hình không gian thuần túy
TRANSCRIPT
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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= + = =
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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
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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
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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.
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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
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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
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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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