phương pháp giải một số dạng toán và bài tập
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PHNG PHP GII MT S DNG TON V BI TP
PHNG PHP GII MT S DNG TON V BI TPVn 1. ng thng vung gc vi mt phng.
Cch chng minh ng thng d vung gc vi mp():
Cch 1: Ta chng minh d vung gc vi hai ng thng a v b ct nhau nm trong ().
Cch 2: Ta chng minh d song song vi mt ng thng d vung gc vi ().
Cch chng minh ng thng a v b vung gc:
Cch 1: Ta chng minh gc gia hai t bng .
Cch 2: Ta chng minh a//c m cb.
Cch 3: Ta chng minh tch v hng ca hai vect ch phng .Cch 4: Ta chng minh a vung gc vi mt mp() cha ng thng b.
Kt qu: + Nu hai ng thng phn bit cng vung gc vi mt mp th song song.
+ Nu hai mp phn bit cng vung gc vi mt ng thng th song song.
Bi 1. Hai tam gic cn ABC v DBC nm trong hai mp khc nhau to nn t din ABCD. Gi I l trung im ca BC.
a) Chng minh BCAD.
b) Gi AH l ng cao ca tam gic ADI. Chng minh AH(BCD).
Bi 2. Cho t din ABCD c ABC l tam gic cn ti A vi AB = a, BC = . Gi M l trung im ca BC. V AHMD.
a) Chng minh AH(BCD).
b) Cho AD = .Tnh gc gia hai ng thng AC v DM.
c) Gi G1, G2 l trng tm ca tam gic ABC v DBC. Chng minh G1G2(ABC).
Bi 3. Cho hnh chp S.ABC c SA vung gc vi y. Gi H, K l trc tm ca tam gic ABC v SBC. Chng minh rng:
a) SC vung gc vi mp(BHK). b) HK vung gc vi mp(SBC).
Bi 4. Cho hnh chp S.ABCD c y l hnh thoi tm O. Bit SA = SC v SB = SD.
a) Chng minh SO (ABCD) v ACSD.
b) Gi I, J l trung im ca BA, BC. Chng minh IJ (SBD).
Bi 5. Cho hnh chp S.ABCD c y l hnh vung cnh a. SAB l tam gic u v SC = a. Gi H, K l trung im ca AB, AD.
a) Chng minh SH (ABCD). b) Chng minh AC SK v CK SD.
Bi 6. Cho hnh lng tr ABC.ABC. Gi H l trc tm ca tam gic ABC v bit rng AH (ABC). Chng minh rng:
a) AABC v AABC.
b) Gi MM l giao tuyn ca hai mp(AHA) v (BCCB) trong M BC v M BC. Chng minh t gic BCCB l hnh ch nht v MM l ng cao ca hnh ch nht .
Bi 7. Cho hnh chp S.ABCD c y l hnh thoi tm O, bit SB = SD.
a) Chng minh (SAC) l mp trung trc ca on thng BD.
b) Gi H, K l hnh chiu ca A trn SB, SD. Chng minh SH = SK, OH = OK v HK//BD.
Chng minh (SAC) l mp trung trc ca HK.
Bi 8. Cho hai hnh ch nht ABCD, ABEF nm trn hai mp khc nhau sao cho ACBF. Gi CH v FK l hai ng cao ca tam gic BCE v ADF. Chng minh:
a) ACH v BFK l cc tam gic vung. b) BFAH v ACBK.
Bi 9. Cho hnh chp S.ABC c SA y, tam gic ABC cn ti B. Gi G l trng tm ca tam gic SAC v N l im thuc cnh SB sao cho SN = 2NB. Chng minh
a) BC (SAB). b) NG (SAC).Bi 10. Cho hnh chp S.ABCD c y l hnh vung cnh bng a, SAB l tam gic u, SCD l tam gic vung cn nh S. Gi I, J l trung im ca AB v CD.
a) Tnh cc cnh ca tam gic SIJ v chng minh SI (SCD), SJ (SAB).
b) Gi SH l ng cao ca tam gic SIJ. Chng minh SH AC v tnh di SH.
c) Gi M l im thuc BD sao cho BM SA. Tnh AM theo aAM theo a.
Bi 11. Cho hnh chp S.ABCD c SA y v SA = a, y ABCD l hnh thang vung ng cao AB = a, BC = 2a. Ngoi ra SC BD.
a) Chng minh tam gic SBC vung.
b) Tnh theo a di on AD.
c) Gi M l mt im trn on SA, t AM = x, vi . Tnh di ng cao DE ca tam gic BDM theo a v x. Xc nh x DE ln nht, nh nht.
Bi 12. Cho hnh chp S.ABC c SA y v SA = 2a, tam gic ABC vung ti C vi AB = 2a, BAC = . Gi M l mt im di ng trn cnh AC, H l hnh chiu ca S trn BM.
a) Chng minh AH BM.
b) t AM = x, vi . Tnh khong cch t S ti BM theo a v x. Tm x khong cch ny l ln nht, nh nht.
Bi 13. Cho tam gic ABC c BC = 2a v ng cao AD = a. Trn ng thng vung gc vi mp(ABC) ti A ly im S sao cho SA = a. Gi E, F l trung im SB, SC.
a) Chng minh BC (SAD).
b) Tnh din tch ca tam gic AEF.
Bi 14. Cho hnh lng tr ABC.ABC c y l tam gic u cnh a. cnh bn AA = a v vung gc vi y.
a) Gi I l trung im ca BC. Chng minh AI BC.
b) Gi M l trung im ca BB. Chng minh AM BC.
c) Gi K l mt im trn on AB sao cho KB = v J l trung im ca BC. Chng minh AM (MKJ).
Bi 15. Cho t din ABCD c DA (DBC) v tam gic ABC vung ti A. K DI BC.
a) Chng minh BC (AID).
b) K DH AI. Chng minh DH (ABC).
c) t ,,. Chng minh .
d) Gi s AD = a, . Tnh BC v .
Bi 16. Cho hnh chp S.ABC c y l tam gic u cnh a, SA = SB = .
a) K SH (ABC). Chng minh H l tm ng trn ngoi tip tam gic ABC.
b) Tnh di SH theo a.
c) Gi I l trung im BC. Chng minh BC (SAI).
d) Gi l gc gia SA v SH. Tnh .
Bi 17. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA (ABCD). Gi I , M l trung im ca SC v AB. Cho SA = a.
a) Gi O l giao im ca AC v BD. Chng minh IO (ABCD).
b) Tnh khong cch t I n CM.
Bi 18. Cho hnh chp S.ABCD c y l hnh ch nht tm O, SA (ABCD).
a) Gi H, K l hnh chiu ca A trn SB, SD. Chng minh SC (AHK).
b) K AJ (SBD). Chng minh J l trc tm ca tam gic SBD.
Bi 19. Cho hnh chp SABCD c y l hnh vung cnh a; SA vung gc vi y. Gi M, N l hnh chiu ca A trn SB, SD.
a) Chng minh MN//BD v SC vung gc vi mp(AMN).
b) Gi K l giao im ca SC vi mp(AMN). Chng minh AMKN c hai ng cho vung gc.
Bi 20. Cho hnh chp S.ABC c SA = SB = SC, tam gic ABC vung ti A. Gi I l trung im ca BC. Chng minh:
a) BC (SAI).
b) SI (ABC).
Bi 21. Cho t din ABCD c DA (ABC). Gi AI l ng cao v H l trc tm ca tam gic ABC. H HK DI. Chng minh:
a) HK BC.
b) K l trc tm ca tam gic DBC.
Bi 22. Cho tam gic ABC vung ti C. Trn ng thng d vung gc vi mp(ABC) ti A, ly im S di ng. Gi D, F l hnh chiu ca A trn SB, SC. Chng minh: AF SB.
Bi 23. Cho hnh chp S.ABC c SA = SB = SC = a, , , .
a) Chng minh tam gic ABC vung.
b) Xc nh hnh chiu H ca S trn mp(ABC). Tnh SH theo a.
Bi 24. Cho hnh chp S.ABCD, y ABCD l t gic c ABD l tam gic u, BCD l tam gic cn ti C c . SA y.
a) Gi H, K l hnh chiu vung gc ca A trn SB, SD. Chng minh SC (AHK).
b) Gi C l giao im ca SC vi mp(AHK). Tnh din tch t gic AHCK khi AB = SA = a.
Bi 25. Cho tam gic ABC u cnh a, d l ng thng vung gc vi (ABC) ti A. Gi H, K l trc tm ca tam gic ABC v SBC. Chng minh HK (SBC).
Bi 26. Cho hnh vung ABCD. Gi H, K l trung im AB, AD. Trn ng thng vung gc vi mp(ABCD) ti H, ly im S (khc H). Chng minh:
a) AC (SHK).
b) CK SD.
Bi 27. Cho hnh chp S.ABC c y l tam gic vung ti B, SA y. H AH SB, AK SC.
a) Chng minh cc mt bn ca hnh chp l cc tam gic vung.
b) Chng minh SHK l tam gic vung.
c) Gi D l giao im ca HK v BC. Chng minh AC AD.
Bi 28. Cho hnh chp S.ABCD c y l hnh vung cnh tm O, AB = SA = a, SA y. Gi (P) l mt phng qua A v vung gc vi SC, (P) ct SB, SC, SD ti H, I, K.
a) Chng minh HK//BD.
b) Chng minh AH SB, AK SD.
c) Chng minh t gic AHIK c hai ng cho vung gc. Tnh din tch AHIK theo a.
Bi 29. Cho hnh chp S.ABCD, y ABCD l hnh ch nht c AB = a, BC = , mt bn SBC vung ti B, SCD vung ti D c SD = .
a) Chng minh SA (ABCD) v tnh SA.
b) ng thng qua A vung gc vi AC, ct CB, CD ti I, J. Gi H l hnh chiu ca A trn SC, K v L l giao im ca SB, SD vi mp(HIJ). Chng minh AK (SBC) v AL (SCD).
c) Tnh din tch t gic AKHL.
Bi 30. Cho tam gic MAB vung ti M nm trong mp(P). Trn ng thng vung gc vi (P) ti A ly hai im C, D nm hai pha i vi (P). Gi C l hnh chiu ca C trn MD, H l giao im ca AM v CC.
a) Chng minh CC (MBD).
b) Gi K l hnh chiu ca H trn AB. Chng minh K l trc tm ca tam gic BCD.Vn 2. Hai mt phng vung gc.
Cch chng minh hai mt phng vung gc vi nhau:
Cch 1: Ta chng minh mp ny cha mt ng thng vung gc vi mp kiaD on: - Thng l nhng ng thng nm trong mt phng ny v vung gc vi giao tuyn ca hai mt phng
- ng thng y song song vi mt ng thng vung gc vi mt phng kia.
- ng thng y l giao tuyn ca hai mt phng cng vung gc vi mt phng cn chnCch 2: Ta chng minh gc gia chng l .
Cch chng minh ng thng d vung gc vi mp():
Cch 3: Nu hai mp cng vung gc vi mt mt phng th ba th giao tuyn (nu c) ca chng cng vung gc vi mt phng ny. Cch 4: Nu hai mp vung gc vi nhau, mt ng thng nm trong mp ny m vung gc vi giao tuyn th vung gc vi mp kia.
Kt qu: +
+ Nu hai mp(P) v (Q) vung gc vi nhau, im A thuc (P) th mi ng thng qua
A v vung gc vi (Q) u nm trong (P). Hnh lng tr ng. Hnh hp ch nht. Hnh lp phng.
Hnh chp u v hnh chp ct u.
Hnh chp u l hnh chp c y l a gic u v chn ng cao trng vi tm ca y.
Ch . + Cn phn bit hai khi nim Hnh chp u v hnh chp c y l a gic u.
+ Hnh chp u c cc cnh bn bng nhau.
+ Hnh chp u c gc gia cc cnh bn v mt y bng nhau.
Dng 1. Chng minh s vung gc.
Bi 1. Cho t din ABCD c ABC l tam gic vung ti B v AD (ABC). Chng minh (ABD) (BCD).
Bi 2. Cho t din ABCD c AB (BCD). Trong tam gic BCD v cc ng cao BE v DF ct nhau ti O. Trong mp(ACD) v DK AC. Gi H l trc tm ca tam gic ACD.
a) Chng minh (ACD) (ABE) v (ACD) (DFK).
b) Chng minh OH (ACD).
Bi 3. Cho hnh chp S.ABCD c y l hnh ch nht. Tam gic SAB u v vung gc vi y. Gi H l trung im ca AB. Chng minh (SAD) (SAB).
Bi 4. Cho hnh chp S.ABC c y ABC l tam gic vung ti C, SAC l tam gic u v nm trong mp vung gc vi (ABC). Gi I l trung im ca SC.
a) Chng minh (SBC)(SAC). b) Chng minh (ABI)(SBC).
Bi 5. Cho tam gic ABC vung ti A. V BB v CC cng vung gc vi mp(ABC).
a) Chng minh (ABB)(ACC).
b) Gi AH, AK l ng cao ca cc tam gic ABC v ABC. Chng minh hai mp(BCCB) v (ABC) cng vung gc vi mp(AHK).
Bi 6. Cho hnh chp S.ABCD c y l hnh thoi cnh a v SA = SB = SC = a.
a) Chng minh (SBD) (ABCD). b) Chng minh tam gic SBD vung.
Bi 7.(gc gia hai mp = ) Cho hnh chp S.ABCD c y ABCD l hnh thoi tm I, c cnh bng a v ng cho BD = a. SC = v vung gc vi (ABCD). Chng minh (SAB) (SAD).
Bi 8. Cho tam gic ABC vung ti B. on thng AD(ABC). Chng minh (ABD)(BCD).
V ng cao AH ca tam gic ABD, chng minh AH(BCD).
Bi 9. Cho hnh lp phng ABCD.ABCD cnh a. Chng minh AC (ABD) v (ACCA)(ABD).
Bi 10. Cho hnh chp S.ABC c y l tam gic u, SAy. Gi H, K l trc tm ca tam gic ABC v DBC. Chng minh:
a) (SAH)(SBC). b) (CHK)(SBC).
Bi 11. Cho t din u ABCD. Gi H l hnh chiu ca A trn (BCD) v O l trung im ca AH. Chng minh cc mp(OBC), (OCD), (OBD) i mt vung gc vi nhau.
Bi 12. Cho tam gic u ABC. Trn ng thng d vung gc vi mp(ABC) ti A ly im S. Gi D l trung im ca BC.
a) Chng minh (SAD)(SBC).
b) K CIAB, CKSB. Chng minh SB(ICK).
c) K BMAC, MNSC. Chng minh SCBN.
d) Chng minh (CIK)(SBC) v (MBN)(SBC).
e) MB ct CI ti G, CK ct BN ti H. Chng minh GH(SBC).
f) Chng minh 6 im B, C, I, K, M, N cch u D.
Bi 13. Cho hnh chp S.ABC c y l tam gic vung ti A, SHy vi H thuc on BC.
a) Chng minh (SBC)(ABC).
b) K HIAB, HKAC. T gic AIHK c c im g?
c) Chng minh (SHI)(SAB) v (SHK)(SAC).
d) K HMSI, HNSK. Chng minh HM(SAB) v HN(SAC).
Bi 14. Cho hnh chp S.ABCD c cc mt bn SAB v SAD cng vung gc vi (ABCD). Bit ABCD l hnh vung v SA = AB. Gi M l trung im ca SC. Chng minh:
a) (SAC) (SBD). b) (SAD) (SCD). c) (SCD) (ABM).
Bi 15. Cho hnh chp S.ABCD c y l hnh thoi tm O. Hai mp(SAC) v (SBD) cng vung gc vi y.
a) Chng minh (SAC)(SBD).
b) T O k OKBC. Chng minh BC(SOA).
c) Chng minh (SBC)(SOK).
d) K OHSK. Chng minh OH(SBC).
Bi 16. Cho hnh vung ABCD. Gi S l im trong khng gian sao cho SAB l tam gic u v nm trong mp vung gc vi y.
a) Chng minh (SAB)(SAD) v (SAB)(SBC).
b) Tnh gc gia hai mp (SAD) v (SBC).
c) Gi H, I l trung im ca AB, BC. Chng minh (SHC)(SDI).
Bi 17. Cho hnh chp S.ABCD c y l hnh vung cnh a. Hai mp(ASB) v (SAD) cng vung gc vi y.
a) Chng minh SA(ABCD).
b) Chng minh (SAC)(SBD).
c) Cho SA = 2a. K AH(SBC). Tnh AH?
Bi 18. Cho hnh chp S.ABCD c y l hnh ch nht tm O, AB < BC, AB = a. Hai mp(SAD) v (SAD) cng vung gc vi y.
a) Chng minh SA(ABCD).
b) Chng minh (CSB)(SAB).
c) t , . Chng minh .
Bi 19. Cho hnh chp S.ABCD c SAy, y ABCD l hnh ch nht. H AHSB, AKSD. Chng minh:
a) (SBC)(SAB). b) (AHK)(SAC).
Bi 20. Cho tam gic u ABC cnh a. Gi D l im i xng vi A qua BC. Trn ng thng vung gc vi (ABC) ti D ly im S sao cho SD = . Chng minh:
a) (SAB) (SAC). b) (SBC)(SAD).
Bi 21. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy. Gi M, N l cc im thuc BC v CD sao cho BM = , . Chng minh (SAM)(SMN).
Bi 22. Cho hnh chp S.ABCD c y l hnh thoi tm O, cnh a, SB = SD = a, BD = . Hai mp(SAC) v (SBD) cng vung gc vi y.
a) Chng minh tam gic SAC vung ti S. b) Chng minh (SBC)(SCD).
Bi 23. Trong mp(P), cho hnh thoi ABCD vi AB = a, AC = . Trn ng thng vung gc vi mp(P) ti giao im O ca hai ng cho AC v BD, ly im S sao cho SB = a. Chng minh:
a) Tam gic ASC vung. b) (SAB)(SAD).
Bi 24. Cho hnh chp tam gic u S.ABC, cc cnh y c di bng a, M, N l trung im ca SB, SC. Bit (AMN)(SBC). Tnh theo a din tch tam gic AMN.
Dng 2. Tm Thit din ca hnh chp s dng quan h vung gc.
Cch xc nh mp() i qua im A v vung gc vi ng thng d:
Cch 1: + K ng thng a qua A v vung gc vi d.
+ Tm ng thng b ct a v bd.
Khi , mp(a,b) chnh l mp() cn dng.
Cch 2: S dng kt qu di.
Cch xc nh mp() cha t a v vung gc vi ng thng mp():
+ Chn mt im A trn t a.
+ K ng thng qua A v vung gc vi mp().
Khi , mp(a,b) chnh l mp() cn dng.
Kt qu: + Nu mt ng thng v mt mp cng vung gc vi mt ng thng th song song.
Bi 25. Cho hnh chp t gic u S.ABCD, cnh y bng a, cnh bn bng . Chng minh (SBC)(SAB).
Bi 26. Cho t din SABC c ba nh A, B, C to thnh tam gic vung cn nh B v AC = 2a, c cnh SAmp(ABC) v SA = a.
a) Chng minh (SAB)(SBC).
b) Gi AH l ng cao ca tam gic SAB. Chng minh AH(SBC).
c) Tnh di on AH.
d) T trung im O ca on AC v OK(SBC). Tnh di on OK.
Bi 27. Cho hnh hp ch nht ABCD.ABCD. Gi O l giao im ca AC v BD. K CKBD.
a) Chng minh CKBD.
b) Chng minh (CBD)(CCK).
c) K CHCK. Chng minh CH(CBD).
Bi 28. Cho tam gic ACD v BCD nm trong hai mp vung gc vi nhau. AC = AD = BC = BD = a v CD = 2x. Gi I, J l trung im ca AB, CD.
a) Chng minh IJ AB v CD.
b) Tnh AB v IJ theo a v x.
c) Xc nh x (ABC) (ABD).
Bi 29. Cho hnh chp S.ABCD c y l hnh vung, SAy. Gi M l trung im ca BC. Tm N trn CD (SAM)(SMN).
Bi 30. Cho hnh chp S.ABCD c y l hnh vung tm O v c cnh SAy. Gi s () l mp qua A v vung gc vi cnh SC, () ct SC ti I.
a) Xc nh giao im K ca SO vi mp().
b) Chng minh (SBD)(SAC) v BD//().
c) Xc nh thit din ca hnh chp ct bi mp().
Bi 31. Cho hnh chp S.ABCD c y l hnh thang vung ti A, hai y l AD = 2a, BC = a. Bit AB = a, SA = v SAy.
a) Chng minh (SAC)(SDC).
b) Dng thit din ca hnh chp khi ct bi mp(P) cha AB v vung gc vi mp(SDC). Tnh din tch thit din theo a.
Bi 32. Cho t din ABCD c AB = BC = a, AC = b, DB = DC = x, AD = y. Tm h thc lin h gia a, b, x, y :
a) (ABC)(BCD). b) (ABC)(ACD).
Bi 33. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy. Gi M, N l hai im thuc cc cnh BC, CD sao cho BM = x, DN = y. Tm h thc lin h gia a, x v y (SAM) (SMN).
Bi 34. Cho hnh chp t gic u S.ABCD c AB = SA = a. Gi I, K l trung im ca AB, CD. Mt mp(P) qua CD v vung gc vi (SAB) ct SA, SB ti M v N.
a) Chng minh (SIK)(SAB).
b) (P) ct hnh chp theo thit din l hnh g? Tnh din tch thit din theo a.
Bi 35. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = v vung gc vi y.
a) Chng minh (SCD)(SAD).
b) Ct hnh chp bi mp(P) cha AB v vung gc vi (SCD). Tnh theo a din tch thit din .
Bi 36. Cho hnh chp S.ABCD c y l hnh vung tm O, cnh a, SAy v SA = . Gi M l mt im thuc on AO sao cho AM = x, .
a) Gi H l hnh chiu ca M trn (SBC). Tnh MH.
b) Mp(P)AC ti M ct hnh chp theo mt a gic. Trnh by cch dng thit din ny.
c) Tm x din tch a gic ln nht.
Bi 37. Trong mp(P) cho tam gic ABC vung ti A vi AB = a, , SB(ABC) v SB = 2a.
a) Chng minh (SAC)(SAB).
b) Ly im M thuc on AB sao cho BM = x, 0 < x < a. Qua M dng mp(Q) song song vi AC v SB.Tnh din tch thit din ca (Q) vi hnh chp. Tm x din tch ny ln nht.
Vn 3. Gc. I. Gc gia hai ng thng.
Cch xc nh gc gia hai ng thng cho nhau a v b:
Chn im O thch hp, ri k hai ng thng i qua im O: a//a v b//b.
Cc phng php tnh gc:
+ S dng h thc lng trong tam gic:
nh l sin: nh l cos:
+ Tnh gc theo vect ch phng:
Ch . +
+
+ Nu a v b song song hoc trng nhau th .
Bi 1. Cho hnh chp S.ABC c SA = SB = SC = AB = AC = a, BC = . Tnh gc gia hai ng thng SC v AB.
Bi 2. Cho t din ABCD. Gi M, N l trung im ca BC v AD.
a) Tnh gc gia AB v DM, bit ABCD l t din u cnh bng a.
b) Tnh gc gia AB v CD, bit AB = CD = 2a v MN = .
c) Tnh gc gia AB v CD, bit AB = CD = 2a v MN = .
d) Tnh gc gia AB v CD, bit AB = 2a, CD = v MN = .
Bi 3. Cho t din ABCD c AB = AC = AD v , . Chng minh:
a) ABCD.
b) Nu I, J l trung im ca AB v CD th IJAB, IJCD.
Bi 4. Cho hnh chp S.ABCD c y l hnh thoi, SA = SB v SABC. Tnh gc gia hai ng thng SD v BC.
Bi 5. Cho t din ABCD. Gi I, J, H, K l trung im ca BC, AC, AD, BD. Hy tnh gc gia hai ng thng AB v CD trong cc trng hp:
a) T gic IJHK l hnh thoi c ng cho IH = IJ.
b) T gic IJHK l hnh ch nht.
Bi 6. Cho hnh hp ABCD.ABCD c cc cnh bng a (hnh hp thoi), , .
a) Tnh gc gia cc cp ng thng AB vi AD v AC vi BD.
b) Tnh din tch cc hnh ABCD v ACCA.
c) Tnh gc gia ng thng AC v cc ng thng AB, AD, AA.
Bi 7. Cho hnh hp ch nht ABCD.ABCD c AB = a, BC = b v AA = c.
a) Tnh gc gia hai ng thng AD v BC.
b) Tnh gc gia hai ng thng AB v AC.
Bi 8. Cho hnh chp S.ABC c SA = SB = SC = a v cc tam gic SAB, SBC, SCA vung ti S. Gi M l trung im BC. Tnh gc gia AC v SM.
Bi 9. Cho hnh chp S.ABCD c tt c cc cnh u bng a, y l hnh vung. Gi N l trung im SB. Tnh gc gia AN v CN, AN v SD.
Bi 10. Cho t din ABCD c cc tam gic ABD v DBC l cc tam gic u cnh a. Cho AD = .
a) Chng minh ADBC.
b) Tnh gc gia hai ng thng AB v CD.
II. Gc gia ng thng v mt phng.
Cch xc nh gc gia ng thng d v mp(P):
+ Xc nh hnh chiu d ca d trn mp(P) (Bng cch tm hnh chiu ca im B trn mp(P)).
+ Gc gia d v hnh chiu d chnh l gc gia ng thng d v mp(P).
Hnh 2
Ch : Xc nh hnh chiu ca mt im ln mt mt phng (hnh 2)
- Dng mt ng thng qua im y v vung gc vi mt phng
+ Dng ng thng qua A v song song vi ng thng a bit m a (P)
+ Dng mt phng (Q) qua A m (Q) (P)
+ Dng ng thng qua A m cng song song vi 2 mt phng (Q) v(R), trong (Q) v (R) cng vung gc vi (P)
- Xc nh giao im ca ng thng va dng v mt phng
- Giao im trn chnh l hnh chiu ca im ln mt
hnh 2: B l hnh chiu vung gc ca im A ln (P) Ch . + .
+ Nu hoc th .
+ Tnh cht ca trc ng trn:
a) N. Gi s O l tm ng trn ngoi tip a gic . ng thng i qua O v vung gc vi mp cha a gic gi l trc ng trn ngoi tip a gic cho.
b) Tnh cht: Nu th S thuc trc ng trn ngoi tip a gic .
Do , hnh chiu ca S trn mp cha a gic l tm ng trn O.Bi 1. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy v SA = . Tnh gc gia ng thng SC v mp(ABCD).
Bi 2. Cho hnh chp S.ABC c y l tam gic vung ti A, BC = a, SA = SA = SC = . Tnh gc gia ng thng SA v mp(ABC).
Bi 3. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy v SA = . Tnh gc gia:
a) SC v (ABCD). b) SC v (SAB).
a) AC v (SBC). d) SB v (SAC).
Bi 4. Cho hnh chp S.ABC c SAy, y l tam gic vung ti B. Bit . t . Gi I l hnh chiu ca B trn SC. Xc nh gc gia BI v mp(SAC) l .
Bi 5. Cho hnh chp S.ABCD c y l hnh thoi. Bit SD = , tt c cc cnh cn li u bng a.
b) Chng minh (SBD) l mt phng trung trc ca AC v SBD l tam gic vung.
c) Xc nh gc gia SD v mp(ABCD).
Bi 6. Cho tam gic ABC vung ti A, AB = a nm trong mp(P), cnh AC = v to vi (P) mt gc . Tnh gc gia BC v (P).
Bi 7. Cho hnh chp S.ABC c SA = SB = SC = v y ABC l tam gic u cnh a.
a) Gi H l hnh chiu ca S trn mp(ABC). Tnh SH.
b) Tnh gc gia SA v (ABC).
Bi 8. Cho hnh chp S.ABCD c SA vung gc vi y, y ABCD l hnh vung cnh a, bit gc gia SC v mt y l . Tnh s o gc:
a) Gia SC v (SAD).
b) Gia SC v (SAD).
Bi 9. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = v vung gc vi y.
a) Tnh gc gia BS v CD
b) Tnh gc gia SC v (ABCD).
c) Tnh gc gia SC v (SAB), SB v (SAC), AC v (SBC).
Bi 10. Cho hnh chp S.ABCD c SA vung gc vi y, y ABCD l hnh thang vung ti A, hai y l AD = 2a, BC = a. Bit SA = 2a, AB = a.
a) Chng minh SCD l tam gic vung.
b) Tnh gc gia SD v (SAC).
Bi 11. Cho hnh chp t gic u S.ABCD c cnh y bng a, tm O. Gi M, N l trung im ca SA, BC. Bit gc gia MN v (ABCD) l .
a) Tnh MN v SO.
b) Tnh gc gia MN v (SBD).
Bi 12. Cho hnh lng tr ng ABC.ABC c y l tam gic u cnh a. Bit BC hp vi mp(ABBA) gc .
a) Tnh AA.
b) Gi M, N l trung im ca AC v BB. Tnh gc gia MN v mp(BAC).
Bi 13. Cho hnh lng tr ng ABC.ABC c y ABC l tam gic vung cn ti A. Gi M, N l trung im ca AB v BC. Bit MN = a v MN hp vi y gc v mt bn (BCCB) gc .
a) Tnh cnh bn v cc cnh y ca lng tr theo a v .
b) Chng minh .
III. Gc gia hai mt phng.
Cch xc nh gc gia hai mt phng ct nhau theo giao tuyn c:
Cch 1:
+ Chn im I thch hp trn giao tuyn c.
+ Qua I v hai ng thng vung gc vi giao tuyn c v ln lt nm trong hai mp cho.Cch 2:
+Xc nh giao tuyn c ca hai mt phng
+ Dng mt phng (R) vung gc vi c. Gi s (R) ct hai mt phng theo giao tuyn a, b.
+ Tnh gc ca hai ng thng a v b. T suy ra gc ca hai mt phng.
Cch 3: + Dng hai ng thng a v b ln lt vung gc vi hai mt phng;
+ Tnh gc ca a v b. T suy ra gc ca hai mt phng
Ch . +
+ Nu hoc th .
+
Bi 1. Cho t din ABCD c AD(BCD) v AB = a. Bit BCD l tam gic u cnh 2a. Tnh gc gia hai mp(ACD) v (BCD).
Bi 2. Cho hnh chp t gic u S.ABCD c cnh y bng a, mt bn hp vi mt y gc . Tnh gc gia cc mt phng:
a) (SAB) v (SCD).
b) (SAB) v (SBC).
Bi 3. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy, SA = x. Tm x hai mp(SBC) v (SCD) to vi nhau gc .
Bi 4. Cho tam gic u ABC cnh a nm trong mp(P). Trn cc ng thng vung gc vi (P) v t B v C ly cc on BD = , CE = nm cng mt bn i vi (P).
a) Chng minh tam gic ADE vung. Tnh din tch tam gic ny.
b) Tnh gc gia hai mp(ADE) v (P).
Bi 5. Cho lng tr ABCD.ABCD c y ABCD l hnh vung tm O, cnh a, AA = a v AO(ABCD). Tnh gc hp bi:
a) Cnh bn v mt y.
b) Cnh bn v cnh y.
c) (BDDB) v (ABCD); (ACCA) v (ABCD).
Bi 6. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAB l tam gic u v (SAB)(ABCD). Tnh gc gia:
a) (SCD) v (ABCD).
b) (SCD) v (SAD).
Bi 7. Cho hnh chp S.ABCD c y l hnh thang vung ti A v D, c AB = 2a, AD = DC = a, c cnh SAy v SA = a.
a) Chng minh (SAB)(SCD) v (SAC)(SCB).
b) Gi l gc gia hai mp(SBC) v (ABCD). Tnh .
Bi 8. Cho t din SABC, hai mp(SAB) v (SBC) vung gc vi nhau v SA(ABC), SB = , , .
a) Chng minh BCSB. Tm im cch u 4 im S, A, B, C.
b) Xc nh hai mp(SAC) v (SCB) to vi nhau gc .
Bi 9. Cho hnh ch nht ABCD c tm O, BA = a, BC = 2a. Ly im S trong khng gian sao cho SO(ABCD), t SO = h. Gi M, N l trung im ca AB, CD.
a) Tnh gc gia (SMN) vi (SAB) v (SCD). Tm h thc lin h gia h v a (SMN) vung gc vi cc mp(SAB), (SCD).
b) Tnh gc gia hai mp(SAB) v (SCD). Tnh h theo a hai mo vung gc.
Bi 10. Cho hnh chp S.ABCD c y l hnh ch nht, BA = a, BC = 2a, cnh bn SAy v SA = a. Tnh:
a) Cc gc gia cc mp cha cc mt bn v mp y ca hnh chp.
b) Gc gia hai mp cha hai cnh bn lin tip hoc hai mt bn i din ca hnh chp.
Bi 11. Cho tam gic cn ABC, AB = AC = a, . Xt hai tia cng chiu Bt, Ct v vung gc vi mp(ABC). Ly im B thuc Bt, C thuc Ct sao cho BB = 3CC. Cho BB = a. Tnh gc gia hai mp(ABC) v (ABC), Tnh din tch tam gic ABC.
Bi 12. Cho hai mp(P) v (Q) vung gc vi nhau theo giao tuyn d. Ly hai im c dnh A, B thuc d sao cho AB = a. Gi SAB l tam gic u trong (P), ABCD l hnh vung trong (Q).
a) Tnh gc gia mp(SCD) vi cc mp(P) v (Q).
b) Gi O1 l giao im ca B1C v A1D, trong B1, D1 l trung im ca SA, SB. Gi H1 l giao im ca ng cao SH ca tam gic SAB vi mp(A1B1CD). Chng minh SO1 vung gc vi SA v CD. Tnh gc gia mp(A1B1O1) vi cc mp(P) v (Q).
Bi 13. Cho hnh chp S.ABCD c y l na lc gic u ni tip ng trn ng knh AB = 2a, SA = v vung gc vi y.
a) Tnh gc gia hai mp(SAD) v (SBC).
b) Tnh gc gia hai mp(SCD) v (SBC).
Bi 14. Cho hnh lng tr ng ABC.ABC c y l tam gic u cnh a, AA = a. Tnh gc gia hai mp(ABC) v (BCA).
Bi 15. Cho hnh chp S.ABC c y l tam gic vung cn nh B, AB = a, SA = 2a v vung gc vi y. Gi M l trung im ca AB. Tnh gc gia hai mp(SCM) v (ABC).
Bi 16. Cho hnh chp S.ABC c y l tam gic vung nh B, AB = a, BC = , SA = 2a v vung gc vi y. Gi M l trung im ca AB.
a) Tnh gc gia hai mp(ABC) v (SBC).
c) Tnh gc gia hai mp(SCM) v (ABC).
Bi 17. Cho hnh chp S.ABCD c SA = SB = SD = , y l hnh thoi cnh a v .
a) Chng minh (SAC)(ABCD) v SBBC.
b) Tnh gc gia hai mp(SBD) v (ABCD).
Bi 18. Cho hnh chp S.ABCD c SAy, hai mt bn (SBC) v (SCD) hp vi nhau gc . Mt y ABCD c AB = AD = a, CB = CD = . DADC v BABC. Tnh gc gia:
a) SC v (ABCD).
b) (SBD) v (ABCD).
Bi 19. Cho hnh chp M.ABC c y l tam gic u cnh a.
a) Tnh gc gia hai mp(ABC) v (MBC) khi bit din tch tam gic MBC = .
b) Cho MA = a. Tnh gc gia hai mp(MBC) v (MAB).
Vn 4. Khong cch.
I. Khong cch t mt im n ng thng, n mt phng khong cch t ng thng n mt phng song song khong cch gia hai mt phng song song.
Khong cch t im A n ng thng , n mp(P):
+ , H l hnh chiu ca A trn .+ , H l hnh chiu ca A trn mp(P).
Khong cch gia hai ng thng song song: l khong cch t mt im nm trn ng thng ny n ng thng kia.
Khong cch gia hai mt phng song song: l khong cch t mt im nm trn mt phng ny n mt phng kia.
Cch xc nh hnh chiu ca im A trn mp(P):+ Chn mt ng thng .
+ Dng mp(Q) qua A v vung gc vi a. Gi s (Q) ct (P) theo giao tuyn l b.
+ Trong (Q), v AHb.
Khi , H l hnh chiu ca A trn mp(P).
Ch . + Bi ton tm cc khong cch ni trn thc cht l tm hnh chiu ca mt im trn ng thng hay mt phng.
+ vi
Bi 1. Cho hnh chp S.ABCD c y l hnh vung tm O, cnh a, SAy v SA = a. Gi I, J l trung im ca SC v AB.
a) Chng minh IO(ABCD).
b) Tnh khong cch t I n CJ.
Bi 2. Cho hnh chp S.ABCD c y l na lc gic u ni tip trong ng trn ng knh AD = 2a, SAy v SA = .
a) Tnh khong cch t A, B n (SCD).
b) Tnh khong cch t AD n (SBC).
Bi 3. Cho hnh hp thoi ABCD.ABCD c cc cnh u bng a v . Tnh khong cch gia hai mp y (ABCD) v (ABCD).
Bi 5. Cho hnh chp S.ABC c y l tam gic vung ti A vi BC = 2a, . Gi M l trung im BC. Bit SA = SB = SC = .
a) Tnh chiu cao ca hnh chp.
b) Tnh khong cch t S n AB.
Bi 6. Cho hnh chp t gic u c y l hnh vung tm O, AB = 2a, SA = 4a. Tnh:
a) Khong cch t O n (SAB).
b) Khong cch t A n (SCD).
Bi 7. Cho hnh lng tr ng ABC.ABC c AA = a, y ABC l tam gic vung ti A c BC = 2a, AB = .
a) Tnh khong cch t AA n mp(BCCB).
b) Tnh khong cch t A n (ABC).
c) Chng minh AB(ACCA) v tnh khong cch t A n (ABC).
d) Gi M, N, P l trung im ca AA, BB, CC. Tnh khong cch gia hai mp(BMN) v (ACP).
Bi 8. Cho t din DABC, ABC l tam gic vung cn ti B, AC = 2a, DAC l tam gic u v (DAC)(ABC). Gi O l trung im ca AC. Tnh cc khong cch :
a) T D n (ABC).
b) T O n (DBC).
Bi 9. Cho tam gic ABC vi AB = 7cm, BC = 5cm, AC = 8cm. Trn ng thng vung gc vi mp(ABC) ti A ly im S sao cho AS = 4cm. Tnh khong cch t O n BC.
Bi 10. Cho gc vung xOy v im M nm ngoi mp cha gc vung. Bit OM = 23cm v khong cch t M ti hai cnh gc vung Ox, Oy u bng 17cm. Tnh khong cch t M n mp cha gc vung.
Bi 11. Cho tam gic ABC vung ti A, cnh AB = a v nm trong mp(P), cnh AC = v to vi (P) gc .
a) Tnh khong cch t C ti (P).
b) Tnh gc to bi BC v (P).
Bi 12. Cho hnh chp S.ABC c SAy, SA = 2a, tam gic ABC vung ti C vi AB = 2a, . Gi M l im di ng trn AC, H l hnh chiu ca S trn BM.
a) Chng minh AHBM.
b) t AM = x, . Tnh khong cch t S n BM theo a v x. Tm x khong cch ny ln nht, nh nht.
Bi 13. Cho tam gic ABC cn nh A c . Ly im S nm ngoi mp cha tam gic sao cho SA = a. Tnh khong cch t A ti (SBC).
Bi 14. Cho t din DABC c ABC l tam gic vung ti A, AB = a, AC = 2a. Cc mt (DAB) v (DAC) cn hp vi (ABC) gc , mp(DBC)(ABC).
a) Tnh khong cch t D n mp(ABC) theo a v .
b) Tm s o khi bit d = . Khi hy tnh khong cch t C n (DAB).
Bi 15. Cho hnh chp u S.ABC c y ABC l tam gic u tm O, AB = a, mt bn hp vi mt y gc . Tnh cc khong cch:
a) T O n (SAB).
b) T C n (SAB).
Bi 16. Cho hnh chp S.ABC c v SA = SB = SC = a. Gi I l trung im ca AC.
a) Chng minh SI(ABC).
b) Tnh khong cch t S n mp(ABC).
Bi 17. Cho hnh chp S.ABC c y l tam gic cn ti A, AB = a, , SA = SB = SC = . Tnh chiu cao ca hnh chp.
Bi 18. Cho hnh chp S.ABC c y l tam gic vung cn ti B, AC = 2a, SA = a v vung gc vi y.
a) Chng minh (SAB)(SBC).
b) Tnh chiu cao ca hnh chp.
c) Gi O l trung im ca AC. Tnh khong cch t O n (SBC).
Bi 19. Trong mp(P) cho tam gic ABC vung ti A c BC = 2a, . Dng hai on BB = a, CC = 2a cng vung gc vi mp(P) v cng mt bn vi (P). Tnh khong cch t:
a) C n mp(ABB).
b) Trung im BC n mp(ACC).
c) B n mp(ABC).
d) Trung im BC n mp(ABC).
Bi 20. Cho hnh chp S.ABC c y l tam gic u cnh a, mt bn SBC vung gc vi y. Gi M, N, P l trung im ca AB, SA, AC.
a) Chng minh (MNP)//(SBC).
b) Tnh khong cch gia hai mp(MNP) v (SBC).
Bi 21. Cho hnh chp S.ABCD c y l hnh thang vung ng cao AB = a, BC = 2a, SA = a v vung gc vi y. Ngoi ra, cn c SCBD.
a) Chng minh tam gic SBC vung.
b) Tnh di AD.
c) Gi M l im trn SA sao cho AM = x, . Tnh khong cch t D n BM theo a v x. Tm x khong cch ny ln nht, nh nht.
Bi 22. Cho hnh chp S.ABCD c y l hnh vung tm O, cnh a, SAy v SA = .
a) Tnh khong cch t A ti mp(SBC).
b) Tnh khong cch t O n mp(SBC).
c) Tnh khong cch t trng tm ca tam gic SAB n mp(SAC).
Bi 23. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = a v SAy.
a) Gi I l trung im ca SD. Chng minh AI(SCD).
b) Tnh khong cch t trng tm ca tam gic SBC n mp(ABCD).
Bi 24. Cho hnh chp S.ABCD c y l hnh ch nht vi AB = a, AD = 2a, SA = a v SAy. Gi I, M l trung im ca SC, CD.
a) Tnh khong cch t A n (SBD).
b) Tnh khong cch t I n (SBD).
c) Tnh khong cch t A n (SBM).
Bi 25. Cho hnh thoi ABCD tm O, cnh a v AC = a. T trung im H ca AB dng SH(ABCD) vi SH = a.
a) Tnh khong cch t H n (SCD).
b) Tnh khong cch t O n (SCD).
c) Tnh khong cch t A n (SBC).
Bi 26. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = a v vung gc vi y.
a) Tm trn mp(ABCD) im I cch u ba im S, B, C. Tnh SI v khong cch t I n (SBC).
b) Tm trn mp(SBC) im J cch u ba im B, C, M vi M l trung im ca CD. Tnh JB.
Bi 27. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = 2a v SAy.
a) Tnh khong cch t A n (SBC), t C n (SBD).
b) Gi M, N l trung im ca AB, AD. Tnh khong cch t MN n (SBD).
Bi 28. Cho hnh chp S.ABCD c SA = 2a v y, y l hnh thang vung ti A v B, AB = BC = a, AD = 2a.
a) Tnh khong cch t A, B n (SCD).
b) Tnh khong cch t AD n (SBC).
Bi 29. Cho hnh chp S.ABCD c y l hnh vung cnh a. mt bn SAB l tam gic cn ti S v (SAB)mt y, cnh bn SC to vi mt y gc . Tnh:
a) Chiu cao ca hnh chp S.ABCD.
b) Khong cch t chn ng cao hnh chp n mp(SCD).
c) Din tch thit din ca hnh chp khi ct bi mp trung trc ca on BC.
Bi 30. Cho hnh chp S.ABCD c y l hnh thoi vi , BD = a, SAy, gc gia mp(SBC) v mp y l . Tnh:
a) ng cao ca hnh chp.
b) Khong cch t A n (SBC).
Bi 31. Cho hnh chp S.ABCD c SAy, y l hnh thoi tm O, SA = AC = 2a, . Tnh:
a) Khong cch t O n SC.
b) Khong cch t D n SB.
Bi 32. Cho hnh hp ch nht ABCD.ABCD c AB = a, BC = b, CC = c.
a) Tnh khong cch t AA n mp(BDDB).
b) Gi M, N l trung im ca AA, BB. Tnh khong cch t MN n (ABCD).
c) Chng minh (ABD)//(CBD). Tnh khong cch gia hai mp ny.
Bi 33. Cho hnh hp ch nht ABCD.ABCD c AB = a, AD = b, AA = c.
a) Tnh khong cch t im B n mp(ACCB).
b) Tnh khong cch gia hai mp(ABC) v (ACD), trong trng hp a = b = c.
Bi 34. Cho hnh lng tr ABC.ABC c tt c cc cnh bn v cnh y u bng a. Cnh bn ca lng tr to vi mt y gc v hnh chiu vung gc ca A trn mp(ABC) trng vi trung im ca BC.
a) Tnh khong cch gia hai mt y ca lng tr.
b) Chng minh mt bn BCCB l hnh vung.
Bi 35. Cho hnh lng tr ABC.ABC c y ABC l tam gic u tm O, cnh a, hnh chiu ca C trn mp(ABC) trng vi tm ca y. Cnh bn CC hp vi mt y (ABC) gc . Gi I l trung im ca AB. Tnh cc khong cch:
a) T O n CC.
b) T C n IC.
c) T C n AB.
Bi 36. Cho hnh lng tr ABC.ABC, y ABC l tam gic u cnh a, hnh chiu ca C trn (ABC) trng vi tm O ca y. Cnh CC hp vi mt y (ABC) gc . Tnh khong cch t C n mp(ABBA).
II. Khong cch gia hai ng thng cho nhau.
Phng php tnh khong cch gia hai ng thng cho nhau:
+ Tnh thng qua khong cch gia ng thng v mp song song.
+ Tnh thng qua khong cch gia hai mp song song cha hai ng thng cho.
+ Tnh di ng vung gc chung.
Cch xc nh ng vung gc chung ca hai ng thng cho nhau:Cch 1: Xc nh mt phng (P) cha a v song song vi b
Tnh khong cch t 1 im bt k trn a n (P)
Cch 2: Xc nh cp mt phng (P), (Q) ln lt cha a, b m chng song song vi nhau
Tnh khong cch t mt im bt k thuc mt phng ny ti mt phng kiaCch 3: Tnh di on vung gc chung:
* Xc nh on vung gc chung:
TH1: Nu ab
Xc nh mt phng (P) cha a (hoc cha b), vung gc vi ng thng b
T giao tuyn A ca a v (P). K AB vung gc vi b, Bb
- Khi AB chnh l on vung gc chng.
TH2: Nu a khng vung gc vi b
Cch 1:
Dng mt phng (P) cha a v song song vi b
Xc nh hnh chiu b ca b ln mt phng (P)
Gi B l giao im ca a v b. Trong mt phng(b,b) k BA vung gc vi b ti A
Khi AB chnh l on vung gc chung.
Cch 2:
Dng mt phng (P) vung gc vi a, ct a ti O.
Xc nh hnh chiu b ca b ln mt phng (P).
T O k OH vung gc vi b ti H.
T H k HB song song vi a, ct b tai B. T B k BA song song vi OH, ct a ti A.
Khi AB chnh l on vung gc chung
Nhn xt: AB=OH
* Tnh di on vung gc chungBi 1. Cho hnh hp ch nht ABCD.ABCD c AB = a, AD = b, AA = c. Tnh khong cch gia hai ng thng BB v AC.
Bi 2. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy v SA = a. Tnh khong cch gia hai ng thng:
a) SB v AD. b) BD v SC.
Bi 3. Cho hnh chp tam gic u S.ABC c cnh y bng 3a, cnh bn bng 2a. Gi G l trng tm ca tam gic ABC.
a) Tnh chiu cao ca hnh chp.
b) Tnh khong cch gia hai ng thng AB v SG.
Bi 4. Cho hnh chp S.ABCD c y l hnh thoi tm O, cnh a, gc A bng v c ng cao SO = a.
a) Tnh khong cch t O n (SBC).
b) Tnh khong cch gia hai ng thng SB v AD.
Bi 5. Cho hnh lng tr ng ABC.ABC c y l tam gic u cnh a v AA = . Gi O, O l trung im ca AB, AB.
a) Chng minh .
b) Tnh khong cch gia hai ng thng AB v BC.
Bi 6. Cho hnh lp phng ABCD.ABCD.
a) Chng minh BC(ABCD).
b) Xc nh v tnh di on vung gc chung ca AB v BC.
Bi 7. Cho hnh hp ch nht ABCD.ABCD c AB = AA = a, AC = 2a.
a) Tnh khong cch t D n (ACD).
b) Tm ng vung gc chung ca cc ng AC v CD. Tnh khong cch gia hai ng y.
Bi 8. Cho t din OABC c OA, OB, OC i mt vung gc vi nhau v OA = OB = OC = a. Gi I l trung im ca BC. Hy dng v tnh di on vung gc chung ca:
a) OA v BC. b) AI v OC.
Bi 9. Cho hnh chp S.ABC c SA = 2a v y, y l tam gic vung cn ti B vi BA = a. Gi M l trung im ca AC. Dng v tnh di on vung gc chung ca hai ng thng SM v BC.
Bi 10. Cho hai tam gic cn ABC v ABD c chung y BC v nm trn hai mp khc nhau.
a) Chng minh ABCD.
b) Xc nh ng vung gc chung ca AB v CD.
Bi 11. Cho t din ABCD c ABC l tam gic u cnh a, ADBC, AD = a v khong cch t D n BC l a. Gi H l trung im BC v I l trung im ca AH.
a) Chng minh BD(ADH) v DH = a.
b) Chng minh DI(ABC).
c) Xc nh v tnh di on vung gc chung ca AD v BC.
Bi 12. Cho hnh chp S.ABC c y l tam gic vung C, SAy, AC = a, BC = b, SA = h. Gi M, N l trung im ca AC, SB.
a) Tnh di MN.
b) Tm h thc lin h gia a, b, h MN l on vung gc chung ca AC v SB.
Bi 13. Cho hnh chp S.ABC c y l tam gic vung ti C, CA = b, CB = a, SA = h v SAy. Gi D l trung im ca AB. Tnh:
a) Gc gia hai ng thng AC v SD.
b) Tnh khong cch gia hai ng thng AC v SD.
c) Tnh khong cch gia hai ng thng BC v SD.
Bi 14. Cho hnh chp S.ABC c SAy, y l tam gic u cnh a, SA = . Tnh khong cch gia hai ng thng SB v AC.
Bi 15. Cho hnh chp S.ABCD c y l hnh vung tm O, AB = a. ng cao SO ca hnh chp vung gc vi mt y v SO = a. Tnh khong cch gia hai ng thng SC v AB.
Bi 16. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = h v y. Dng v tnh di on vung gc chung ca:
a) SB v CD. b) SC v BD. c) SC v AB.
Bi 17. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA = SB = SC = SD = . Gi I, K l trung im ca AD, BC.
a) Chng minh (SIK)(SBC).
b) Tnh khong cch gia hai ng thng SB v AD.
Bi 18. Cho hnh chp S.ABCD c y l hnh ch nht, AB = 2a, BC = a. Cc cnh bn hnh chp bng nhau v bng .
a) Tnh chiu cao ca hnh chp.
b) Gi E, F l trung im ca AB, CD; K l im bt k thuc AD. Chng minh khong cch gia hai ng thng EF v SK khng ph thuc vo v tr ca K, hy tnh khong cch theo a.
Bi 19. Cho hnh vung ABCD cnh a, I l trung im AB. Dng IS(ABCD) sao cho . Gi M, N, P l trung im ca BC, SD, SB. Hy dng v tnh di on vung gc chung ca cc cp ng thng sau:
a) AB v SD. b) SA v BD. c) NP v AC. d) MN v AP.
Bi 20. Cho hnh chp S.ABCD c y l hnh vung cnh a, SAy v SA = a. Tnh :
a) Khong cch t S n mp(ACD) vi A l trung im SA.
b) Khong cch gia AC v SD.
Bi 21. Cho hnh chp S.ABCD c y l hnh vung cnh a, tm O. Gi H l trung im ca AD, SHy, SAD l tam gic u.
a) Chng minh
b) Xc nh v tnh di on vung gc chung ca hai ng thng SH v BC.
c) Gi M, N, K l trung im ca SA, SC, AB. Xc nh v tnh di on vung gc chung ca cc cp ng thng sau:
+ MN v BD. + DM v NK.
Bi 22. Cho lng tr ABC.ABC c cc mt bn u l hnh vung cnh a. Gi D, E, F l trung im ca cc cnh BC, AC, CD. Dng v tnh di on vung gc chung ca:
a) AB v BC. b) DE v AB. c) AB v BC. d) DE v AF.
Bi 23. Cho hnh hp ng ABCD.ABCD c y l hnh thoi cnh a, gc A bng , gc ca ng cho AC v mp y bng .
a) Tnh ng cao ca hnh hp .
b) Tm ng vung gc chung ca AC v BB. Tnh khong cch gia hai ng thng .
A
B
B
Hnh 1
d
d'
EMBED Equation.DSMT4
PAGE 1
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