Hình Không Gian Thuần Túy

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Hnh hc khng gian cin trong cc kthi tuyn sinh i hc(chnh thc)

Trch tthi tuyn sinh i hc khi A-2014

Cho hnh chp S.ABCD c y ABCD l hnh vung cnh bng a,3

2

aSD = . Hnh

chiu vung gc ca S ln mt phng y (ABCD) l trung im ca cnh AB. Tnhtheo a thtch khi chp S.ABCD v khong cch tA n mt phng (SBD).

Hng dn gii

Gi H l trung im ca AB, suy ra ( )SH ABCD .

Do : SH HD . Ta c

( )2 2 2 2 2SH SD DH SD AH AD a= = + =

Suy ra3

.

1. .

3 3S ABCD ABCD

aV SH S = =

Gi K l hnh chiu vung gc ca H ln BD v E

l hnh chiu vung gc ca H ln SK. Ta c

( )BD HK

BH SHKBD SH

Suy ra BD HE m ( )HE SK HE SBD

Ta c: 2

.sin4

aHK HB KBH= = . Suy ra

2 2

.

3

HS HK aHE

HS HK

= =

+

Do : ( )( ) ( )( )2

; 2 ; 33

ad A SBD d H SBD HE = = =

Trch tthi tuyn sinh i hc khi B-2014Cho lng trABC.ABC c y l tam gic u cnh a. Hnh chiu vung gc caA ln mt phng (ABC) l trung im ca cnh AB, gc gia ng thng AC vmt phng y bng 600. Tnh theo a thtch ca khi lng trABC.ABC vkhong cch tim B n mt phng (ACCA).

Hng dn gii

Gi H l trung im ca AB, ( )'A H ABC v 0' 60A CH =

Do

3

' . tan ' 2

aA H CH A CH= =

. Do th

tch kh

i l

ng

trl3

. ' ' '

3 3

8ABC A B C

aV =

Gi I l hnh chiu vung gc ca H ln AC; K l hnhchiu vung gc ca H ln AI. Suy ra

( )( ), ' 'HK d H ACC A=

Ta c: 3

.sin4

aHI AH IAH= = ;

2 2 2 2

1 1 1 52 3 13

' 9 26

aHK

HK HI HA a= + = =


2/16



Do : ( )( ) ( )( )3 13

; ' ' 2 ; ' ' 213

ad B ACC A d H ACC A HK = = =

Trch tthi tuyn sinh i hc khi D-2014Cho hnh chp S.ABC c y ABC l tam gic vung cn ti A, mt phng bn SBC ltam gic u cnh a v mt phng (SBC) vung gc vi mt phng y. Tnh theo a thtch ca khi chp S.ABC v khong cch gia hai ng thng SA, BC.

Hng dn gii

Gi H l trung im ca BC, suy ra2 2

BC aAH = =

( )3

,2

aSH ABC SH = v

21.

2 4ABC

aS BC AH

= =

Thtch ca khi chp l3

.

1 3.

3 24S ABC ABC

aV SH S

= =

Gi K l hnh chiu vung gc ca H ln SA, Suyra HK SA .

Ta c ( )BC SAH BC HK

Do : HK l ng vung gc chung ca BC v SA.

Ta c2 2 2 2

1 1 1 16

3HK SH AH a= + = . Do : ( )

3;

4

ad BC SA HK = =

Trch tthi tuyn sinh i hc khi A-2013

Cho hnh chp S.ABC c y l tam gic vung ti A, 030ABC = , SBC l tam gic u cnh a vmt bn SBC vung gc vi y. Tnh theo a thtch ca khi chp S.ABC v khong cch tim C n mt phng (SAB)

Hng dn gii

Gi H l trung im ca BC, suy ra SH BC . M ( )SBC

vung gc vi ( )ABC theo giao tuyn BC, nn

( )SH ABC

Ta c:

0

0

3; sin 30 ;

2 2

3.cos30

2

a aBC a SH AC BC

aAB BC

= = = =

= =

Do :3

.

1. .

6 16S ABC

aV SH AB AC = =

Tam gic ABC vung ti A v H l trung im ca BC nn HA HB= . M ( )SH ABC , suy ra

.SA SB a= = Gi I l trung im ca AB, suy ra SI AB


3/16



Do :2

2 13

4 4

AB aSI SB= = . Suy ra : ( )( ) . .

3 6 39;

. 13

S ABC S ABC

SAB

V V ad C SAB

S SI AB

= = =

Trch tthi tuyn sinh i hc khi B-2013Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a. Mt SAB l tam gic u v nmtrong mt phng vung gc vi mt phng y. Tnh thtch khi chp S.ABCD v tnh khongcch tA n mt phng (SCD) theo a.

Hng dn giiGi H l trung im ca AB, suy ra SH vung gc vi AB

v3

2

aSH = .

M mt phng (SAB) vung gc vi mt phng (ABCD)theo giao tuyn AB, nn ( )SH ABCD .

Do :3

.

1 3.

3 6S ABCD ABCD

aV SH S = =

Do AB song song vi CD v H thuc AB nn( )( ) ( )( ), ,d A SCD d H SCD=

Gi K l trung im ca CD v I l hnh chiu vung gcca H trn SK. Ta c: HK CD .

M SH CD ( )CD SHK CD HI . Do : ( )HI SCD

Suy ra: ( )( ),d A SCD2 2

. 21

7

SH HK aHI

SH KH

= = =

+

Trch tthi tuyn sinh i hc khi D-2013Cho hnh chp S.ABCD c y ABCD l hnh thoi cnh a, cnh bn SA vung gc vi y, 0120BAD = , M l trung im ca cnh BC v 045SMA = . Tnh theo a th tch ca khi chpS.ABCD v khong cch tD n mt phng (SBC)

Hng dn gii 0120BAD ABC ABC= u

33 3

2 2ABCD

a aAM S = =

SAM vung ti A c 045SMA SAM = vung ti

A3

2

aSA AM = =

Do :3

.

1.

3 4S ABCD ABCD

aV SA S = =

Do AD song song vi BC nn ( )( ) ( )( ), ,d D SBC d A SBC = Gi H l hnh chiu vung gc ca A trn SM

Ta c: ( )AM BC

BC SAMSA BC

( ) ( )( ),BC AH AH SBC d A SBC AH =

Ta c: ( )( )

2 6 6

,2 4 4

AM a aAH d D SBC= = =

Trch tthi tuyn sinh Cao ng-2013Cho lng tru ABC.ABC c AB = a v ng thng AB to vi y mt gc bng


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600. Gi M v N ln lt l trung im ca cc cnh AC v BC. Tnh theo a thtch ca

khi lng trABC.ABC v di MNHng dn gii

( ) ' 'AA ABC A BA l gc gia AB vi y.

Suy ra: 0' 60 ' . tan ' 3A BA AA AB A BA a= = =

Do 3

. ' ' '3'.4

ABC A B C ABCaV AA S

= =

Gi K l trung im ca cnh BC.Suy ra MNK vung ti K, c

, ' 32 2

AB aMK NK AA a= = = =

Do : 2 213

2

aMN MK NK= + =

Trch tthi tuyn sinh i hc khi A-2012Cho hnh chp S.ABC c y l tam gic u cnh a. Hnh chiu vung gc ca S ln mtphng (ABC) l H thuc cnh AB sao cho 2HA HB= . Gc gia hai ng thng SC v mtphng (ABC) bng 060 . Tnh thtch ca khi chp S.ABC v tnh khong cch gia haing thng SA v BC theo a

Hng dn gii

Ta c: SCH l gc gia SC v mt phng (ABC). Suy ra 060SCH =

Gi D l trung im ca cnh AB. Ta c:3

,

6 2

a aHD CD= =

2 2 07 21, . tan 60

3 3

a aHC HD CD SH HC= + = = =

2 3

.

1 1 21 3 7. . .

3 3 3 4 12S ABC ABC

a a aV SH S

= = =

KAx song song vi BC, gi N v K ln lt l hnh chiuvung gc ca H ln Ax v SN. Ta c BC song song vi mt

phng (SAN) v3

2BA HA=

Nn ( ) ( )( ) ( )( )3

, , .2

d SA BC d B SAN d H SAN = =

Ta cng c: ( )Ax SHN Ax HK . Do : ( ) ( )( ),HK SAN d H SAN HK =

0

2 2

2 3 . 42, .sin 60 ,

3 3 12

a a SH HN aAH HN AH HK

SH HN

= = = = =

+

vy ( )42

,8

ad SA BC =

Trch tthi tuyn sinh i hc khi B-2012Cho hnh chp tam gic u S.ABC vi 2SA a= , AB a= . Gi H l hnh chiu vung gcca A ln cnh SC. Chng minh SC vung gc vi mt phng ( )ABH . Tnh thtch ca

khi chp S.ABH theo aHng dn gii


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Gi D l trung im ca cnh AB v O l tm ca tam gic

ABC. Ta cAB CD

AB SO

nn ( ) ,AB SCD Do AB SC

Mt khc SC AH , Suy ra ( )SC ABH

Ta c:3

2

aCD = ,

3

3

aOC = nn 2 2

33

3

aSO SC OC = =

Do :2. 11 1 11

.4 2 8

ABH

SO CD a aDH S AB DH

SC

= = = =

Ta c: 2 27

4

aSH SC HC SC CD DH = = = . Do :

3

.

1 7 11.

3 96S ABH ABH

aV SH S

= =

Trch tthi tuyn sinh i hc khi D-2012Cho hnh hp ng ABCD.ABCD c y l hnh vung, tam gic AAC vung cn

'A C a= . Tnh thtch ca khi tdin ABBC v khong cch tim A n mt phng

(BCD) theo aHng dn gii

Tam gic AAC vung cn ti A v 'A C a= nn

'2

aA A AC= = . Do : ' '

2

aAB B C= =

3

' ' '

1 1 2' '. ' '. . '

3 6 48ABB C ABB

aV B C S B C AB BB

= = =

Gi H l chn ng cao ktA ca tam gic AAB. Tac

( )'

'

AH A B

AH A BCAB BC

. Ngha l :

( ) ( )( )' , 'AH BCD AH d A BCD =

Ta c:2 2 2

1 1 1

'AH AB AA= + Do : ( )( )

6, '

6

ad a BCD AH = =

Trch tthi tuyn sinh Cao ng khi A-2012Cho khi chp S.ABC c y ABC l tam gic vung cn ti A, 2,AB a SA SB SC= = = .

Gc gia ng thng v mt phng (ABC) bng 060 . Tnh thtch ca khi chp S.ABCtheo a.

Hng dn giiGi H l trung im ca BC HA HB HC = = Kt hp vi githit

,SA SB SC SH BC SHA SHB SHC = = = =

( )

060

SH ABC

SAH

=

Tam gic ABC l tam gic vung cn ti A.

2 2AC AB a BC a AH a= = = = Tam gic SHA vung

30

.

1 1 3tan60 3 . . .

3 2 3S ABC

aSH AH a V AB AC SH = = = =


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Gi O;R ln lt l tm v bn knh ca mt cu ngoi tip hnh chp S.ABC. Suy ra Othuc ng thng SH, nn O thuc mt phng (SBC). Do : R l bn knh ng trn

ngoi tip tam gic SBC. Xt tam gic SHA ta c:0

2sin 60

SHSA a SBC = = l tam gic u

c di cnh bng 2a. Suy ra :0

2 2 3

2sin 60 3

a aR = =

Trch tthi tuyn sinh i hc khi A-2011Cho hnh chp S.ABC c y ABC l tam gic vung cn ti B, 2 ;AB BC a= = hai mt

phng ( )SAB v ( )SAC cng vung gc vi mt phng (ABC). Gi M l trung im ca

AM; mt phng qua SM v song song vi B, ct AC ti N. Bit gc gia hai mt phng(SBC) v (ABC) bng 060 . Tnh thtch ca khi chp S.BCNM v khong cch gia haing thng AB v SN theo a

Hng dn giiHai mt phng (SAB) v (SAC) cng vunggc vi (ABC) ( )SA ABC .

AB BC SB BC SBA l gc gia haimt phng (SBC) v mt phng

(ABC) 060 . tan 2 3SBA SA AB SBA a = = =

Mt phng qua SM v song song vi BC, ctAC ti N. //MN BC v N l trung im ca

AC. ;2 2

BC ABMN a BM a= = = =

Din tch :( ) 23

2 2BCNM

BC MN BM aS

+= = .

Thtch 3.1

. 33

S BCNM BCNM V S SA a= =

Kng thng i qua N, song song vi AB. H ( ) ( )//AD D AB SND

( ) ( )( ) ( )( ); , ,d AB SN d AB SND d A SND = = .

H ( ) ( ) ( )( ),AH SD H SD AH SND d A SND AH =

Tam gic SAD vung ti A:AH SD

AD MN a

= = ( )

2 2

. 2 39,

13

SA AD ad AB SN AH

SA AD

= = =+

Trch tthi tuyn sinh i hc khi B-2011

Cho lng trABCD.ABCD c y ABCD l hnh chnht, , 3AB A AD a= = . Hnhchiu vung gc ca im A1ln mt phng (ABCD) trung vi giao im ca AC v BD.Gc gia hai mt phng ( )1 1ADD A v (ABCD) bng

060 . Tnh thtch ca khi lng tr

cho v khong cch tin 1B n mt phng ( )1A BD theo a.

Hng dn giiGi O l giao im ca AC v BD. ( )1A O ABCD

Gi E l trung im ca AD1

OE AD

A E AD

Suy ra 1A EO l gc gia hai mt phng ( )1 1ADD A v (ABCD) 0

1 60A EO =


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Suy ra: 1 1 13

. tan tan2 2

AB aA O OE A EO A EO= = =

Din tch y 2. 3ABCD

S AB AD a= =

Thtch3

. ' ' ' ' 1

3

2ABCD A B C D ABCD

aV S A O= =

Ta c ( )

( )( ) ( )( )1 1 1 1

1 1 1

// //

, ,

B C A D B C A BD

d B A BD d C A BD CH

= =

Suy ra ( )( )1 12 2

. 3

2

CD CB ad B A BD CH

CD CB

= = =

+

Trch tthi tuyn sinh i hc khi D-2011Cho hnh chp S.ABC c y ABC l tam gic vung ti B, 3 , 4BA a BC a= = , mt

phng (SBC) vung gc vi mt phng (ABC). Bit 2 3SB a= v 030 .SBC = Tnh th

tch ca kh

i chp S.ABC v kho

ng cch t

im B

n m

t ph

ng (SAC) theo a.Hng dn gii

H( ) ( )

; .sin 3

SH BC SBC ABC

SH BC SH SB SBC a

= =

Din tch: 212

. 6ABC

S BA BC a= =

Thtch 3.1

. 2 33

S ABC ABC V S SH a= =

H ( ) ( )

( ) ( )( )

( )( ) ( )

,

, .

.cos 3 4

, 4 ,

HD AC D AC HK SD K SD

HK SAC HK d H SAC

BH SB SBC a BC HC

d B SAC d H SAC

=

= = =

=

Ta c 2 23

5 ; .5

HC aAC BA BC a HC BC BH a HD BA

AC= + = = = = =

2 2

. 3 7

14

SH HD aHK

SH HD

= =

+

.

Vy ( )( )6 7

, 47

ad B SAC HK = =

Trch tthi tuyn sinh Cao ng khi A-2011Cho hnh chp S.ABC c y ABC l tam gic vung cn ti B, AB = a, SA vung gc vimt phng (ABC), gc gia hai mt phng (SBC) v (ABC) bng 030 . Gi M l trung imca cnh SC. Tnh thtch ca khi chp S.ABM theo a

Hng dn gii


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Ta cSA BC

SB BC AB BC

Do : gc gia hai mt phng (SBC) v

(ABC) bng 030SBA =

. .

1 1. .

2 12S ABM S ABC

V V SA AB BC = =

0 3; . tan 303

aBC AB a SA AB= = = =

Vy3

.

3

36S ABM

aV =

Trch tthi tuyn sinh i hc khi A-2010Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a. Gi M v N ln lt l trungim ca cc cnh AB v AD; H l giao im ca N v DM. Bit SH vung gc vi mt phng

(ABCD) v 3SH a= . Tnh thtch ca khi chp S.CDNM v khong cch gia hai ngthng DM v SC theo a.

Hng dn giiThtch ca khi chp S.CDNM

2

2 2 22

1 1. .

2 2

5

8 4 8

CDNM ABCD AMN S S S SBC

AB AM AN BC BM

a a aa

=

=

= =

Vy

31 5 3.3 24

SCDNM CDNM

a

V S SH = =

Khong cch gia hai ng thng DM v SC. ADM DCN ADM DCN DM CN = = kt hp vi iu kin( )DM SH DM SHC

H ( )HK SC K SC HK l on vung gc chung ca DM v SC.

Do : ( ),d DM SC HK =

Ta c : ( )

2

2 2

2

5 2 3

, 19. 2 3

19

CD aHC

CN a

d DM S C SH HC aHK

SH HC

= =

=

= =

+

Trch tthi tuyn sinh i hc khi B-2010:Cho hnh lng trtam gic u ABC.ABC c AB a= , gc gia hai mt phng (ABC) v(ABC) bng 060 . Gi G l trng tm ca tam gic ABC. Tnh thtch ca khi lng trcho v bn knh mt cu ngoi tip tdin GABC theo a.

Hng dn gii


9/16



Thtch khi lng tr.Gi D l trung im ca BC ta c:

0' ' 60BC AD BC A D ADA =

Ta c: 23 3

' .tan ' ;2 4

ABC

a aAA AD ADA S= = =

Do :

3

. ' ' '

3 3' 8

ABC A B C ABC

a

V S AA= =

Bn knh mt cu ngoi tip tdin GABCGi H l trng tm ca tam gic ABC, suy ra:

( )// ' //GH AA GH ABC

Gi I l tm ca mt cu ngoi tip tdinGABC, ta c I l giao im ca GH vi ngtrung trc ca AG trong mt phng (AGH.Gi E l trung im ca AG, ta c:

2.

2

GE GA GAR GI

GH GH

= = =

Ta c2

2 2 2' 3 7; ;3 2 3 12

AA a a aGH AH GA GH AH = = = = + =

Do :27 2 7

2.12 12

a aR

a= =

Trch tthi tuyn sinh i hc khi D-2010Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, cnh bn SA = a; hnh chiu

vung gc ca nh S ln mt phng (ABCD) l im H thuc on AC,4

ACAH = . Gi

CM l ng cao ca tam gic SAC. Chng minh M l trung im ca SA v tnh thtchca khi tdin SMBC theo a

Hng dn giiChng minh M l trung im ca SA.

2 22 14;4 4

a aAH SH SA AH= = =

2 23 2; 2

4

aHC SC SH HC a SC AC= = + = =

Do : tam gic SAC cn ti C, Suy ra M l trung imca SATnh thtch ca khi tdin SBCM.M l trung im ca SA suy ra

. .

1 1

2 2SCM SCA SBCM B SCA S ABC

S S V V V = = =

31 14

6 48SBCM ABC

aV S SH = =

Trch tthi tuyn sinh Cao ng khi A-2010Cho hnh chp S.ABCD c y ABCD l hnh vung cnh a, mt phng (SAB) vung gcvi mt phng y, SA = SB, gc gia ng thng SC v mt phng y bng 045 . Tnhthtch ca khi chp S.ABCD theo a.


10/16



Hng dn giiGi I l trung im ca AB. Ta c

.SA SB SI AB= M hai mt phng(SAB) v mt phng (ABCD) vung gcvi nhau nn suy ra ( )SI ABCD

Gc gia SC v mt phng (ABCD bng 045SCI = , Suy ra

2 2 5

2

aSI IC IB BC = = + =

Thtch ca khi chp l3

.

1 5.

3 6S ABCD ABCD

aV SI S = = (n vthtch)

Trch tthi tuyn sinh i hc khi A-2009:Cho hnh chp S.ABCD c y ABCD l hnh thang vung ti A v D; 2AB AD a= = , CD a= ;gc gia hai mt phng (SBC) v (ABCD) bng 060 . Gi I l trung im ca cnh AD. Bit hai

mt phng (SBI) v (SCI) cng vung gc vi mt phng (ABCD). Tnh thtch ca khi chpS.ABCD theo a

Hng dn gii

( ) ( )

( ) ( ) ( )

SIB ABCDSI ABCD

SIC ABCD

K

( ) ( ) 060IK BC K BC BC SIK SKI =

Din tch hnh thang ABCD : 23ABCD

S a=

Tng din tch cc tam gic ABI v CDI

bng2

3

2

a, suy ra

23

2IBC

aS

=

( ) 2 2 2 3 5 3 15

5 . tan5 5

IBCS a a

BC AB CD AD a IK SI IK SKIBC

= + = = = = =

Thtch ca khi chp S.ABCD:3

1 3 15.

3 5ABCD

aV S SI = =

Trch tthi tuyn sinh i hc khi B-2009:Cho hnh trtam gic ABC.ABC c 'BB a= , gc gia ng thng BB v mt phng

(ABC) bng060 ; tam gic ABC vung ti C v

060BAC = . Hnh chiu vung gc ca B lnmt phng (ABC) trng vi trng tm ca tam gic ABC. Tnh thtch ca khi tdin AABCtheo a

Hng dn gii


11/16



Gi D l trung im ca AC v G l trngtm ca tam gic ABC ta c

( ) 0' ' 60B G ABC B BG =

3' '.sin '32

4

2

aB G BB B BG

aBD

aBG

= =

=

=

Tam gic ABC c:

3,

2 2 4

B AB ABBC AC CD

= = =

Ta li c:2 2 2 2

2 2 2 3 9 3 13 9 3;4 16 16 26 104

ABC

AB AB a a aBC CD BD AB S

+ = + = = =

Thtch ca khi tdin AABC:3

' '

1 9' .

3 208A ABC B ABC ABC

aV V B G S

= = =

Trch tthi tuyn sinh i hc khi D-2009:Cho hnh lng trng ABC.ABC c y ABC l tam gic vung ti B,; ' 2 ; ' 3AB a AA a A C a= = = . Gi M l trung im ca on thng AC, I l giao im ca AM v

AC. Tnh theo a thtch ca khi tdin IABC v khong cch tim A n mt phng(IBC)

Hng dn gii

H ( ) ( )IH AC H AC IH ABC ; IH l ng

cao ca tdin IABC.

Suy ra2 2 4

// ' '

' ' 3 3 3

IH CI aIH AA IH AA

AA CA = = = =

2 2 2' ' 5; 2AC A C A A a BC AC AB a= = = =

Din tch tam gic ABC: 21

.2

ABCS AB BC a

= =

Vy thtch ca khi tdin IABC:31 4

.3 9

ABC

aV IH S

= =

H ( )' 'AK A B K A B . V ( )' 'BC ABB A nn AK BC Suy ra ( )AK IBC

Khong cch tA n mt phng (IBC) l AK

'

2 22 '. 2 5

' 5'

AA BS AA AB aAKA B A A AB

= = =

+

Trch tthi tuyn sinh Cao ng khi A-2009:Cho hnh chp tgic u S.ABCD c , 2.AB a SA a= = Gi M, N v P ln lt l trung imca cc cnh SA, SB v CD. Chng minh rng ng thng MN vung gc vi ng thngSP. Thnh theo a thtch ca khi tdin AMNP

Hng dn gii


12/16



Ta c MN song song vi CD v SP vung gc viCD suy ra MN vung gc vi SPGi O l tm ca y ABCD. Ta c :

2 2 6

2

aSO SA OA= =

32

.

1 1 1 1 6

. .4 8 8 3 48AMNP ABSP S ABCDa

V V V SO AB= = = =

Trch tthi tuyn sinh i hc khi A-2008:Cho lng trtam gic ABC.ABC c di cnh bn bng 2a, y ABC l tam gic vung ti

A, , 3AB a AC a= = v hnh chiu vung gc ca nh A ln mt phng (ABC) l trung imca cnh BC. Tnh theo a thtch ca khi chp A.ABC v tnh cosin ca gc gia hai ngthng AA, BC

Hng dn giiGi H l trung im ca cnh BC. Suy ra

( )

2 2

'

1 13

2 2

A H ABC

AH BC a a a

= = + =

Do : 2 2 2 2 2' ' 3 3 ' 3A H A A AH a a A H a= = = =

Vy3

'.

1'

3 2A ABC ABC

aV A H S

= = (n vthtch)

Trong tam gic vung ABH c:2 2' ' ' ' 2HB A B A H a= + = nn tam gic BBH cn ti

B

t l gc gia hai ng thng AA v BC th 'B BH = . Vy1

cos2.2 4

a

a = =

Trch tthi tuyn sinh i hc khi B-2008:Cho hnh chp S.ABCD c y ABCD l hnh vung cnh 2a, SA a= , 3SB a= v mt phng(SAB) vung gc vi mt phng y. Gi M, N ln lt l trung im ca cc cnh AB, BC.Tnh theo a thtch ca khi chp S.BMDN v tnh cosin ca gc gia hai ng thng SM vDN.

Hng dn giiGi H l hnh chiu vung gc ca S ln AB, suy ra ( )SH ABCD . Do , SH l ng cao

ca hnh chp S.BMDNTa c: 2 2 2 2 23SA SB a a AB+ = + = nn tam gic SAB l tam gic vung ti S. Suy ra

.2

ABSM a= = Do tam gic SAM l tam gic u, suy ra

3

2

aSH =

Din tch ca tgic BMDN l 21

22

BMDN ABCDS S a= =

Thtch ca khi chp S.BMDN l

31 3

3 3BMDNa

V SH S = =(vtt)


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KME song song vi DN ( )E AD

Suy ra2

aAE = . t l gc gia hai ng thng

SM v DN. Ta c ( ),SM ME = . Theo nh l ba

ng vung gc ta c : SA AE

Suy ra:2 2 5 ,

2

aSE SA AE = + =

2 2 5

2

aME AM AE= + =

Tam gic SME l tam gic cn ti E nn

52cos55

2

SME

a

a

=

= =

Trch tthi tuyn sinh i hc khi D-2008:Cho lng trng tam gic ABC.ABC c y ABC l tam gic vung, AB BC a= = , cnhbn ' 2AA a= . Gi M l trung im ca cnh BC. Tnh theo a thtch ca khi lng trABC.ABC v khong cch gia hai ng thng AM, BC

Hng dn giiTgithit ta suy ra tam gic ABC l tam gic vung cn ti B

Thtch ca khi lng trl 2 3. ' ' '1 2

' 2. .2 2

ABC A B CV AA BC a a a= = = (vtt)

Gi E l trung im ca BB. Khi mt phng (AME)song song vi BC nn khong cch gia hai ng thng

AM, BC bng khong cch gia BC v mt phng(AME)

Nhn thy, khong cch tB n mt phng (AME) bngkhong cch tC n mt phng (AME)Gi h l khong cch tB n mt phng (AME). Do tdin BAME c BA, BM,BE i mt vung gc vinhau nn:

2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 4 2 7 7

7

ah

h BA BM BE h a a a a= + + = + + = =

Vy: khong cch gia hai ng thng BC v AM bng7

7

a

Trch tthi tuyn sinh Cao ng-2008:Cho hnh chp S.ABCD c y ABCD l hnh thang, 090 ;BAD ABC AB BC a= = = = , 2AD a= ,

SA vung gc vi y v 2SA a= . Gi M, N ln lt l trung im ca SA v SD. Chng minhrng BCNM l hnh chnht v tnh thtch ca khi chp S.BCNM theo a.

Hng dn giiTa c: MN l ng trung bnh ca tam gic SAD, suy ra MN song song vi AD v

1

2MN AD=

//MN BCBCNM

MN BC

=

l hnh bnh hnh (1)


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Mt khc

( ){ ( )2BC AB

BC SAB BC BMBC SA

T(1) v (2) ta suy ra BCNM l hnh chnhtTa c:

. .2 2BCNM BCM S BCNM S BCMS S V V = = 3

. .

1 1 1 1. . . .

3 6 6 2 6S BCM C SBM SBM SAB

aV V CB S CB S CB SA AB

= = = = =

Vy3

.3

s BCNM

aV = (vtt)

Trch tthi tuyn sinh i hc A-2007Cho hnh chp S.ABCD c y l hnh vung cnh a, mt bn SAD l tam gic u v nm trongmt phng vung gc vi y. Gi M, N, P ln lt l trung im ca cc cnh SB,BC,CD .Chng minh AM vung gc vi BP v tnh thtch ca khi tdin CMNP

Hng dn giiGi H l trung im ca AD. Do tam gic SAD ltam gic u nn SH vung gc vi AD.Do mt phng (SAD) vung gc vi mt phng(ABCD) nn SH vung gc vi BP (1).Xt hnh vung ABCD ta c:

( )2CDH BCP CH BP =

T(1) v (2) ta suy ra ( )BP SHC

V

( ) ( )

( )

////

//

MN SC

AMN SHCAN CH

BP AMN BP AM

KMK vung gc vi mt phng (ABCD), (Kthuc vo mt phng (ABCD)). Ta c:

1.

3CMNP CNP

V MK S = .

V2 3

1 3 1 3;

2 4 2 8 96CNP CMNP

a a aMK SH S CN CP V= = = = = (vtt)

Trch tthi tuyn sinh i hc B-2007Cho hnh chp tgic u S.ABCD c y ABCD l hnh vung cnh a.. E l im i xngca D qua trung im ca SA, M l trung im ca AE, N l trung im ca BC. Chng minhMN vung gc vi BD v tnh theo a khong cch gia hai ng thng MN v AC

Hng dn giiGi P l trung im ca SA. Ta c MNCP l hnh bnh hnh nn MN song song vi mtphng (SAC). Mt khc, BD vung gc vi mt phng (SAC) nn BD vung gc vi MN


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V MN song song vi mt phng (SAC)nn

( ) ( )( )

( )( )

, ,

1 1 2;

2 4 4

d MN AC d N SAC

ad B SAC BD

=

= = =

Vy ( ) 2; 4a

d MN AC =

Trch tthi tuyn sinh i hc D-2007Cho hnh chp S.ABCD c y l hnh thang 090 ,ABC BAD= = ; 2BA BC a AD a= = = . Cnh

bn SA vung gc vi mt phng y v 2SA a= . Gi H l hnh chp vung gc ca A lnSB. Chng minh tam gic SCD l tam gic vung v tnh theo a khong cch tH n mtphng (SCD)

Hng dn giiGi I l trung im ca AD. Ta c: IA ID IC a CD AC= = = . Mt khc, CD SA , Suy raCD vung gc SC nn tam gic SCD l tam gic vung ti C.

Trong tam gic vung SAB ta c:2 2 2

2 2 2 2 2

2 2

2 3

SH SA SA a

SB SB SA AB a a= = = =

+ +

Gi 1 2,d d ln lt l khong cch tB vH n mt phng (SCD) th

22 1

1

2 2

3 3

d SHd d

d SB= = =

Ta c:

.1

2

3

1 1.

2 2

B SCD BCD

SCD SCD

BCD

V SA S d

S S

S AB BC a

= =

= =

2 2 2 2 2 21 1. . 22 2

SCDS SC CD SA AB BC IC ID a= = + + + =

Suy ra 12

ad = . Vy khong cch tH n mt phng (SCD) l 2 1

2

3 3

ad d= =

+Ti liu c su tp v bin son li bi thy gio Nguyn Quc Tun kt hp vi trung tm gio vinQuc Tun a ch157 ng Vn Ng- Thnh phHu-in thoi: 0905671232-0989824932. L ni quytnhng gio vin ging dy v luyn thi y hc c uy tn trn a bn thnh phHu. Lun c nhng chnhsch v nhng phng php ging dy cng nhtnh cp nht hng u. Lun mcc lp, cc nhm dy hccht lng cao vi chi ph r. c bit hng li c thng ngn ti liu trn Xuctu.com v hng trmVideo Tutorial bi ging c cp pht min ph cho hc vin ti trung tm cng nhcng ng hc sinh.+ c bit trong nm hc 2013-2014, trung tm mra chng trnh khuyn hc nhsau:

- Min ph n hc mt tun khng nh cht lng- Gim ngay 20% hc ph thng u tin khi n hc- Tng ngay 20% hc ph thng u tin khi cc hc vin khc gii thiu 1 hc vin n hc- c sging dy trc tip ca thy c gio y kinh nghim luyn thi- Phng hc thong mt, yn tnh tuyt i.-

c php hc tng cng khi cha hiu bi

n tham quan v ng k hc ti a chtrn hoc tm hiu thng qua sin thoi: 0905671232hoc website http://xuctu.com

Trn trng v chc cc em hc sinh sc khe v may mn


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