BI TP : XC NH TM V TNH BN KNH CA MT CU NGOI TIP HNH A DIN:

I.KiN THC CN NH:Mt cu (S) ngoi tip hnh a din (H) khi cc nh ca (H) nm trn mt cu -Mt hnh chp ni tip trong mt cu (S) khi v ch khi y ca hnh chp l mt a gic ni tip c. -Mt hnh lng tr ng ni tip trong mt cu (S) khi v ch khi y ca hnh lng tr l cc a gic ni tip c. -Hnh t din ,Lng tr u , hnh chp u v cc khi a din u u ni tip c :Ch :Trong khng gian tp hp ca nhng im cc u cc nh ca mt a gic l mt ng thng vung gc vi mt phng cha a gic ti tm ca a gic .Trong khng gian tp hp nhng im cch u 2 im A v B l mt phng trung trc ca AB.Dng 1: HNH A DIN C CC MT L NHNG TAM GIC VUNG C CHUNG CNH HUYN:Phng php:Gi I l trung im ca cnh huyn chung.Tm ca mt cu l I v bn knh l na cnh huyn Th d :Cho hnh chop S.ABC c y ABC l tam gic vung cn ti B v SA vung gc vi y .Gi H v K l hnh chiu vung gc ca A ln SB v SC.Chng minh hnh a din AHKBC ni tip c trong mt cu (S) , tm tm v bn knh ca (S) theo a ,Vi SA=AB= a

BBi tp:

1.Cho hnh chop S.ABCD c y l hnh vung cnh a , Hai mt bn SAB v SAD vung gc vi y , hai mt bn cn li to vi y gc 600.a.Xc nh tm v tnh bn knh mt cu ngoi tip hnh chop S.ABCD.

BI GIISA (ABC)=>SA BC BC AB=>BC (SAB)=>(SAB) (SBC)AHSB=>AH(SBC)=>AH CH=>AHC vung ti HAKC vung ti KABC vung ti B=>Hnh a din AKCBH ni tip trong mt cu ng knh AC , tm I l trung im ca AC v bn

knh R= 2

2a

b.Gi B ; C l hnh chiu ca A ln SB v SDCl ;D l giao im ca DS v mp(ABC). Xc nh tm v tnh bn knh ca mt cu ngoi tip hnh a din ABCDBCD.2.Xc nh tm v tnh bn knh mt cu ngoi tip khi bt din u cnh a.3.Cho 2 ng thng cho nhau (d) v (d) c on vung gc chung l AA (A thuc (d) v A thuc (d). Gi (P) l mp qua AA v vung g vi (d) .Cho bit AA=a .Mt ng thng (l) song song vi (P) ct (d) v (d) ti M v M .Hnh chiu vung gc ca M ln (P) l l N.Xc nh tm I ca mt cu i qua 5 im A ;A ; M ; M ; N. bit b=AM v =(d;d)S:

222

2

222

1

2

1

bcosacos

M'Acos

bAMma

AMAA'MA'NtaivuongMN'A

b'M'AMNvaAMN

)P(vigocvuongCung()'d//(MN:HD

bcosacos r knhban

MA'iemtrungla O tamco(S)cauMat

2

(2)ASAI

(1)CIBIAI

(S)caumat cuatamlaIGoi

SCOSBOSAO(ABC).SO

BC.cuaiemtrung laM

ABCcuatamtrong la OABCcuatamlaOGoi

60

Dng 2: Hnh chp S.A1A2AnGi O l tm ca a gic y , v I l tm ca mt cu (S) .Dng (d) (A1A2An)=>IA1=IA2=.=IAn => I thuc (d).IA1 =IS => I thuc mt phng trung trc ca SA1.Vy I l giao im ca mp trung trc ca SA1 v (d).

Th d 1:Cho hnh chp tam gic u c cnh y l a v cnh bn to vi y cc gc 600 .Tm tm v bn knh ca mt cu (S) ngoi tip hnh chp trn

Ra

a

aSI

a

cos

aSAvaatan

aSO

aAMOAma

cos

OASAvaOAtanSOSAOvaOtaivuongSAO

SO

SASI

SO

SK

SA

SIdangongSOA;SKI

SAIKSAcuaiemtrunglaK Goi

SO(d)I

mp(SAO)trongSAcuatrctrung(d)lavi)d(I)(SOI)(

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2

18

12

3

32

603

360

3

3

3

3

3

2

6060

2

21

2

00

00

2

Th d 2: Cho hnh chp S.ABC c y ABC l tam gic u cnh b v SA =a vung gc vi y .Xc nh tm v tnh bn knh mt cu ngoi tip hnh chp trn.

6

129

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baIAR

abOAIOIAOtaivuongIOM

bbAMOA

a

2

SAIOnhatchhnhlaOIJAgiacT

SAIJSAcuaiemtrungla JGoi

mptrong)(d'(d)I

mptrong

SAcua)(d'trctrungngvethuocI)(

.)d(I)(

(2)SIAI

(1)CIBIAI

ABC.S tiep ngoaicaumat tamlaIGoi

d)mp(SA,mp(d)//SAO.tai(ABC) vi

gocvuong(d)dng,ABCgiactamcuatamlaOGoi

Th du 3:Cho hnh chop O.ABC biet AOB=900 ;BOC=600 va COA=1200 v OA =OB =OC =a.a.Chng minh tam giac ABC vuong.b.Tm tam va tnh ban knh mat cau ngoai tiep hnh chop O.ABCGIAI

ABCgiactamcuatamlaH

ACcuaiemtrungHlaGoi.BtaivuongABCaCA;aBC;aAB

32

aOIRa

OH

OH

OAOI

OH

OJ

OA

OI

dangongOAH;OIJ

OAIJOAcuaiemtrungla JGoi

(d)OHIVay

mp(OAH)trongOA

cua(d)trctrungngI)(;OHI)(

)(IOIA

)(ICIBIA

)S(caumattamlaIGoi

)ABC(OHOCOBOA

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2

21

2

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Dng 3: Xc nh tm v tnh bn knh ca mt cu ngoi tip lng tr ngPhng php:Gi O v O l 2 tm ca 2 y.-Ni OO =>OO hai y.Gi I l tm mt cu ngoi tip lng tr ng

=>

)(I'AIA

)(I'A....I'AI'AIA.....IAIA nn2

1

11

2121

(1)=>IOO (2)=>I(d) (vi (d) l ng trung trc ca AA1 trong mp (AA1 ;OO)=> I l giao im ca (d) v OO=>I l trung im ca OO,Th d : Cho hnh lng tr ng ABC.ABC c y ABC l tam gic vung ti A v AC = b , C=600.ng cho BC ca mt bn BBCC to vi mp (AACC) gc 300.a.Tnh th tch ca khi lng tr . b. Xc nh tm v tnh bn knh mt cu (S) ngoi tip hnh lng tr.GII:

bcos

bBCvab'BC

bACtan60ABAtaivuongABC

ABsin30

ABBC'Ataivuong'BAC

.30BA'

C)C'mp(AA'lenBC'cuachieu hnhlaAC'

C)C'(AA'BAAA'BA;ACBA

)C'B'(A'OO'va(ABC)OO'//AA'OO'

C'.B'A'vaABCcuatamlaO';O

C'B'vaBCcuaiemtrunglalt lanO'vaO1.Goi

0

0

0

260

32

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2

0

228412 222 bbbb'CC

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1 3

bBC'

RvaBC'cuaiemtrunglaITam

BC' knhngcaumat trongtiepnoiC'B'ABC.A'truLang

A'taivuongC'BA';Atai vuongBAC'; CtaivuongBCC'

: khacCach

b2

BC'IBROO'cuaiemtrunglaI

O)O'mp(AA'trongAA'cuatrctrungngla(d)(vi)d('OOI

IA'AI

IC'IB'IA'

CIBIAI

trulangtiepngoaiScaumat tam laIGoi.

b'CC.AC.AB'CC.dtABCV

BAI TAP:1.Cho hnh chop tam gic u c y ABC l tam gic cn ti A vi AB =AC =a BAC = . SA = a3 v SA vung gc vi y.Xc nh tm v tnh bn knh mt cu ngoi tip hnh chop trn.2.Cho hnh chop t gic u S,ABCD c chiu cao SO =2a , gc gia cnh bn v y l . Xc nh tn v tnh bn knh mt cu ngoi tip hnh chop S,ABCD.3.Cho hnh chp S.ABC c y ABC l tam gic u cnh 2a , mt bn SAB tam gic vung cn ti S v nm trong mt phng vung gc vi (ABC).Xc nh tm v tnh bn knh ca mt cu ngoi tip hnh chop trn.4.Cho t din OABC c 3 cnh OA ; OB; OC i mt vung gc v OA = a ; OB = b ;OC =c. Xc nh tm v tnh bn knh ca mt cu ngoi tip hnh t din trn.5.Cho hnh vung ABCD cnh a .Trn ng vung gc vi (ABCD) dng t tm O ca hnh vung ly mt im S sao cho OS = a/2. Xc nh tm v tnh bn knh ca mt cu ngoi tip hnh chop S.ABCD6.Cho hnh chp S.ABC c y l tam gic ABC bit AB =5a ; BC =4a v CA = 3a..Trn ng vung gc vi (ABC) dng t A ly mt im S sao cho (SBC) to vi y gc 450 . Xc nh tm v tnh bn knh ca mt cu ngoi tip hnh chop trn.7.Cho t din SABC c SBC v ABC l 2 tam gic u cnh a v SA= a2a.Tm tm v bn knh ca mt cu (S) ngoi tip t din SABCb.Gi O l trung im ca BC .Trn tia i ca tia AO ly im D sao cho OD=OA.Tnh cc cnh ca tdin S,BCD.