[123doc.vn] Khai Thac 73 Noi Dung Tu 1 Bai Toan Hinh Hoc Doc

7/24/2019 [123doc.vn] Khai Thac 73 Noi Dung Tu 1 Bai Toan Hinh Hoc Doc

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bi: Cho hnh chp SABCD c yABCD l hnh vung tm O cnh a. SA

(ABCD), SA = 3a . Gi H, I, K lnlt l hnh chiu vung gc ca A trnSB, SC, SD v J l hnh chiu ca Btrn SC. Gi M, N, P, Q ln lt ltrung im ca AB, AD, BC, SC.

D'

P

Q

N

M

J

I

K

H

O

A B

D C

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A. Chng minh ng thng vung gc vi mt phng

1) BC ( SAB) 2) CD ( SAD) 3) AH ( SBC) 4) AK ( SCD) 5) SC ( AHK)

6) BD (SAC) 7) SC ( AIK) 8) HK (SAC) 9) OM (SAB) 10) ON ( SAD)11) BC (OPQ) 12) AB (OMQ) 13) AD (ONQ) 14) SC ( JBD)

B. Chng minh hai ng thng vung gc1) BC SB 2) CD SD 3) BD SO 4) BD SC 5) AH SC6) AK SC 7) AI HK 8) DJ SC

C. Chng minh hai mt phng vung gc1) (SBC) ( SAB) 2) (SCD) ( SAD) 3) (AHK) (SBC) 4) (AHK) ( SCD) 5) (SBD) (SAC)

6) (AHK) (SAC) 7) (OQM) (SAB) 8) (OQN) (SAD) 9) (OPQ) ( (SBC) 10) (SAC) ( JBD)

11) (SBC) ( JBD) 12) (SCD) (JBD)

D. Tnh khong cch t1 im n 1 mt phng1) C; (SAB) 2) C; (SAD) 3) A; (SBC) 4)A; (SCD) 5)A; (SBD)6) O; (SAB) 7)O; (SAD) 8)O; (SBC) 9)O; (SCD) 10)S; (AHK)11)S; (JBD) 12)Q; (ABCD)

E. Tnh khong cch t1 im n 1 ng thng1) A; SC 2)O; SC 3)O;SB 4)O;SD 5)


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F. Tnh khong cch gia 2 ng thng1) AD; SC 2)AB; SC 3)BC; SA 4)CD; SA 5)AB; SO6) CD; SO 7)BC; SD 8)AD; SB

G. Tnh gc gia 1 ng thng v 1 mt phng1)SB; (ABCD) 2)SC; (ABCD) 3)SD; (ABCD) 4)SO; (ABCD) 5) SC; (SAB)6) SC;( SAD) 7)SO;(SAB) 8)SO;(SAD) 9) SA;(SCD) 10)SA;(SBC)

H. Tnh gc gia 2 mt phng1)(SBC); (ABCD) 2)(SCD); (ABCD) 3) (SBD); (ABCD) 4) (SBC); (SAB) 5) (SCD); (SAD)6) (SCD); (SAB) 7) (SBC); (SCD) 8) (SBD); (SCD) 9) (SBD); (SBC)

K.Cc cu hi mang tnh tng hp

Cho hnh chp SABCD c y ABCD l hnh vung tm O cnh a. SA (ABCD), SA = 3a . Gi H,I, K, ln lt l hnh chiu vung gc ca A trn SB, SC, SD v J l hnh chiu ca B trn SC. Chngminh rng1) AH,AK,AI cng nm trn mt mt phng.b) Tgic AKIH c hai ng cho vung gc2)Tnh din tch thit din ct hnh chp bi mt phng i qua A v vung gc vi SC3) Tnh thtch khi chp S.AKIH4)Tnh din tch thit din ct bi hnh chp v mt phng i qua BD v vung gc vi SC ti J.5) Tnh thtch khi chp S.BDJ6) Gi G l giao im ca BN v AC.Tnh thtch khi chp QAGB.8)Tnh thtch tdin C.JDB

9) Giscc mt phng (ASB),(ASD) v (ABD) ln lt to vi mt phng (SBD) cc gc a,b.c.Chng minh rng:

2 2 2

2 2 2 2

) os os os 1.

) SBD ASB ASD ABD

a c a c b c c

b S S S S

LI GII

A. Chng minh ng thng vung gc vi mt phng1) BC ( SAB) 2) CD ( SAD) 3) AH ( SBC) 4) AK ( SCD) 5) SC ( AHK)6) BD (SAC) 7) SC ( AIK) 8) HK (SAC) 9) OM (SAB) 10) ON ( SAD)

11) BC (OPQ) 12) AB (OMQ) 13) AD (ONQ) 14) SC ( JBD)

1)

BC AB ( g/t hnh vung), BC SA ( SA ( ABCD),BC ( ABCD)) BC ( SAB)2) CD AD ( g/t hnh vung), CD SA ( SA ( ABCD),CD ( ABCD)) CD ( SAD)3) AH SB ( gt), AH BC ( BC ( SAB) (cu 1)) AH ( SBC)4)

AK SD ( gt), AK CD ( CD ( SAD) (cu 2)) AK ( SCD)5) AH ( SBC) (do cu 1) AH SC,AK ( SCD) ( do cu 2) AK SCSC ( AHK)6) BD AC ( g/t hnh vung), BD SA ( SA ( ABCD),BD ( ABCD)) BD ( SAC)


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7) AK ( SCD) ( do cu 2) AK SC, AI SC (GT) SC ( AIK)

8) SAB = SAD ( c.g.c) SB = SD v SB ASD , AH SB v AK SD ( cmt) c

SAH = SAK ( cnh huyn, gc nhn) SH = SK SH SK

SB SD HK // BD.Mt khc ta li

c BD ( SAC) ( cu 6) nn HK ( SAC)9)

OM l ng trung bnh ca tam gic ABC nn OM // BC, BC ( SAB) (cmt) OM(SAB).10)

ON l ng trung bnh ca tam gic ABD nn ON// AB //CD, CD ( SAD) (cmt)ON(SAD).

11)

OP l ng trung bnh ca tam gic BDC OP // CD,BC CD (gt hnh vung) BC OPOQ l ng trung bnh ca SAC OQ // SA,SA ( ABCD) OQ ( ABCD) BC OQBC ( OPQ)

Hoc c thchng minh:OQ v PQ ln lt l cc ng trung bnh ca cc tam gic SAC v SBC nn ng thi c OQ// SA V PQ // SB ( OPQ ) // ( SAB) m BC ( SAB ) (cu 1) BC ( OPQ).12) AB AD ( gt hv), AB SA ( SA ( ABCD) AB ( SAD)OQ v OM ln lt l cc ng trung bnh ca cc tam gic SAC v ABC nn ng thi c

OQ // SA V OM // BC//AD ( OMQ ) // ( SAD) li c AB ( SAD) ( cmt) AB ( OMQ)13) AD AB ( gt hv), AD SA ( SA ( ABCD) AD ( SAB)OQ v ON ln lt l cc ng trung bnh ca cc tam gic SAC v ABD nn ng thi c OQ// SA V ON//AB ( ONQ ) // ( SAB) li c AD ( SAB) ( cmt) AB ( OMQ)14) SC ( AHK) ( cu 5)) A,H,I,K ng phng ( AHIK) SC SC IH .Trong mp (SBC) c HI SC, BJ SC BJ // HI, li c BD // HK ( JBD) // ( AHIK), ta lic ( AHIK) SC ( cmt) nn SC (JBD).

B. Chng minh hai ng thng vung gc1) BC SB 2) CD SD 3) BD SO 4) BD SC 5) AH SC6) AK SC 7) AI HK 8) DJ SC

1)

BC (SAB) ( cu 1 phn A), SB (SAB) BC SB.2) CD (SAD) ( cu 2 phn A), SD (SAD) CD SD.3) BD (SAC) ( cu 6 phn A), SO (SAC) BD SO4) BD (SAC) ( cu 6 phn A), SC (SAC) BD SC5) AH (SBC) ( cu 3 phn A), SC (SBC) AH SC6) AK (SCD) ( cu 4 phn A), SC (SCD) AK SC7) AI ( SAC) , HK ( SAC ) ( cu 8 phn A) HK AI8) SC ( JDB) ( cu 14 phn A), DJ ( JDB) DJ SC.

C. Chng minh hai mt phng vung gc

1) (SBC) ( SAB) 2) (SCD) ( SAD) 3) (AHK) (SBC) 4) (AHK) ( SCD) 5) (SBD) (SAC)6) (AHK) (SAC) 7) (OQM) (SAB) 8) (OQN) (SAD) 9) (OPQ) ( (SBC) 10) (SAC) ( JBD)

11) (SBC) ( JBD) 12) (SCD) (JBD)1) BC (SAB) ( cu 1 phn A), BC (SBC) (SBC) (SAB)2) CD (SAD) ( cu 2 phn A), CD (SCD) (SCD) (SAD)3) AH (SBC) ( cu 3 phn A), AH (AHK) (AHK) (SBC)4)

AK (SCD) ( cu 4 phn A), AK (AHK) (AHK) (SCD)5) BD (SAC) ( cu 6 phn A), BD (SBD) (SBD) (SAC)


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6) SC (AHK) ( cu 5 phn A), SC (SAC) (AHK) (SAC)7) OM ( SAB) ( cu 9 phn A), OM (OQM )(OQM) ( SAB).8) ON ( SAD)( cu 10 phn A), ON (ONQ) ( ONQ) (SAD).9) BC ( OPQ)( cu 11 phn A) , BC (SBC) ( OPQ) (SBC).10)SC ( JBD)( cu 14 phn A) , SC (SAC) ( SAC) (JBD)11) SC ( JBD)( cu 14 phn A) , SC (SBC) ( SBC) (JBD).12)SC ( JBD)( cu 14 phn A) , SC (SCD) ( SCD) (JBD).

D. Tnh khong cch t1 im n 1 mt phng1) C; (SAB) 2) C; (SAD) 3) A; (SBC) 4)A; (SCD) 5)A; (SBD)6) O; (SAB) 7)O; (SAD) 8)O; (SBC) 9)O; (SCD) 10)S; (AHK)11)S; (JBD) 12)Q; (ABCD)

1)

CB ( SAB) ( cu 1 phn A) d( C,(SAB) = CB = a.2) CD ( SAD) ( cu 2 phn A) d( ,(SAD) = CD = a.3) AH ( SBC) ( cu 3 phn A) d( A,(SBC) = AH.

2 2 2 2 2 2 2

1 1 1 1 1 1 4 3

23 3

a

AHH SA AB AH a a a

4) AK ( SCD) ( cu 4 phn A) d( A,(SCD) = AK

5) (SAC) ( SBD) (cu 5 phn C.) (SAC) ( SBD) = SO , hAE SO AE (SBD)

SAO vung ti A nn c2 2 2 2 2 2

1 1 1 1 2 7

3 3E SA AO a a a

d( A,(SBD) = AE =21

7

a

6)OM (SAB) ( cu 9 phn A) d( O,(SAB) ) = OM =2

a

7)ON (SAD) ( cu 10 phn A) d( O,(SAB) ) = ON =2

a

8)(OPQ) ( (SBC) ( cu 9 phn C), (OPQ)( (SBC) = PQ, OPQ vung ti O nn hAF PQ th AF

(SBC) d( O,( SBC) ) = AF.

2 2 2 2 2 2

1 1 1 4 4 16 3

4AF 3 3

aAF

OP OQ a a a ,

9)Dthy d( O,(SCD) = d( O,(SBC) =3

4

a

2 2 2 2 2 2 2

1 1 1 1 1 1 4 3

23 3

aAK

K SA AD AH a a a


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10) Cu 1 phn A c c BC (SAB) ( SBC) (SAB) m ( SAB)(SBC ) = SB. Trong mtphng ( SAB) c AH SB ( SAB) ( SBC) AH SC.

Cu 2 phn A c c CD (SAD) ( SCD) (SAD) m ( SAD)(SCD ) = SD. Trong mtphng ( SAD) c AK SD ( SAD) ( SCD) AK SC.AK ( AHK)

SC AK, SC AI SC( AKI) SC ( AHK ) = I d( S, (AHK) ) = SITam gic SBC vung ti B, tam gic SHI vung ti I, hai tam gic ny ng dng

Tnh ton SB = 2 2 2SA AB a , SC = 2 2 2 23 2 5SA AC a a a

*)SH.SB = 2SA SH =2 23 3

2 2

SA a a

SB a

*)SIHSBC nn ta c

3.2. 3 52

55

aa

SI SH SH SB aSI

SB SC SC a

Vy d( S,(AHK) =3 5

5

a

11)Tnh d(S,(JBD)?

SJBSBC nn c2 24 4 5

55

SB a aSJ

SC a

12)OQ l ng trung bnh ca SAC nn OQ =1

2SA a

E. Tnh khong cch t1 im n 1 ng thng1) A; SC 2)O; SC 3)O;SB 4)O;SD 5)

1) Ta c AI SC (gt) SAC vung ti A nn h I SC

2 2 2 2 2 2

1 1 1 1 1 53 2 6I SA AC a a a

Vy d( A,SC) = AI =30

5

a

2)

V O l trung im AC nn d( O,SC ) =1 30

OJ ( , )2 10

ad A SC= =

3) SO =2

2 2 5

2

aSA AO

22

2

aOB d(O,SB) =

2 2

OS. 15

6

OB a

SO OB=

+

4) d(O,CD) = d(O,SB) =15

6

a

F. Tnh khong cch gia 2 ng thng1) AD; SC 2)AB; SC 3)BC; SA 4)CD; SA 5)AB; SO6) CD; SO 7)BC; SD 8)AD; SB


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1) AD// BC (gt hnh vung) (SBC) //AD d( AD,SC) = d( A , (SBC)) = AH =3

2

a= ( Cu 3

phn A)

2) AB // CD (SCD) // AB d( AB,SC) = d( A, (SCD)) = AK =3

2

a

3) AB SA,AB BC nn d( BC,SA) = AB = a4)

AD SA,AD CD nn d( CD,SA) = AD = a5) NP//ABSO ( SNP) //AB d( AB,SO) = d( A, ( SNP))

HAN SN ,NP // CD m DC (SAD) nn NP ( SAD) AN NP AN (SNP)d( AB,SO) = d( A, ( SNP) = AN

Tnh2 2 2 2 2 2

1 1 1 1 4 13

' 3 3A N SA A N a a a= + = + = AN=

39

3

a

6)HDD SN DD // AN nn DND = ANN DD = AN

d( CD,SO ) = DD = AN =39

3

a

7)BC//AD BC // ( SAD ) cha SD d( BC,SD ) = d( BC,(SAD) = d( C,(SAD) ) = CD = a.

8)AD// BC (gt hnh vung) (SBC) //AD d( AD,SB) = d( A , (SBC)) = AH =3

2

a= ( Cu 3

phn A)

G. Tnh gc gia 1 ng thng v 1 mt phng1)SB; (ABCD) 2)SC; (ABCD) 3)SD; (ABCD) 4)SO; (ABCD) 5) SC; (SAB)6) SC;( SAD) 7)SO;(SAB) 8)SO;(SAD) 9) SA;(SCD) 10)SA;(SBC)

1)

SA (ABCD) (gt) AB l hnh chiu ca SB trn ( ABCD)

( , ( ))SB A B CD = 0t an 3 60

SASB A SB A SBA

A B = = =

2) SA (ABCD) (gt) AC l hnh chiu ca SC trn ( ABCD) ( ,( ))SC A B CD =

0

6tan

2

SASCA SCA

A C = =

3) SA (ABCD) (gt) AD l hnh chiu ca SD trn ( ABCD) ( ,( ))S D A B CD =

0t an 3 60SA

SDA SDA SDA

A D

= = =

4) SA (ABCD) (gt) AO l hnh chiu ca SO trn ( ABCD)

( ,( ))S O A B CD =

tan 6SA

SOA SOA aA O

= =

5) BC ( SAB) SB l hnh chiu ca SC trn ( SAB) ( , ( )) ( ,SC SA B SC SB CSB= =

1tan2 2

BC aCS B

SB a= = =


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6) CD ( SAD) SD l hnh chiu ca SC trn ( SAD) ( ,( )) ( , )SC SA D SC SD CSD= =

1tan2 2

CD aCS B

SD a= = =

7) OM ( SAB) SM l hnh chiu ca SO trn ( SAB) ( ,( )) ( , )SO SA B SO SM OSM = =

t an

OM

OSM SM= , OM = 2

a

,SM =

22 2 2 13

3 4 2

a a

SA A M a+ = + =

8)ON ( SAD) SN l hnh chiu ca SO trn ( SAD)

( , ( )) ( , )SO SA D SO SN OSN = =

t an ON

OSNSN

= , OM =2

a,SN=

22 2 2 133

4 2

a aSA A N a+ = + =

9) AK ( SCD) SK l hnh chiu ca SA trn ( SCD) ( , ( )) ( , )SA SCD SA A K A SK = =

t an A K

A SKSK

= , SK=3

2

a,AK =

3

2

a 01tan 303

A KA SK A SK

SK = = =

10) AH ( SBC) SH l hnh chiu ca SA trn ( SBC) ( , ( )) ( , )SA SBC SA A H A SH = =

t an A H

A SHSH

= , SH=3

2

a,AH =

3

2

a 01tan 30

3

A HA SH A SH

SH = = =

H. Tnh gc gia 2 mt phng1)(SBC); (ABCD) 2)(SCD); (ABCD) 3) (SBD); (ABCD) 4) (SBC); (SAB) 5) (SCD); (SAD)6) (SCD); (SAB) 7) (SBC); (SCD) 8) (SBD); (SCD) 9) (SBD); (SBC)

1) (SBC) (ABCD) = BC ,BCAB ( gt hv) (1)

BCSA(do SA ( ABCD) ,BC AB ( gthv) BC (SAB) BC SB (2)

T(1) v (2) ta c (( ), ( )) ( , )SBC ABCD AB SB SBA v tan 03 60SA

SBA SBA

AB

2) (SCD) (ABCD) = CD ,CDAD ( gt hv) (1)CDSA(do SA ( ABCD) ,CD AD ( gthv) CD (SAD) CD SD (2)

T(1) v (2) ta c (( ), ( )) ( , )SCD ABCD AD SD SDA v tan 03 60SA

SDA SDAAD

3) (SBD) (ABCD) = BD ,BDAC ( gt hv) (1) SAB = SAD ( c.g.c) SBD cn ti S v O l trung im BD SO BD (2)

T(1) v (2) ta c (( ), ( )) ( , )SBD ABCD AO SO SOA v tan 6SA

SDAAO

4) SA( ABCD) SA BC, BCAB BC ( SAB) . Li c BC ( SBC) ( SBC) (

SAB) hay 0

(( ), ( )) 90SAB SBC .5)

SA( ABCD) SA CD, CDAB CD ( SAD) . Li c CD ( SCD) ( SCD) (

SAD) hay 0(( ), ( )) 90SAD SCD .

6) SA( ABCD) SA CD, CDAB CD ( SAD) .Li c AKSD, AK CD(do CD(SAD))AK ( SCD) (1)SA( ABCD) SA AD, ADAB AD ( SAB)(2)


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T(1) v (2) ta c (( ), ( )) ( , )SCD SAB AD AK DAK v do 0 0tan 3 60 30SDA SDA DAK

7) Ta c (SBC) ( SCD) = SC , SC ( JBD) (cmt) (( ), ( )) 2SBC SCD BJD BJO

*) Tam gic OBJ vung ti J c tan 15

3

OBBJO

JO .

8) AK ( (SCD), AE ( (SBD) (( ), ( )) ( , )SCD SBD AK AE EAK , cos 2 77

AEEAKAK

9) AH ( (SBC), AE ( (SBD) (( ), ( )) ( , )SBC SBD AH AE EAH , cos 2 7

7

AEEAH

AH

K.Cc cu hi mang tnh tng hp

Bi 1:Cho hnh chp SABCD c y ABCD l hnh vung tm O cnh a. SA (ABCD), SA = 3a .Gi H, I, K, ln lt l hnh chiu vung gc ca A trn SB, SC, SD v J l hnh chiu ca B trn SC.Chng minh rng1) AH,AK,AI cng nm trn mt mt phng.2) Tgic AKIH c hai ng cho vung gc

3)Tnh din tch thit din ct hnh chp bi mt phng i qua A v vung gc vi SC4) Tnh thtch khi chp S.AKIH5)Tnh din tch thit din ct bi hnh chp v mt phng i qua BD v vung gc vi SC ti J.6) Tnh thtch khi chp S.BDJ7) Gi G l giao im ca BN v AC.Tnh thtch khi chp QAGB.8)Tnh thtch tdin C.JDBBi gii:1)Trong phn A tcu 1),2) 3),4) cho ta kt lun SC AH, SC AK nn SC ( AHK )Tgithit ta cng c SC AK, SC AI SC ( AKI ) , qua A chc mt mt phng duy nhtvung gc vi SC vy ( AKH ) ( AKI) AH,AK,AI cng nm trm mt phng qua A v vung gcvi SC.

2) Ta chng minh c SAB = SAD SB = SD v ASB DSB sau chng minh c SHA = SKA SH = SK HK // BD chng minh BD (SAC) nn HK (SAC), AI ( SAC) HK AI.

3)V qua A chc mt phng duy nht vuong gc vi SC nn (AHK) SC = I vy thit din chnhl tgic AKIH.

SB = SD = 2a, SH = SK =3

2

a, SC = 5a , SI =

3 5

5

a,BD = 2a

. 3 2

4

SH BD aHK

SB

C din tch2

1 1 30 3 2 15. . .2 2 5 4 20AKIH

a a aS AI HK

4) Cch 1:

SI =3 5

5

a,

23 15

20AKIHa

S nn2 3

.

1 1 3 5 3 15 3 3. . . .

3 3 5 20 20S AKIH AKIH a a a

V S SI

Cch 2:

SB = SD = 2a, SH = SK =3

2

a, SC = 5a , SI =

3 5

5

a


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. ..

9 9. .

16 16S AHK

S AHK SABD

S ABD

V SA SH SKV V

V SA SB SD

..

.

27 27. .

20 20S IKH

S IHK SABD

S BCD

V SI SH SK V V

V SC SB SD

3 3

. .

9 27 9 3 3 3( ) .16 80 10 6 20S AKIH S ABD

a aV V

5)

Din tch thit din JBD l tng din tch hai tam gic JOB v JOD

M OJ =30

( , )10

ad O SC ,

2

2

aOD vy

21 30 2 15OJ. OJ. .

2 10 2 10JOD JBDa a a

S OD S OD

6) Cch 1:

SJ =54 5

5

a

2 3

.

1 1 15 4 5 2 3. . .

3 3 10 5 15S BJD JBDa a a

V S SJ

7)

Dthy G l trng tm ca tam gic ABD

G

D'

Q

N

A B

D C

S

32.

1 1 3. . 33 2 6S ABC

aV a a .Li c

3.

..

1 3. .

2 12S AQB

S AQB

S ABC

V SA SQ SB aV

V SA SC SB

G l trng tm ABD nn GO =1 1 1 1 2

( )3 6 6 2 3

AO AC CG AC AC

.. .

.

2 1 1 1. . .

3 2 3 3C QBG

C QBG S ABC

S ABC

V CG CQ CBV V

V CA CS CB

3. . .

1 1 1 3(1 )2 3 6 36Q ABG S ABC S ABC

aV V V


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J

O

A B

D C

S

8)

Ta c SJ =4 5

5

a,SC = 5a nn CJ =

5

5

a

.

.

1. .

5C JBD

S BCD

V CD CJ CB

V CD CS CB ,

3

. .

1 3

2 6S BCD S ABCDa

V V

Vy

3

.

3

30C JBDa

V

Ta bit AE ( SBD)Xt php chiu vung gc ln mt phng (SBD) ta c

ES S

ESD S

EBD S

.cos (1)

.cos (2)

.cos (3)

B A B

A B

A B

S S a

S S b

S S c

Mt khc ln lt xt cc php chiu vung gc ln cc mt phng (SAB),(SAD), (ABD) ta c

S

SD

BD

.cos (1')

.cos (2 ')

.cos (3')

A B SBD

A SBD

A SBD

S S a

S S b

S S c

Thvo htrn ta c

2S

2SD

2

BD

.cos (1")

.cos (2")

.cos (3")

E B SBD

E SBD

E SBD

S S a

S S b

S S c

Cng cc vca hcui ta c 2 2 2 2 2 2( os os os ) os os os 1SBD SBDS S c a c b c c c a c b c c

b) Tcu a) v h(1),(2),(3) ta c2 2 2AS

2 2 2AS

2 2 2

.cos

.cos

.cos

B SBD

D SBD

ABD SBD

S S a

S S b

S S c

Cng cc vv do kt qucu a) ta c 2 2 2 2) SBD ASB ASD ABDb S S S S


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P

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N

M

J

I

K

H

O

A B

D C

S

N'

E

Bi 2 :Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, SA (ABCD) v SA =2a.Trn cnh AD ly im M sao cho AM = x ( 0< x a ).

a) Tnh khong cch tim M n mt phng (SAC).b) Nu MH AC ti H.Tm vtr ca M thtch khi chp SMCH ln nht.


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H

O

A D

B C

S

M

HMH AC , do SA ( ABCD) v MH (ABCD) nnSA MH MH (SAC) D( M , ( SAC)) = MHMH // OD

2.. ax 22

2

ax

AM MH AM ODMH

AD OD AD a

2 2.

. .2 2.

.S AHM S AHM S AODS AOD

V AM AH x xV V

V AD AO a a

.. .

.

2( ). .S MCD S AHM S AOD

S ACD

V DS DC DM a x a xV V

V DS DC DA a a

2

. . . . .2

2

. . .

2( )(2 )

2(2 ) .

2

S MHC S ACD S AHM S DMC S AOD

S AOD S AOD S AOD

x a xV V V V V

a a

x x

x x a aV V Va a

Vy thtch ca khi chp S.MGC ln nht bng

32

.

1 1 3. . 3

4 3 12S AODa

V a a khi v chkhi

2 1x x x

a M Da a a

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