen giao tuyen cua chiing qua M va

song song voi A D , giao tuyen nay cat

C D tai diem Q

T u o n g t u : >M.VV >: , n'

• Mat phang (a) va mat phang (SAB) c6 M la diem chung va ( a ) // S A ,

nen giao tuyen cua chung la M N voi M N // SA va N e SB

• Mat phSng (a) va mat ph^ng (SBC) c6 N la diem chung va (a) //AD// BC,

nen giao tuyen cua chiing la NP voi NP // BC va P € S C ' 4' ' ( 2 ) =

• Mat phJing (a) va mat ph^ng (SCD) c6 2 la diem chung la P, Q

Vay: giao tuyen ciia chimg la PQ

Suy ra: thiet dien can tim la MNPQ

T u (1) va (2) thi MQ // PN Vay M N P Q la hinh thang

b) Tim qui tich giao diem I ciia M N va PQ khi M di dpng tren doan A B :

Gioi han qui tich : Khi M = A thi I s S Con khi M = B thi I = SQ

c) Tinh di#n tich ciia thiet di^n theo a va x:

I Ta c6: S M N P Q = S - S = S^^AD - SAINI' (Vi AIMQ = A S A D (c.g.c))

Tinh S ^ A D : Ta CO A S A D vuong can tai A, do do: Sg^D = 2 '

Tinh S ^ N p : Xet tam giac SBC, tam giac SBS) va tam giac S A B c6:

Vgy: dien tich thiet dien: SMNI'Q ^x^ " ^ ( a ^

De S M N P Q ~ = > - ( a ^ - x 2 ) = — « x^ « x^ ~ « x =

Cho tii dien A B C D c6 A B = A C = C D = a va AB vuong goc voi C D Lay M thupc doan A C voi A M = x(0 < x < a ) Mat phcing (a) qua M va song song

voi AB, C D cat BC, BD, A D Ian lupt tai N, P, Q (r^tV'x hU

a) Chung minh M N P Q la hinh chCr nhat , , " rt-' i' >>

b) Tinh dien tich M N P Q theo a va X , -^'-.u./r fn,.,M^Z,.,i\

c) Tinh X de dien tich M N P Q Ion nha't Tinh gia trj Ion nha't do , _ d) Djnh X de M N P Q la hinh vuong^

N P = ( a ) n ( B C D ) ( a ) // C D

• M Q // C D

• NP // C D