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Chuyn Hnh hc 11 - thyTrn Vit Knh Quan hsong song
Hocmai.vnNgi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 1 -
I. Chng minh ba im M, N, P thng hng:
+ Tm trong hnh v hai mt phng khc nhau: mp (ABC), mp (ABC) cha M, N, P.
+ Chng minh: M, N, P ( )ABC
M, N, P ( ' ' ') A B C
Theo tin TNG GIAO suy ra ( ) ( ' ' ') ABC A B C ng thng t i qua M, N, P M, N, P
thng hng.
II. Tm giao tuyn hai mt phng (P) v mp (Q) cho trc:
+ Tm trong hnh vhai im A, B nm trn mp (P) v mp (Q) (c chng minh).
+ S dng tin Tng giao ( ) ( )mp P mp Q Dt AB
II. Tm giao im ca ng thng v mt phng (P):
+ Chn trong hnh v mt phng (Q) cha d.
Tm giao tuyn ( ) ( )t mp P mp Q
+ Tm : ( )( )
M d M d t Cm M d P
M P
Gio vin : Trn Vit Knh
Ngun : Hocmai.vn
BI 02.
CC DNG TON C BN
TI LIU BI GING
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Chuyn Hnh hc 11 - ThyTrn Vit Knh Quan hsong song
Hocmai.vnNgi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 1 -
Bi 1:Cho hnh bnh hnh ABCD . S l im khng thuc (ABCD) ,M v N ln lt l trung im ca
on AB v SC.
a. Xc nh giao im I = AN (SBD)
b. Xc nh giao im J = MN (SBD)
c. Chng minh I , J , B thng hng
Bi 2:
Cho t din SABC.Gi L, M, N ln lt l cc im trn cc cnh SA, SB v AC sao cho LM
khng song song vi AB, LN khng song song vi SC.
a. Tm giao tuyn ca mp (LMN) v (ABC)
b. Tm giao im I = BC ( LMN) v J = SC ( LMN)
c. Chng minh M , I , J thng hng.
Bi 3:
Cho bn imA ,B ,C , D khng cng nm trong mt mt phng.
a. Chng minhAB v CD cho nhau
b. Trn cc on thng AB v CD ln lt ly cc imM, Nsao cho ng thngMNct ng
thngBD tiI. Hi imI thuc nhng mt phng no? X giao tuyn ca hai mp (CMN) v ( BCD)
Bi 4. Cho t dinABCD ,M l mt im bn trong tam gicABD , N l mt im bn trong tam
gicACD . Tm giao tuyn ca cc cp mt phng sau:
a. (AMN) v (BCD)
b. (DMN) v (ABC )
Bi 5: Cho t gic ABCD v S (ABCD). Gi M , N l hai im trn BC v SD.a. Tm giao im I = BN ( SAC)
b. Tm giao im J = MN ( SAC)
c. Chng minh C , I , J thng hng
Ngun : Hocmai.vn
BI GING 02.
CC DNG TON C BN
BI T P T LUY N
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Chuyn Hnh hc 11 - ThyTrn Vit Knh Quan h song song
Hocmai.vn Ngi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 1 -
Bi 1:Cho hnh bnh hnh ABCD . S l im khng thuc (ABCD) ,M v N ln lt l trung im caon AB v SC .a. Xc nh giao im I = AN (SBD)
b. Xc nh giao im J = MN (SBD)c. Chng minh I , J , B thng hng
Gii:
a. Xc nh giao im I = AN (SBD ) Chn mp ph (SAC) AN Tm giao tuyn ca (SAC ) v (SBD)
( SAC) (SBD) = SO Trong (SAC), gi I = AN SO
I ANI SO m SO ( SBD) I ( SBD)
Vy: I = AN ( SBD)b. Xc nh giao im J = MN (SBD) Chn mp ph (SMC) MN
Tm giao tuyn ca (SMC ) v (SBD)S l im chung ca (SMC ) v (SBD)Trong (ABCD) , gi E = MC BD ( SAC) (SBD) = SE
Trong (SMC), gi J = MN SEJMNJ SE m SE ( SBD) J ( SBD)
Vy J = MN ( SBD)c. Chng minh I , J , B thng hngTa c : B l im chung ca (ANB) v ( SBD)
I SO m SO ( SBD) I ( SBD) I AN m AN (ANB) I (ANB)
I l im chung ca (ANB) v ( SBD) J SE m SE ( SBD) J ( SBD)
BI GING 02.
CC DNG TON C BN
P N BI T P T LUY N
IJ
E
A
BC
M
ND
S
O
J
E
I
O
S
C
N
M B
A
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Chuyn Hnh hc 11 - ThyTrn Vit Knh Quan h song song
Hocmai.vn Ngi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 2 -
J MN m MN (ANB) J (ANB) J l im chung ca (ANB) v ( SBD)
Vy : B , I , J thng hngBi 2:Cho t din SABC.Gi L, M, N ln lt l cc im trn cc cnh SA, SB v AC sao cho LMkhng song song vi AB, LN khng song song vi SC.a. Tm giao tuyn ca mp (LMN) v (ABC)
b. Tm giao im I = BC ( LMN) v J = SC ( LMN)c. Chng minh M , I , J thng hng.
Gii:a. Tm giao tuyn ca mp (LMN) v (ABC)Ta c : N l im chung ca (LMN) v (ABC)Trong (SAB) , LM khng song song vi AB
Gi K = AB LMK LM m LM (LMN ) K (LMN )K AB m AB ( ABC) K ( ABC)
b. Tm giao im I = BC ( LMN) Chn mp ph (ABC) BC Tm giao tuyn ca (ABC ) v (LMN)
(ABC) ( LMN) = NK Trong (ABC), gi I = NK BC
I BCI NK m NK (LMN ) I (LMN)
Vy : I = BC ( LMN)Tm giao im J = SC ( LMN) Trong (SAC), LN khng song song vi SC
Gi J = LN SC
J SCJ LN m LN (LMN ) J (LMN)
Vy : J = SC ( LMN)c. Chng minh M , I , J thng hngTa c : M , I , J l im chung ca (LMN) v ( SBC)Vy : M , I , J thng hng
Bi 3:Cho bn imA ,B ,C , D khng cng nm trong mt mt phng.a. Chng minhAB v CD cho nhau
b. Trn cc on thng AB v CD ln lt ly cc im M, Nsao cho ng thng MNct ng
thngBD tiI. Hi imI thuc nhng mt phng no? X giao tuyn ca hai mp (CMN) v ( BCD)
Gii :a. Chng minh AB v CD cho nhau :Gi sAB v CD khng cho nhauDo c mp () chaAB v CDA ,B ,C , D nm trong mp () mu thun gi thuytVy : AB v CD cho nhaub. im I thuc nhng mp : I MN m MN(ABD ) I (ABD ) I MN m MN(CMN ) I (CMN ) I BD m BD (BCD ) I (BCD )X giao tuyn ca hai mp (CMN) v ( BCD) l CI
K
JI
S
C
M
L
N
B
A
M
I
C
B D
N
A
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Chuyn Hnh hc 11 - ThyTrn Vit Knh Quan h song song
Hocmai.vn Ngi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 3 -
Bi 4.Cho t dinABCD , M l mt im bn trong tam gicABD , N l mt im bn trong tamgicACD . Tm giao tuyn ca cc cp mt phng sau:a. (AMN) v (BCD)
b. (DMN) v (ABC )Gii :
a. Tm giao tuyn ca (AMN) v (BCD)Trong (ABD ) , gi E = AMBD
EAM m AM ( AMN) E ( AMN)EBD m BD ( BCD) E ( BCD)
E l im chung ca mp ( AMN) v (BCD )Trong (ACD ) , gi F = ANCD
FAN m AN ( AMN) F ( AMN)FCD m CD ( BCD) F ( BCD)
Fl im chung ca mp ( AMN) v (BCD )Vy:EFl giao tuyn ca mp ( AMN) v (BCD )
b. Tm giao tuyn ca (DMN) v (ABC)Trong (ABD ) , gi P = DMAB
PDM m DM ( DMN) P (DMN )PAB m AB ( ABC) P (ABC)
P l im chung ca mp ( DMN) v (ABC )Trong (ACD) , gi Q = DNAC
Q DN m DN ( DMN) Q ( DMN)Q AC m AC( ABC) Q ( ABCA)
Q l im chung ca mp ( DMN) v (ABC )Vy:PQ l giao tuyn ca mp ( DMN) v (ABC )
Bi 5.Cho t gic ABCD v S (ABCD). Gi M , N l hai im trn BC v SD.a. Tm giao im I = BN ( SAC)
b. Tm giao im J = MN ( SAC)c. Chng minh C , I , J thng hng
Giia. Tm giao im I = BN ( SAC) Chn mp ph (SBD) BN Tm giao tuyn ca (SBD ) v (SAC)
Trong (ABCD), gi O = AC BD (SBD) ( SAC) = SO
Trong (SBD), gi I = BN SOI BN
B
C
ED
F
N
M
Q
P
A
O
J
K
I
M
N
A
D
CB
S
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Chuyn Hnh hc 11 - ThyTrn Vit Knh Quan h song song
Hocmai.vn Ngi trng chung ca hc tr Vit Tng i t vn: 1900 58-58-12 - Trang | 4 -
I SO m SO (SAC ) I (SAC)Vy : I = BN ( SAC)b. Tm giao im J = MN ( SAC) : Chn mp ph (SMD) MN Tm giao tuyn ca (SMD ) v (SAC)
Trong (ABCD), gi K = AC DM
(SMD) ( SAC) = SK Trong (SMD), gi J = MN SK
J MNJ SK m SK (SAC ) J (SAC)
Vy : J = MN ( SAC)c. Chng minh C , I , J thng hng:Ta c : C , I , J l im chung ca (BCN ) v (SAC)Vy : C , I , J thng hng
Ngun : Hocmai.vn