Khng g l khng th phi khng bn ?

Tm giao tuyn ca 2 mt phngPhng php: Tm hai im chung ca hai mt phng . Cc lu quan trng: A d A( ) 1) A, B, C ( ABC ) 2) d ( ) 3) Mun ni 2 ng thng a v b ct nhau. Ta cn tm mt phng (P) no cha c a v b, sau xem 2 ng thng a, b c song song khng. Nu a v b cng thuc mt (P) v khng song song th mi chc chn chng ct nhau. Bi 1 Cho t gic ABCD c cc cp cnh i khng song song v S l mt im khng nm trong mt phng (ABCD). Tm giao tuyn ca : a) (SAC) v (SBD) b) (SAB) v (SCD). c) (SAD) v (SBC). Gii a) S l im chung th nht ca (SAC) v (SBD). Trong mp(ABCD) gi O = AC BD . Ta c O AC O ( SAC ) O BD O ( SBD ) Suy ra O l im chung th 2 ca (SAC) v (SBD). Vy SO l giao tuyn ca (SAC) v (SBD). b) S l im chung th nht ca (SAB) v (SCD). Trong mp(ABCD) gi M = AB CD . Ta c M AB M ( SAB ) M CD M ( SCD )

S

Suy ra M l im chung th 2 ca (SAC) v A M (SBD). Vy SM l giao tuyn ca (SAC) v (SBD). Cu c gii tng t. Ta tm giao im ca AD v O D BC, sau l lun tng t nh cu a, b. C Bnh lun: Hnh v: Khi cho hnh chp c y l t gic thng(cc cp cnh i khng song song), ta cn v cc cp cnh i AB v CD, cp cnh i AD v BC sao cho khi ko di ra th chng phi ct nhau. V nh rng khi v hnh chp ta nn v y trc, hnh dung ng no nt t, nt lin, sau ta chn nh hp l. Hnh v cng r rng, d nhn th bn cng d thy cch lm. Cch trnh by: Lu cch k hiu im thuc ng thng, im thuc mt phng u dng k hiu , ng cha trong mt th dng k hiu . Ngoi ra bn cn phi nm 3 vn cp trn. Bi 2. (trang 23 cng trng). Cho t din ABCD. Gi I, J ln lt l trung im ca AD v BC.c

B

Khng g l khng th phi khng bn ?

a) Tm giao tuyn ca (IBC) v (JAD). b) Gi M l im trn cnh AB, N l im trn cnh AC sao cho MN khng song song vi BC. Tm giao tuyn ca (IBC) v (DMN). A Gii I AD I ( JAD ) . a) Ta c AD ( JAD ) N M I ( IBC ) . Suy ra I l im chung th nht ca M (JAD) v (IBC). J BC Q J ( IBC ) Tng t BC ( IBC ) B M J ( JAD ) . Suy ra J l im chung th nht caI K P D

(JAD) v (IBC). Vy IJ l giao tuyn ca (JAD) v (IBC). b) Trong mp(ACD) gi K = IC ND . K ND K ( DMN ) . Ta c ND ( DMN )

J J

C

K IC K ( IBC ) . IC ( IBC ) Suy ra K l im chung ca (DMN) v (IBC). Trong mp(ABD) gi P = IB MD . Tng t ta cng chng minh c P l im chung ca (DMN) v (IBC). Vy KP l giao tuyn ca (DMN) v (IBC). Nhn xt: Cu b th hin r tm quan trng ca vic xt 2 ng thng cng nm trong mt phng trc khi xt chng c ct nhau hay khng. Nu gi Q l giao im ca MN v BC th Q l im chung th 3 ca hai mt (IBC) v (DMN). M giao tuyn ca hai mt phng l duy nht mt ng thng. T suy ra ba im chung K, P, Q phi thng hng. y chnh l phng php chng minh nhiu im thng hng trong khng gian. Dng ton ny s c trnh by r phn sau. Bi 3. Cho hnh chp S.ABCD c y ABCD l hnh thang, y ln AB. Trn SD ly im M. a) Tm giao tuyn ca (MBC) v (SAC). b) Tm giao tuyn ca (MBC) v (SAD). S Gii. a) Ta c C l im chung th nht. Trong mp(ABCD) gi O = AC BD . Trong mp(SBD) gi N = MB SO . N SO N M N ( SAC ) . SO ( SAC ) A B

O D C

K

Khng g l khng th phi khng bn ?

N MB N ( MBC ) . MB ( MBC ) Suy ra N l im chung th hai ca (SAC) v (MBC). Vy NC l giao tuyn ca (SAC) v (MBC). b) Ta c M SD M ( SAD ) .

V K BC K ( MBC ) . Suy ra K l im chung th 2 ca (SAD) v (MBC). Vy MK l giao tuyn ca (SAD) v (MBC). Nhn xt: Hnh v: bi ny cho hnh chp c y l hnh thang, bn nn v 2 cnh song song nm ngang, v cnh ln nm pha trn. Cu a kh hn cu b. Bn nh rng nguyn tc tm im chung ca hai mt phng (P) v (Q) l phi tm trong (P) ng thng a, trong (Q) ng thng b sao cho a v b cng nm trong mt phng th 3. Trong cu a ny, bn chn ng thng MB trong (MBC), trong mp(SAC) rt kh chn v khng c ng no trong mp(SAC) li cng nm trong cng mt phng vi MB, bn phi tm ra im O, khi ng thng cn chn l SO s cng nm trong mp(SBD) vi MB Bi 4. Cho hnh chp S.ABCD c y ABCD l hnh bnh hnh. Gi M, N ln lt l trung im ca SB v SD. P l mt im trn SC sao cho SP > PC. Tm giao tuyn ca mt phng (MNP) vi cc mt phng: (SAC), (SAB), (SAD), (ABCD). Gii a) (MNP) v (SAC) S Ta c P SC P ( SAC ) M P ( MNP ) . Suy ra P l im chung th nht. Trong mp(ABCD) gi O = AC BD Trong mp(SBD) gi I = MN SO . Ta c I MN I ( MNP ) I SO I ( SAC ) . v SO ( SAC ) Suy ra I l im chung th 2. Vy IP l giao tuyn ca (MNP) v (SAC).P K M N A O D C Q I B P H

M M ( MBC ) . Suy ra M l im chung th nht ca (SAD) v (MBC). Trong mp(ABCD) gi K = AD BC . K AD K ( SAD )

b) (MNP) v (SAB) Ta c M SB M ( SAB ) v M ( MNP ) . Suy ra M l im chung th nht. Trong mp(SAC) gi K = IP SA Ta c K SA K ( SAB ) ,

Khng g l khng th phi khng bn ?

K IP K ( MNP ) . Suy ra K l im chung th hai. V IP ( MNP ) Vy MK l giao tuyn chung ca (MNP) v (SAB). c) (MNP) v (SAD) KN l giao tuyn chung. d) (MNP) v (ABCD) Do SP > PC nn trong mp(SAC) gi Q = IP AC , v trong (SAB) gi H = KM AB . Lp lun tng t ta cng c QH l giao tuyn ca (MNP) v (ABCD). Nhn xt : Bn cn lu gi thit SP > PC c vai tr g, nu SP < PC hoc SP = PC th bi ton s nh th no. Nu suy ngh k iu ny th bn s hiu r hn tm quan trng ca phn lu s 3 trn. Nu gi P l giao ca AD v KN th P cng l im chung ca (MNP) v (ABCD). Khi ta cng c 3 im H, P, Q thng hng. Cui cng, bn thy mp(MNP) nh mt nht dao ct hnh chp S.ABCD to thnh t gic MNPK. T gic MNPK c gi l thit din to bi (MNP) vi hnh chp S.ABCD.