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Tuynchnvpn: LuynthithiHccacctrngtrong ncnm2012.

Mn:HNHHCKHNGGIAN M n H N H K H N G A I(laisacctvdn)HNHCHPBi1.ChohnhchpSABCDcyABCDlhnhvungcnh a ,tamgicSABu,tam gicSCDvungcntiS.GiI,J,KlnltltrungimcacccnhAB,CD,SA. Chngminhrng (SIJ )^(ABCD .TnhthtchkhichpK.IBCD. ) Gii.T githittac:S

K

AB ^SI AB^ (SIJ) AB ^ IJ Do AB (ABCD ( )^ (ABCD . ) SIJ )D

A K ' I H C J

B

( )^(ABCD SIJ ) ) SH ^ (ABCD ( ) (ABCD = IJ SIJ ) +Goi KlhnhchiuvunggccaK ln (ABCD)khi KK'//SH do Kltrungim SAnn Kltrung 1 im AH&KK' = SH . 2 1 Ttac:V K.IBCD = KK'. IBCD S 3 a 3 1 a Dthy: SI = SJ = CD = IJ = a DSIJ vung tiSv:SI 2 +SJ2 = IJ2 2 2 2 SI. SJ a 3 a 3 hthcSI.SJ=SH.IJ SH = = KK '= IJ 4 8 (IB+ CD). BC 3 2 a Tac = IBCD lhnhthangvungtaiBvCnnSIBCD = 2 4 3 a . 3 Thayvotac V .IBCD = K 32+K SH ^ IJ do

Bi2. Chohnhchp S .ABCD cylhnhthangvungti A v B vi BC lynh.Bit rngtam gic SAB ltamgicuccnhvidibng 2a vnmtrong mtphng vunggcvimty, SC =a 5 vkhongccht D timtphng ( SHC) bng 2a 2 (y H ltrung im AB ).Hytnhthtchkhichptheo a .

S

GiiTgithitsuyra SH ^( ABCD ) v2a a 5 A a D

B a C 2a a 45 H a H a 45 A

SH =

2a 3 =a 3 2

TheonhlPythagorastac a C CH = SC 2 - SH 2 =a 2 . 4a C'C Dotamgic HBC vungcnti B v BC =a Gi E = HC ADththtamgic HAE cngvungcnvdo CE = 2a 2 = d ( D HC ) =d ( D( SHC ) ) suyra DE = 2a 2 2 = 4a AD =3a.E a D B 2a 2

Suyra S ABCD =

1 ( BC + DA ) AB =4a 2 (.v.d.t.).Vy 2 1 4 3 a VS . ABCD = SH S ABCD = (.v.t.t.) 3 3

0 Bi3. Chohnhchptgicu S.ABCDccnhbnto viy mt gc60 vcnh y bng a. 1) TnhthtchkhichpS.ABCD. 2) QuaAdngmtphng (P)vunggcviSC. Tnhdintchthitdintobimtphng (P)cthnhchpS.ABCD.

Gii.a) *SABCD = a2 * SBO =600 SO= AOtan600 =S

=M E

B

I F O

A

D

a 2 a 6 . 3 = 2 2 1 * VS. ABCD = SOSABCD . 3 1 a 6 2 C = . . a 3 2 a 3 6 = 6

b) *Gis (P)SC = M V (P)^SC v A (P nn AM ^ SC ) Mtkhc,gi EF =(P (SBD vi E SB F SD th EF//BD v EF quaI vi I = AM SO ) ) (do BD ^SC(P ^ SC nn BD//(P ). ) ) *Tathymtphng (P ct S.ABCD theothitdinltgic AEMF ctnhcht AM ^ EF. ) Do SAEMF = AM. EFa 6 2 VAMltrungtuynca D SAC .MtkhcAOcngltrungtuynca D SAC nnI ltrng tmca D SAC EF SI 2 2 2 2 a *Tac = = EF = BD= BD SO 3 3 3 1 2

*Tathy D SAC = 600,SA= SC ),m AM ^ SC nn AM = . SAC u(vgc

SAEMF

1 1 a 6 2 2 a2 3 a = AM. = . EF . = . 2 2 2 3 3

Bi4.Chohnhchp S.ABCcy ABC ltam gic vungcnnh A, AB =a 2.Gi Il trung im ca cnh BC. Hnh chiu vung gc H ca S ln mt phng (ABC) tha mn uu r uuu r 0 IA = -2IH . Gc gia SC v mt y (ABC) bng 60 . Hy tnh th tch khi chp S.ABC v khongcchttrungimKcaSBnmtphng(SAH). Gii uu r uuu r *Tac IA = -2 HthuctiaicatiaIAv IA =2 IH IHBC = AB 2 =2 a uu r uuu r *Tac IA = -2IH HthuctiaicatiaIAv IA =2IH BC = AB 2 =2 a a 3 a Suyra IA = a,IH = AH = IA + IH = 2 2 a 5 0 Tac HC 2 = AC 2 + AH 2 - 2 AC. AH .cos 45 HC = 20 V SH ^ ( ABC ) ( SC , ( ABC ) )= SCH = 600 SH = HC.tan 60 =

a 15 2

0 Tac HC 2 = AC 2 + AH 2 - 2 AC. AH .cos 45 HC =

a 5 2 a 15 2

0 V SH ^ ( ABC ) ( SC , ( ABC ) )= SCH = 600 SH = HC.tan 60 =

ThtchkhichpS.ABCDl: VS .ABC = S DABC. = SH

1 3

3 a 15 ( dvtt ) 6

BI ^ AH BI ^ ( SAH) BI ^ SH d ( K ,( SAH) ) SK 1 1 1 a = = d ( K , ( SAH ) ) = d ( B,( SAH ) )= BI = 2 2 2 d ( B,( SAH ) ) SB 2

Bi5.Chohnhchp S .ABC cyltamgic ABC vungti B SA vunggcviy, AB =a , SA = BC =2 .Trntiaicatia BA lyim M saocho =a (00 < a 0)SAtoviy(ABC)mtgcbng60 . a TamgicABCvungtiB, =300 .GltrngtmtamgicABC.Haimtphng(SGB)v ACB

(SGC)cngvunggcvimtphng(ABC).TnhthtchhnhchpS.ABCtheoa. GiiGiKltrungimBC.Tac SG ^ ( ABC ) SAG = 600, AG = T AK =3 a . 2

9a 3a 3 SG = . TrongtamgicABCt AB = x AC = 2 x BC = x 3. 4 2 9a 7 1 243 3 Tac AK 2 = AB 2 +BK 2 nn x = .Suyra VS .ABC = SG. ABC = S a (vtt) 14 3 112

Bi9. Cho hnh chp S.ABCD c y l hnh vung cnh a, SA vung gc vi mt phng y v SA=a.GiM,NlnltltrungimcacccnhSB,SDIlgiaoimcaSCvmtphng (AMN).ChngminhSCvunggcviAIvtnhthtchkhichpMBAI.

Gii

S

Chngminh SC ^AI :TacI N

M A D

B

C

AM ^ SB AN ^ SD AM ^ SC AN ^ SC SC ^ (AMN) SC ^ AI AN AM ^ BC ^ CD 1 K IH // BC IH ^(SAB) (v BC ^(SAB) ) VMBAI = SVMAB.IH 3 2 2 2 SA a a a SI.SC = SA 2 SI= = = = 2 2 2 SC 3 SA + AC 3a

SV MAB =

a2 VMBAI 4

SI IH SI.BC a = IH= = SC BC SC 3 3 1 a = SVMAB.IH= 3 36

Bi10: ChohnhchpS.ABCcyltamgicvungtiA,AB=3,AC=4gctobicc o mtbnvybng60 .TnhthtchcakhichpS.ABC Gii. S GiHlhnhchiuca Sln(ABC)M,N,Klnltlhnhchiuca HlnhcnhAB,AC,BC.Khith tchVcakhichpctnh bicngthc C K1 V = S DABC. SH 3 1 m S DABC = AB. AC =6 2

A M

N H

TnhSH. XtcctamgicSHM,SHN, SHKvungtiH, cccgcSMH,SNH,SKH 0 bng 60 doHM=HN=HK=>Hltm ng trnnitiptamgicABC=> B 2 ABC S 0 HM = = 1=>SH=HM.tan60 = 3AB + BC +CA 1 Vy V = 3.6 =2 3 3

Bi11.ChohnhchpS.ABCD,yABCDlhnhthoi.SA=x(0 BD = a 2 SD = BD tan 600 =a 6 Vy VS .ABCD = SD. ABCD = S1 3 a 6 (vtt) 23

C G E A B

)chngminhcBC ^ (SBD),kDH ^ SB=> C1 1 1 a 6 = + DH = 2 2 2 DH SD DB 2

DH ^ (SBC)

)GiEltrungimBC,kGK//DH,KthucHE=>GK ^ (SBC)vGK EG 1 a 6 a 6 = = GK = Vyd(G,(SBC)= GK = DH ED 3 6 6

GiNlimixngcaNquaIthNthucAB,tac: =>N(45)=>PtngthngAB:4x+3y1=0 KhongcchtInngthngAB:d= 4.2 + 3.1 - 12 42 +3

= 2

AC=2.BDnnAI=2BI,tBI=x,AI=2xtrongtamgicvungABIc:1 1 1 = 2 + 2 suyrax= 5 suyraBI = 5 2 d x 4x TtacBthuc(C): ( x - 2) 2 + ( y - 1)2 =5

imBlgiaoimcatAB:4x+3y1=0vingtrntmIbnknh 5 0 Bi15.ChohnhchpS.ABCDcyABCDlhnhthoicnhavcgc =60 ,haimt ABC phng(SAC)v(SBD)cngvunggcviy,gcgiahaimtphng(SAB)v(ABCD) 0 bng 30 .TnhthtchkhichpS.ABCDvkhongcchgiahaingthngSA,CDtheoa. Gii. GiO= AC IBD ,MltrungimABvIltrungimca AM. DotamgicABCltamgicucnhann: CM ^ AB,OI ^ AB vCM =2 a 3 a 3 a 3 , OI = ,S ABCD = 2 4 2

V(SAC)v(SBD)cngvunggcvi(ABCD)nn SO ^( ABCD ) 0 Do AB ^ OI AB ^SI .Suyra: ( SAB ) , ( ABCD ) = ( OI , SI ) = SIO = 30

XttamgicvungSOItac: SO = OI .t an300 =

a 3 3 a . = 4 3 4

3 1 1 a 2 3 a a 3 Suyra: V = .S ABCD.SO = . . = . 3 3 2 4 24 GiJ= OI ICD vHlhnhchiuvunggccaJtrnSI

a 3 v JH ^( SAB ) 2 Do CD / / AB CD / /( SAB ).Suyra:

Suyra: IJ = 2 = OI

d ( SA, CD ) = d CD, ( SAB ) = d J ,( SAB ) = JH

XttamgicvungIJHtac: JH = IJ .s in300 = Vy d ( SA,CD )=a 3 . 4

a 3 1 a 3 . = 2 2 4

Bi16.Trongkhnggian,chotamgicvungcnABCccnhhuyn AB=2a.TrnngthngdiquaAvvunggcvimtphng(ABC)lyimS,saocho 0 mtphng(SBC)tovimtphng(ABC)mtgc 60 .Tnhdintchmtcungoitipt dinSABC.

Gii. Tgithitsuyra D ABC vungtiCkthpvi d ^(SAC ). Suyra BC ^( SAC )0 Do SCA =60 Do D ABC vungtiCvAB=2a

S

AC = BC =a 2

TrongtamgicvungSACtacSA = AC.tan 600 =a 6

A

B

TrongtamgicSABc: SB = SA2 + AB 2 =a 10 C 0 = SAB =90 nntdinSABCnitiptrongmtcungknhSB. Do SCB SuyrabnknhmtcubngSB a 10 = 2 2

VySmc= 4p R 2 =10 a 2 (.V.D.T) p

LNGTRBi1.Cholngtrtamgicu ABC.A1B1C cchncnhubng 5 .Tnhgcvkhong 1cchgiahaingthng AB v BC . 1 1 Gii.Tnhgcvkhongcchgiahaingthng AB v BC . 1 1 Tacylngtrltamgicucnhbng5ccmtbnlhnhvungcnhbng5 AB1 = BC1 =5 2.Dnghnhbnhhnh BDB1C1 DB1 = BC1 = 5 2, BD = C1 B1 =5,AD = CD.sin 600 =5 3

(do D ACD vungti A v BA = BC = BD) a = ( AB1 BC1 ) = ( AB1 DB1)2 5 2 + 5 2 - 5 3 AB12 + DB1 - AD2 1 AB1D cos = AB1D = = nhnt 2 AB1.DB2 4 2.5 2.5 2

(

) (

2

) (

2

)

2

1 a = cosa = .Tathy BC1 / / mp ( AB1 D ) ,AB1 mp ( AB1D ) t AB1D 4 3 B.AB1D V 3 B1.ABC V d ( BC1 , AB1 ) = d ( BC1 , mp ( AB1 D ) ) = d ( B,mp ( AB1D ) )= = 1 dtDAB1D AB1.DB1.sina 2 1 1 cos a = ( a = ( AB1 BC ) ) 4 = = = 5 .ps 1 d ( AB , BC )= 5 AB1. AD1sina 1 .5 2.5 2. 15 1 1 2 2 4

BB1dt ABC D

25 3 5. 4

Bi2. Cholngtrng ABC .A' B 'C'cth Ccmtphng(ABC ' ), ( AB 'C ), ( A'BC)ct O.TnhthtchkhitdinO.ABCtheoV. Gii.GiI=AC AC,J=AB ABA

A'

C'

tchV. nhauti

B' I

J O

H M B

C

(BA'C) (ABC')=BI (BA'C) (AB'C)=CJ Olimcntm GoiO=BI CJ

TacOltrngtmtamgicBAC

GiHlhnhchiucaOln(ABC) Do V ABClhnhchiuvunggcca V BACtrn(ABC)nnHltrngtm V ABCOH HM 1 = = A ' B AM 3 1 1 1 VOABC = OH .SV ABC = A ' B. VABC = V S 3 9 9 Bi3.Cholngtrtamgicu ABC . A ' B ' C' ccnhylavkhongcchtA a nmtphng(ABC)bng .Tnhtheo athtchkhilngtr ABC . A ' B ' C' 2

GiMltrungimBC.Tac:

Gii.GiMltrungimBC,hAHvunggcviAMBC ^ AM BC ^ ( AA ' M ) BC ^ AH BC ^ AA ' a M AH ^ A ' M AH ^ ( A ' BC ) AH = . 2 1 1 1 a 6 Mtkhc: = + AA'= 2 2 2 4 AH A 'A AM 3 3a 2 KL: VABC . A ' B ' C' = . 16

Tac:

Bi4. Cho hnh lng tr ABC .A1B1C c y l tam gic u cnh bng 5 v 1 A1 A = A1B = A1C =5.ChngminhrngtgicBCC1B lhnhchnhtvtnhthtchkhilng 1 tr ABC .A1B1C . 1 Gii.Gi O ltmcatamgicu ABC OA = OB =OC . Ngoi ra ta c A1 A = A1B = A1C =5 A1O l trc ng trn ngoi tip tam gic ABC A1O ^ ( ABC ) AO lhnhchiuvunggcca AA ln mp ( ABC). 1 M OA ^ BC A1A ^BC do AA1 / /BB1 BB1 ^BC hay hnh bnh hnh BCC1B l hnh ch 1 nht.2 5 3 5 6 Tac A1O ^ ( ABC ) A1O ^ CO A1O = CA - CO = 5 - . = 3 2 3 2 1 2 2 2

52 3 5 6 125 2 . = 4 3 4 Bi5.Chohnhlpphng ABCD.A1B1C1D cdicnhbng a.TrncccnhABvCD 1

Thtchlngtr:V = dtDABC. A1O =

lylnltccim M,N saocho BM = CN = x. XcnhvtrimMsaochokhongcch giahaidngthng A1C v MN bng .a 3 Gii.Tac MN / / BC MN / / ( A1 BC ) d ( MN , A1C ) = d ( MN , ( A1BC ) )

D1

C1

Gi H = A1 B AB1 v MK / / HA,K A1B MK =

A1

B1

x 2 2 V A1 B ^ AB1 MK ^ A1B v CB ^ ( ABB1 A1) CB ^ MK .

TD N C

suy

ra

MK ^ ( A1 BC ) MK = d ( MN , ( A1 BC ) ) =d ( MN , A1C )

a x 2 a a 2 a 2 = x = .VyMthamn BM = 3 2 3 3 3 Bi6.Cholngtr ABCAB C cyltamgicABCvungcntiA,BC=2a, AA vunggc 0 vimtphng(ABC).Gcgia ( AB ) v ( BB ) bng 60 .Tnhthtchlngtr ABCAB C . C C GiiTAkAI ^ BC Ilt