Phương pháp giải toán hình học theo chuyên đề part 2

Goi G la trong tam tarn giac A'Bc Tinh the tich khoi lang try da cho va tinh ban kinh mat cau ngoai tiep tu dien GQI I la tam cua tam giac ABC, suy ra GI / / AA' GI ± ABC Gpi J la tam

Phutnig phdp gidi Todn Htnh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

Goi d^;d2 Ian lugt la khoang each tit B,H den mp(SCD)

Ta c6: ASAB - ASHA S A S B S H SA^ 2 S H d, 2 ^ 2^ a 2 3 1

AAj = 2aV5 va BAC = 120*' Gpi M la trung diem cua canh C C j Chiing minh

hai duong thang MB va MAj vuong goc voi nhau Tinh khoang each tu diem

A den m^t phang ( A J B M )

Jiuang ddn gidi Taco: BC = V A B ^ + AC^ -2AB.AC.eosl20° = ayf?

BM = V B C ^ T C M 2 =2>/3a; AjM = ^AjCj^ + C^M^ =3a BAj = 7 A B ^ + A A 7 = \/2Ta; BM^ + A^M^ = BA^^

Suy ra MB 1 MAj Ke C H 1 A B C H 1 (ABAj)

Ta CO CH = AC.sin60° = >/3a

Cty TNHH MTV DWH Khang Viet

fhe tich kho'i tu di^n M A B A :

Jiuang ddn gidi

Gpi M, N Ian lugt la trung diem cua SA, BC

Ke B H l C N t a i H suy ra BH la / / \ v :

khoang each tu Btoi mp (SAC)

Ta CO SNA = 60° la goc giiia hai m|it phing(SBC),(ABC)

Tam giac SAN deu c^nh SA = A N =

Gpi E la diem tren duong thSng AAj sao cho Aj la trung diem cua

ME, taco BM/ZBjE

JiC = V2a,BiE = BM = V B A 2 + A M 2 C E = V C A 2 + A E 2 = ^ y/l3a •i.> •

Phumtg phdpgiai Todtt Hinh hgc theo chuyen de- Nguyen Phu Khanh, Nguyen Tai Thu

B^C^ + BjE^ = CE^ => BjC 1 BjE => BjC ± B M

* T a c o B M / / ( B j C E ) ^ d ( B M , B i C ) = d(M,(BiCE))

Gpi H la t r u n g d i e m cua A j C j ta c6

B H l ( A C q A i )

The tich cua khoi chop Bj CME :

V B ] C M E -= - B , H S 2 • ' l ^ - ^ C M E = — .—a.a = 1 Vsa 1 VSa^

3 2 2 12 Goi I la hinh chieu ciia M len mp(BjEC) ta c6 :

Bai 2.3.9. Cho lang t r y d u n g ABC.A'B'C c6 day A B C la tarn giac vuong,

A B = BC = a , canh ben A A ' = aV2 Gpi M la trung d i e m ciia canh BC Tinh

theo a the tich cua khoi lang t r y ABC.A'B'C va khoang each giiia hai duong

thang A M , B'C

Jiuang dan gidi

T u gia thiet suy ra tam giac ABC v u o n g can tai B B"

The tich kho'i lang try la:

D o t u di?n B A M E c6 BA, B M , BE doi mot v u o n g goc nen:

— - +

- = ^ = > h = h^ BA^ B M ^ BE^ h^ 7

V|y khoang each giira hai d u o n g t h i n g A M va B'C la a^/7

Bai 2.3.10. Cho h i n h chop t u giac deu S.ABCD c6 day la h i n h v u o n g canh A

Gpi E la d i e m d o i x u n g ciia D qua trung d i e m cua SA M la trung d i e m ciJ^

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^ g , N la trung d i e m cua BC C h i i n g mirJi M N vuong goc v a i BD va tinh (theo

3 ) khoang each giiia hai d u o n g t h i n g M N va AC

Jiuang ddn gidi ? ^

Gpi P la t r u n g d i e m ciia SA •-: ; ;, ,

I Ta CO M P la d u o n g trung binh cua tam giac E A D

Bai 2.3,11 Cho h i n h chop SABC c6 tam giac ABC vuong can tai B, AB = BC = 2a,

(SAB) va (SAC) eiing v u o n g goc v o i (ABC) Goi M la trung diem AB, mat phang qua MS song song v o i BC cat A C tai N Biet goc giiia (SBC) va (ABC) bang 60° T i n h the tich khoi chop S.BCNM va khoang each giiia hai duong thang A B v a S N

Jiuang dan gidi ^

Do hai mat phang ( S A B ) va ( S A C ) c a t nhau theo giao tuyen S A va cung v u o n g goc voi ( A B C ) nen S A 1 ( A B C ) , hay S A la d u o n g

jcao cua khoi chop S B C N M

Trong tam giac vuong SAB ta c6 S A = AB tan 60^ = 2a>/3

Vay V s B C N M = ^ S A S B C N M = i 2 a > / 3 ^ = V3a3(dvtt)

Goi P la trung diem cua BC thi AB / /NP, AB (2 ( S P N ) nen AB / / ( S P N ) do

do d (AB, SN) = d (AB; ( S P N ) ) = d (A; ( S P N ) )

Tir A ha A E l N P , E e P N thi \> P N 1 ( S A E ) ;ha A H I S E thi

Bai 2.3.12. Cho lang try ABCD.AiBiCiDi c6 day ABCD la hinh chii nhat,

AB = a, A D = a\/3 Hinh chie'u vuong goc ciia diem A i tren mat phang

(ABCD) trung voi giao diem AC va BD Goc giiia hai mat phang (ADDiAi) va

(ABCD) bang 60° Tinh the tich khoi lang tru da cho va khoang each tir diem Bi

den mSt phSng (AiBD) theo a

Trong do C H la duong cao ciia tam giac vuong B C D

Ta co: CH = , = — - V^y d Bi,(AiBD) = — -

VCD^+CB^ 2 2

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Cty TNHH MTV DWH Khang Viet

0di 2.3.13. Cho hinh chop S.ABC c6 day ABC la tam giac vuong tai B, BA = 3a,

= 4a; mat phang (SBC) vuong goc voi mat phang (ABC) Bie't SB = 2a73 va ggC= 30" Tinh the tich khoi chop S.ABC va khoang each tir diem B deh mat

phang (SAC) theo a s " '• '

Jiic&ng dan gidi

Goi H la hinh chie'u cua S xuo'ng BC

Bai 2.3.14. Cho hinh chop S A B C D c6 day A B C D la hinh vuong canh a Goi

M va N Ian luot la trung diem cua cac canh AB va A D ; H la giao diem cua C N

va D M Bie't SH vuong goc voi mat ph5ng ( A B C D ) va SH = aS Tinh the tich

khoi chop S C D N M va khoang each giiia hai duong thiing D M va SC theo a

^ S C D N M 8 24 (dvtt)

Laithay: DM.CN = i ( 2 D A - D C ) i ( 2 D C - D A ) = D A 2 - D C ^ = 0

2V— / 2' V^y CN 1 D M h:r do SC 1 D M bai vay:

d(SC;DM) = d(H;SC)=^^'^"SC-^"-^^ SH.CH

SC SC VsH^+CH^

Phumig phiifigidi Tiu'ui Ilinh hoc theo chuyen de - Nguyen Phti Khiinh, Nguyen TA't Thu

Laico: CH = ^ ^ ^ ^

D M

Hay ta c6 khoang each can tinh la: ^ay— •

Bdi 2.3.15 Cho hinh lap phuong ABCD.A'B'C'D' c6 canh b^ng a Goi M , N

Ian lugt la trung diem cua AB va B ' C Tinh khoang each giCra hai duang

thing A N va D M

Gpi E la trung diein ciia BC

Dethay AADM = ABAE

Su dung cac cong thlie: /iO : < , • , The tich khoi chop: V = -h.S^^, trong do h la chieu cao, Sj la dien tich day

• Ne'u hinh H duoc taeh thanh hai hinh roi nhau H j , H2 thi V,]^ = Vj^ ~ ^ H 2

• Tren cac duong thang SA, SB, SC ciia hinh chop S.ABC ta lay Ian lugt cac , „, ^, , SA'.SB'.SC',,

diem A ,B ,C Ta co: VS.A'WC = g^SBSC ^'^"^ "

Chijy:Khi xet ti so the tich ciia hai khoi chop thi ta thuong tim each chuyen ve hai khoi chop c6 ehung mat phang day

Vidu 2.4.1.Cho hinh chop tu giac deu S.ABCD Tinh the tich khoi chop biet 1) Canh ben bang a\/5 va mat ben tao vol day mot goc 60"

2) Duong cao cua hinh chop tao voi day mot goc 45" va khoang each giira hai duong thang AB va SC b^ng 2a

Goi O la tarn cua day, ta c6 SO 1 (ABCD) suy ra : Vg yi^g,;^ = -^SCSy^gco

J[ffi gidi

L (ABCD;

1) Goi M la trung diem C D , ta c6: C D 1 (SMO)

Do do goc S M O la goc giua mat ben voi mat day, nen S M O = 60°

I Dat A B = 2x => M O = x,OC = xV2

Trong cac tam giac vuong SOC,SOM ta c6: A = : ; A r ^ i i ;

SO^ = SC^ - OC^ - 5a2 - 2 x ^ SO = O M tan 60" =xS , ,, • ,

149

Phuang phdp gidi Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu

Nen ta c6 phuong trinh: 5a^ - 2x^ = 3x^ =^ x = a

Vay Vs.^3CD 4 '<^-(2x)^ = ^ x ^ = ^ a 3

3 3 2) Goi K la hinh chieu ciia O len AM,

ta CO O K ± (SCD) nen OSK la goc giiia

duong cao SO vol mat ben nen OSK = 45°

Goi N la trung diem AB

Vi du 2.4.2 Cho hinh chop S.ABC c6 day ABC la tam giac vuong

AB = a, AC = aVs , SA 1 (ABC) Tinh the tich cua khoi chop S.ABC

cac truong hop sau

1) Mat phang (SBC) tao vai day mot goc 60°

2) A each mat phang (SBC) mot khoang bang -

4

tai A, trong

Xgigidi

,2

a'Vs

Ta CO BC = 2a, S^pc = - AB.AC = ^

va V< S.ABC = isA.S AABC SA

1) Goi K la hinh chieu cua A len BC,

taco BCl(SAK)

Suy ra SKA = ((SBC), (ABC)) = 60°

Taco: AK='^^^ = ^ BC 6

nen SA = AK tan 60° = | Vay Vg ^BC = ^2

2) Gpi H la hinh chieu cua A len SK, ta c6 AH L (SBC)

150

Cty TNHH MTV DWH Khang Viet

Trong tam giac SAK, ta c6:

X^i du 2.4.3 Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai

va B, AB = BC = a, AD = 4a Tam giac SAD la tam giac deu va nam trong

0iat phang vuong goc voi day Mat phang (SCD) tao vai day mot goc 60° Tinh the tich cua kho'i chop S.ABCD theo a

Suy ra SKH la goc giii-a mat phang

(SCD) voi mat day, do do S K H = 60°

Goi E la hinh chieu aia C len AD, suy ra ABCD la hinh vuong canh a

du 2.4.4 Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a,

SA vuong goc voi day Mat phang (SBD) tao voi day mot goc 60° Goi M, 1^ Ian lugt la hinh chieu cua A len SB, SD Mat phang (AMN) cat SC tai P Tmh the tich khoi chop S.AMPN

JCsngidi

Goi O la tam ciia day, ta c6 BD1 (SOA)

Phtfcmg phdpgiiii Toan Hinh hoc theo chuyen de- Nguyen Phu Khanh, Nguyen Tat Thu

suy ra goc SOA la goc giiia hai mat 5

phing (SBD) va mat day nen SOA = 60° / V s

Trong tam vuong SAO ta c6: / j \ V r \ ,

SA:.AO.tan60" = = ^ 7 3 = ^ / ] ^ » \

BC ± A M => A M 1 (SBC) ri> A M 1 SC / ~'X^=zz-\ V-'

Tuong tu: A N 1 (SCD) => A N 1SC, '- ' " ' o ' ^ ^

Nen AP la duong cao cua hinh chop S.AMPN

Suy ra: Vg^^^p^ = - AP.S^^^,,^,

Ap dung he thiic lugng trong tam giac vuong SAC ta c6:

Vi du 2.4.5. Cho hinh chop S A B C D c6 day A B C D la hinh vuong tam O, SA

vuong goc voi ( A B C D ) , A B = a,SA = aV2 Gpi H , K Ian lupt la hinh chieu

vuong goc cua A tren SB, SD Chung minh: SC 1 ( A H K ) va tinh the tich cua

khoi chop O H A K theo a

Cty TNIIIl MTV DWII Khaug Viet

Gpi G la giao diem cua S O vh K H

thi G la trung diem ciia K H , ma

va OI = — = — = - Suy ra VQ.AHK = 3 OI-S^HK = 3 ' 2 = •

Cdch 2: Gpi E la hinh chieu ciia A tren SO thi A E 1 ( O H K ) nen A E la duong

cao cua hinh chop A O H K

Pi du 2.4.6 Cho hinh chop S.ABC c6 C3c canh day AB = 5 3 , BC = 6 3 , AC = 7a

Cac m3t ben t3o voi dsy mot goc bang nhau V3 bSng 60° Tinh the tich khoi chop S.ABC V 3 tinh khosng each tir A deh mat phSng (SBC) Bie't hinh chieu cua dinh S thupc mien trong t3m gidc ABC _ _ _ _

153

Phuattg phapgiai Todn Hinh hoc theo chuyen de- Nguyen Phii Khanh, Nguyen Tat Thu

GQ'I I la hinh chieu vuong goc ciia S tren ( A B C ) , A ' , B ' , C ' Ian lugt la hinli

chieu cua I tren BC,CA,AB Tu gia thie't suy ra S A l = SB^I = S C I = 60° Cac

tam giac vuong SIA',SIB',SIC' bang nhau nen l A ' = IB' = I C => I la tam du6n»

tron noi tiep tam giac ABC

Goi p la nua chu vi tam giac ABC

5a + 6a + 7a

SAABC = VP(P-BC)(P-AC)(P-AB)

= 79a(9a -6a)(9a -7a)(9a -5a) = sSa^

Goi r la ban kinh duang tron noi tiep

Suy ra Vg^gc = ISLS^BC = ^2V2a.6^/6a = sVSa^

Vi du 2.4.7. Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, SA =

SB = SC = a Tinh SD theo a de khoi chop S.ABCD c6 the tich Ion nha't

Xffigidi

Goi H la hinh chieu ciia S len mat day, ta suy ra H la tam duang tron ngoai

tiep tam giac ABC nen H thuQC BD

Mat khac • g ^ ^ ^ ^ ^ A C l ( S B D ) ^ 0 = B D n A C la hinh chieu cua A

len mat phang (SBD), ma AS = AB = AD = a => O la tam duang tron ngoai tiep

tam giac SBD =i> ASBD vuong tai S Dat SD = x

Ta c6: SH.BD = SB.SD => SH = va S^BCD = | A C B D

Nen V S.ABCD J A C B D = iAB.SD.OA

MaOA^=AB^-:52l = a ^ - ^ 3 a 2 - x 2

Cty TNHH MTV DWH Khang Vigt

Dodo: Vs^ABCD = —.a.x.Vsa^ - x^

Vi dii 2.4.8. Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai

A va D , tam giac SAD deu c6 canh bang 2a, BC = 3a Cac mat ben tao vai day cac goc bang nhau Tinh the tich cua khoi chop S.ABCD

J!gi gidi

GQI I la hinh chieu vuong goc cua S

tren(ABCD), tuong tu nhu <i du tren

ta Cling CO I la tam dudng tron npi tiep

hinh thang ABCD

Vi tu giac ABCD ngoai tiep nen AB + DC = AD + BC = 5a Dien tich hinh thang ABCD la

^idy 2.4.9 Cho hinh chop S.ABC c6 SA = SB = SC = a va ASB =

CSA = y Tinh the tich khoi chop S.ABC theo a, a, (3, y

= a, BSC = P,

155

Phuonig phapgini Todn Hinh hoc theo chuyeit tic - Nguyen Phil Khanh, Nguyen Tat Thu

Tuong tu: B C = 2a cos ^ , C A = 2a cos ^ A

Goi H la hinh chieu ciia S len mat phang day

( A B C ) , ta CO H la tarn duong tron ngoai tiep

la goc giira hai mat phang ( A B C ) va ( A B C ) => C A C = cp

Do do: VsABC = ^SH.S^ABC =

Goi p la nira chu vi tarn giac A B C , ta co: p = a

Vidu 2.4.10.Cho lang try dung A B C A ' B ' C , co day A B C la tam giac vuong

tai A Khoang each tir A A ' d e h ( B C C B ' ) b a n g a, khoang each tir C den

( A B C ) bang b , goc giiia hai mat phSng ( A B C ) va ( A B C ) bang cp

1) Tinh the tich kho'i lang try A B C A ' B ' C theo a,b va (p

2) K h i a = b khong doi, hay xac dinh cp de the tich kho'i lang try A B C A ' B ' C

nho nhat

156

2 ) K h i a = b = > V = — s| 2sin(pcos cp

Dang thiic xay ra khi 2 sin'' 9 = cos'' cp <=> tan (p = o 9 = arctan -j=

Vay khi (p = a r c t a n - ^ thi V dat gia tri nho nhat , ; ;

V2

157

Phucmg phdp gidi Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Vidu 2.4.1-0.Cho lang try dung ABCD.A'B'C'D' c6 day ABCD la hinh thoi

canh 2a Mat phang (B'AC) tao voi day mot goc 2>QP , khoang each tu B de'n

mat phang (D' AC) bang | Tinh the tich khoi h i di^n ACB' D '

JUsfi gidi

Gpi O la giao cua hai duong cheo AC va BD, ta c6 A C 1 ( B ' O B ) = > B ^ = 30"

GQI H la hinh chieu cua B len B'O, suy ra:

Vi du 2.4.11 Cho hinh hpp ABCD.A'B'C'D' c6 cac mat ben va mat (A'BD) hgp

voi day goc 60", biet goc B ^ = 60°,AB = 2a,BD = a\/7 Tinh V a ^ ^ - B ' C i y •

Xgigidi

Gpi H la hinh chieu cua A' tren (ABD), J, K la hinh chieu cua H tren AB,

AD

Ap dyng djnh l i cosin cho AABD: BD^ = AB^ + AD^ - 2AB.AD.cosBAD

=> AD^ 2a.AD Sa^ = 0 o A D = 3a S^^BD = ^AB.AD.sinBAD

-^ Tu gia thiet suy ra hir\ chop A' ABD c6 cac mat ben hqip day goc 60"

Nen H la each deu cac canh cua AABD

1: Neu H nam trong AABD thi H

la tam duong tron npi tiep AABD Goc giiia mat ben (ABB'A')

va day bang A ^ H = 6O''

Gpi r la ban kinh duong tron npi tjg'p AABD thi:

^du 2.4.12 Cho lang tru ABC.A'B'C c6 the tich bang the tich khoi lap

Phuang canh a Tren cac canh AA',BB' lay M , N sao cho — = ^ = -

AA' BB' 3

Gpi E,F Ian lupt la giao diem cua CM vai C A' va CN vai C'B'

1) Mat phang (CMN) chia khoi lang try thanh hai phan Tinh ti so the tich hai phan do

j)Tmh the tich khoi chop C'CEF

Phumig pUiipgidi Toan With hoc titeo chmien de- Nguyen Phii Kluinit, Nguyen Tti't Thu

Pi du 2.4./3 Cho hinh chop deu S.ABCD c6 M , N , E Ian luat la trung diem

cac canh AB, AD, SC Tinh ty so the tich hai phan cua hinh chop dugc cat

boi mat phang ( M N E )

.Cgi gidi

Duong th3ng M N cat BC va

CD tai K va L; EL cat SD tai P; EK

cat SB tai Q Mat phang (MNE) cat

hinh chop theo mat cat la ngu giac

la duang trung binh ^

cua ASOC nen EH = —

Suy ra = VBCDNMQEP = ^ECKL -fV,

GQ'I V 2 la phan the tich SEQMANP ta c6:

KBMQ + ^ L D N p l - Sa^h a^h a^h

16 48

Suy ra V2 = VSABCD - Vj = a^h a^h a^h

X>i dijL 2.4.14 Cho hinh chop S.ABCD c6 day la hinh vuong canh a,

ASC = 90°,SA lap voi day goc a (0° < a < 90°) va mat phang (SAC) vuong

^6c voi mat phang (ABC) Tinh khoang each tu A den (SBC)

J[gigidi

Taco VA.sBc=^d(A,(SBC)).SBcs nen d(A,(SBC)) =

SBCS

Vi (SAC) 1 (ABC) nengpi H la hinh chieu ciia S tren canh AC thi S H 1 (ABC), hinh chieu cua SA tren mat phang (ABCD) la A H nen (SA,(ABCfD)) = SAH = a

Taco ASC = 90° nen SA = AC cos a = 72.a cos a

Vay khoang each can tim la: d(A,(SBC)) =

3. — a cos a sm a yl2.a cos a

-a^ sina.\/2-sin2a V2-sin^a

^idvL 2.4.15 Cho hinh hop dung ABCD.A'B'C'D' c6 day la hinh thoi canh

^ t a m giac ABD la tam giac deu Goi M , N Ian lugt la trung diem cua cac

161

Phuang phdp gidi Todn Hinh liQC theo chuyen de- Nguyen Pku Khdnh, Nguyen Tat Thu

canh B C C ' D ' Tinh khoang each tif D den mat phang (AMN) biet rang

/ ^ / / \ ' ' ^ - \

Goi H la trung diem cua DC thi N H 1 (ABCD),NH = — a nen

re

' ^ D A M N - ^ N A M D - ^ N H S ^ M D " 24 a3

Ke HK 1 A M ta c6 NK i A M Theo djnh li ham so cosin

AM^ = BA^ + BC^ - 2BA.BC.cosl20° - ^a^ => A M - ^ a

22 Vay khoang each tu diem D den mat phang (AMN) la ——a

Cty TNHH MTV DWH Khang Viet

p Bai tap

BOi 2.4.1, Cho hinh chop S.ABCD c6 day ABCD la hinh thoi va AB = BD = a,

SA = a>/3 , SA 1 (ABCD) Ggi M la diem tren canh SB sao choBM = - SB, gia

3

sCr N la diem di dong tren canh AD Tim vi tri ciia diem N de BN 1 D M va

Ichi do tinh the tich cua khoi tii dien BDMN

V M B D N - ^ M L S ^ B N D 1 2a73 a^V3 _2a^

•3" 3 - _ 5 15 '^<^i 2.4.2 Cho hai khoi chop S A B C D va 5 ' A B C D c6 chung day A B C D la f»Ot hinh vuong canh a (S va S' nam ve cimg mpt phia cua ( A B C D ) ) Goi H, K

lugt la trung diem cua A D va B C , biet SH = S'K = h va SH,S'K ciing

Phucmg phdp gidi To&n Hinh hgc theo chuySn AJ- Nguyen Phu Khdnh, Nguyen Tat Thu

vuong goc vai ( A B C D ) Ti'nh the tich phan chung ciia hai khoi chop S A B C D

va S ' A B C D theo a va h

Jiudrng ddn gidi

Tir gia thiet de bai, ta suy ra cac t i i S

giac S D C S S A B S ' la cac hinh binh hanh

Gpi E, F Ian luc^t la tam ciia cac hinh binh

hanh SDCS', S A B S ' Ta c6 phan chung

cua hai khoi chop S A B C D va S ' A B C D

Suy ra VS.BCEF = ^s.gcF + ^ s x E F = • ^ay VABCDEF = Sa^h

Bdi 2,4.3, Cho hinh chop S.ABC c6 day la tam giac vuong tai A, AB = a, AC = 2a

Mat phSng (SBC) vuong goc voi day, hai mat phang (SAB) va (SAC) cimg tao

voi mat phang day goc 60" Tinh the tich kho'i chop S.ABC theo a ^

Jiudrng ddn gidi

Goi H la hinh chieu cua S len BC; E,F Ian lugt

la hinh chieu cua H len AB, AC suy ra S H 1 (ABC)

va HE = HF nen A H la phan giac ciia goc BAC

W = I AB.AC = a2 Vay Vs.A3C - •

Bdi 2.4.4. Cho hinh chop tam giac deu S.ABC Tinh the tich khoi chop S.ABC biet:

1) Canh day bang a va mat ben tao voi day mpt goc 60°

2) Canh ben bang 2a va SA 1 B M , voi M la trung diem SC

164

Cty TNHH MTV DWH Khang Viet Jiitong ddn gidi

Goi O la tam ciia day, I la trung diem B C

1) Ta CO B C 1 (SIO) => slo = ((SBC),(AB"C)) = 60°,

lO = i A I = SO = IOtan60° = - ,

3 6 2 'AABC

^r" w Icr^c l a a^V3 a^Vs

Vay VsABC - 3SO.SAABC = 3 • 2 = '

2) Goi E , F , P Ian lugt la trung diem ciia A B , BS, S M , ta c6:

Tam giac E F P vuong tai F nen EP^ = EF^ + FP^ o - 8a^ x = 2aV2

f'Bdi 2,4.5. Cho lang tru tam giac ABCAjB^Ci c6 ta't ca cac canh bang a, goc tao boi canh ben va mat phang day bang 30° Hinh chieu H ciia diem A tren mat phang ( A i B j C i ) thugc doan thSng BjCj Tinh the tich khoi lang tru ABC.AjBjCj va khoang each giiia hai duong thang A A j va BjCj theo a

Phuontg phdp gidi Todn Hinh hoc theo chuyen de- Nguyen Phii Khdnh, Nguyen Td't Thu

A H = A A j s i n 3 0 " = | , A j H = AApCosSO" =

Ma tarn giac A j B j C j deu

nen H la t r u n g diem ciia B[C|

The tich kho'i lang t r u la:

phang (a) l u o n song song v o i A B va C D T i m v i t r i cua (a) de (a) chia t i i dien

thanh hai phan c6 the tich bang nhau

Jiu&ng dan gidi A

V i cac mat cua t u di?n c6 dien tich bang

nhau nen cac d u o n g cao bang nhau

Mat phang (a) song song v o i A B va CD

A D

A H A D 2 - 2 A H 2

o M a t phang a d i qua trung diem cua canh A D

CtyTNHH MTV DWH Khang Viet

gdi 2.4.7 Cho hinh chop S.ABCD c6 day ABCD la hinh chii nhat AB = a, A D = 2a,

c^nh SA v u o n g goc v o i day, canh SB tao v o i mat phang day m g t goc 60" Tren c^nh SA lay diem M sao cho A M = ^ M a t phSng ( B C M ) cat canh SD tai N Tinh the tich khoi chop S B C M N ,^

JIuang dan gidi

Ha SH 1 B M =^ SH 1 ( B C M N ) : ^ SH

la d u o n g cao cua khoi chop S.BCMN

Do A M H S ~ A M A B nen suy ra:

Bai 2,4.8 Cho t i i dien ABCD c6 A C = A D = aV2 , BC = BD - a, khoang each t u

B den mat phSng (ACD) bang Tinh goc giiia hai mat phang (ACD) va

V3 (BCD) Biet the tich cua khoi t u dien A B C D bang a^Vl5

Phuongphiipgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnit, Nguyen Tat Thu

Mat khac: A E ^ + DE^ = 2a^ =^ AE^, DE^

la hai nghiem cua phuong trinh

Xet A B H E vuong tai H nen sin a = BH 1

• a = 45' 0

BE ^ Vay goc giua hai mp(ACD) va (BCD) la a = 45°

Bdi 2.4.9 Cho hinh chop S.ABCD c6 day ABCD la hinh thoi canh a, BD = a

Tren canh AB lay M sao cho BM = 2AM Goi I la giao diem ctia AC va D M SI

vuong goc voi mat phang day va mat ben (SAB) tao voi day mot goc 60" Tinh

the tich cua khol chop S.IMBC

Jluongddngidi

Goi H la hinh chieu cua I len AB, suy ra AB 1 (SIH) => SHI la goc giCra mat

ben (SAB) va mat day Do do SHI = 60°

sin60'^ sin BDM MD V7 7 ->tanBDM =

(jiem AO

- ^ : ^ - l = ^ = > O I = OD.tanBDM = ^ ^ I la trung cos^BDM 2 4 "

- A A T T T A A r ^ D A I AI.OB aS

Ta CO AAHI - AAOB = => IH = • Suy ra SI = IHtan60° =

OB AB AB 3a

8 SAMI _ A M A I _ 1 1 1 _ 1 _ _ a \ / |

^ Suy ra ((SAB), (ABCD)) = ( S K J I K ) = SKH = 60°

Gpi O la tam ciia day, ta CO AAKH - AAOB ,^ „ ^

Phuang phdp gidi Toiin Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Bai 2.4.11 Cho hinh chop S.ABCD c6 day A B C D la hinh vuong tarn ,)

A B = a G o i M , N Ian lugt la trung d i e m cua cac canh OB, SD M a t phark>

( N A C ) tao v 6 i mat phang day mot goc 60°; hai mat phang (SAM) va (SCl^^j

ciing v u o n g goc v o l mat phang day T i n h the tich cua kho'i chop S.ACN

Jiixang ddn gidi ^

V i hai m a t phang (SAM) va (SCM) cung

v u o n g goc v o i mat phang day nen giao tuyen

S M ciia hai mat phang do v u o n g goc v o i day

Trong tam giac v u o n g N H O , ta c6: N H = O H tan 60° = S H =

•^NACD *'SACD ^ S A C N ^SACD -SH.S A A C D 1 aV6 a^

6 4 • 2

N6_

48

Bai 2.4.12 Cho h i n h chop S.ABC c6 mat phang (SAC) vuong goc v o i mat

phang (ABC), SA = A B = a, A C = 2a va ASC = ABC = 90° T i n h the tich khoi

chop S.ABC va cosin ciia goc giua hai mat phang (SAB), (SBC)

Goi M la t r u n g diem SB v^ (pla goc

giua hai mat phang (SAB) va (SBC) ^

Ta c6: SA = A B = a, SC = BC = aS

cos A M C

A M 1 SB va C M 1 SB, suy ra coscp =

ax/3 ASAC = ABAC => SH - B H = • SB = Jx/6

A M la t r u n g tuyen ASAB nen:

Cty T'iVHH MTV DWH Khang Viet

gjjj 2.4.13 Cho lang t r y A B C A ' B ' C c6 day A B C la t a m giac can

^B = A C = a,BAC = 120°va A B ' vuong goc v o i day ( A ' B ' C ) G o i M , N Ian

\^xQt la t r u n g diem cac canh C C va A ' B ' , mat phang ( A A ' C ) tao v a i mat

phang (ABC) mot goc 30° Tinh the tich khoi lang t r u A B C A ' B ' C va c6 sin cua goc giira hai d u o n g thang A M va C ' N

Suy ra A B ' = B'K.tan30° - ^ The tich kho'i lang t r u :

V = AB'.S AABC - g • Goi E la t r u n g d i e m cua A B ' , suy ra M E / / C ' N , nen ( C ' N , A M ) = ( E M , A M )

Phuong phap gidi Todn Hinh hgc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

Bai 2.4.14 Cho hinh lang tru deu ABC.A'B'C, M la trung diem cua canh CC

Mat phSng (A'B'M) tao voi mat phang (ABC) mpt goc 60" va tam giac A'MB'

di^n tich bang

13 Tinh the tich khoi chop A M A ' B '

Jivcang dan gidi

Gpi N la trung diem cua A'B', ta c6

C N I A ' B '

Matkhac: A ' M = B'N ^ M N 1 A'B',

suy ra A ' B ' l ( M N C ' )

Do do M N C la goc giiia mat phSng

(A'MN) voi (A'B'C)

chieu cua S len mat day trimg voi diem H la trung diem ciia AO Mat phang

(SAD) tao voi day mpt goc 60° va SC = a Tinh Vg^Bj^D va d(AB,SC)

J^it&ng ddn gidi

D a t A B = x,x>0 ^

Ve H K I A D , suy ra A D 1 ( S H K ) ^ S K H la goc giua mat ben (SAD) va

mat day nen SKH = 60°

Taco: HC = - A C = ^ ^ ; HK = - D C = —

4 4 4 4 Trong tam giac vuong SHC, ta c6: SH^ = SC^ - HC^ =a^-^^

4a 3>/5 ' 16Vl5a

675

Bdi 2.4.16 Cho hinh chop S.ABC c6 day ABC la tam giac vuong can tai B,

AB = BC = 2a; hai mat phang (SAB) va (SAC) ciing vuong goc voi mat phang (ABC) Gpi M la trung diem cua AB; mat phang SM va song song voi BC, cat

AC tai N Biet goc giiia hai mat phSng (SBC) va (ABC) bSng 60° Tinh the tich khoi chop S.BCNM va khoang each giiia hai duang thang AB va S N theo a

Jiicang dan gidi

Do hai mat phang ( S A B ) va (SAC) cat nhau theo giao tuye'n SA va cung vuong goc voi ( A B C ) nen SA 1 ( A B C ) , hay SA la duong cao ciia khoi chop S.BCNM

173

Pltumig phlif} gidi Toiin Hiiih hoc theo chuycn de- Nguyen Phu KItdnh, Nguyen Tat Thu '

Trong tam giac v u o n g S A B ta c6 S A = ABtaneo" = 2a73

Bdi 2.4.17, Cho lang try A B C D - A j B i C i D j c6 day A B C D la h i n h c h u nhat

A B = a, A D = asl3 H i n h chieu v u o n g goc cua diem A i tren mat phang (ABCD)

t r u n g v o i giao d i e m A C va BD Goc giua hai mat phang ( A D D J A J ) va (ABCD)

bang 60'^ T i n h the tich khoi lang t r y da cho va khoang each t u d i e m Bi den

Trong d o C H la d u a n g cao cua tam giac v u o n g BCD

Cty TNHH MTV DWH Khang Viet

•pa c6: C H = CD.CB

CD^ + CB'

: ^ V a y d ( B i , ( A , B D ) ) = ^

gCii 2A 18 Cho h i n h chop S.ABC c6 day ABC la tam giac v u o n g tai B, B A = 3a,

p C = 4 a ; mat phang (SBC) v u o n g goc v o i m a t phMng ( A B C ) Biet SB = 2aV3

5 B C = 30'^ T i n h the tich khoi chop S.ABC va khoang each t u d i e m B den in?t phang ( S A C ) theo a

J^udrng ddn gidi ^ GQI H la h i n h chieu ciia S xuong BC

V i (SBC) ± (ABC) nen S H 1 ( A B C )

Ta CO SH = aS

Do do Vs.ABCD =|SH.S^ABC =2a^^/3

Ta CO t a m giac SAC v u o n g tai S

V i SA = aV2T, SC = 2a, A C = 5a

va S^^Q = a V21 nen ta c6 dug-c:

d ( B , ( S A C ) ) = ^ ^ = ; |

'AS AC

Bdi 2.4.19. Cho h i n h chop S A B C D c6 day A B C D la h i n h v u o n g canh a Gpi

M va N Ian lu(?t la t r u n g d i e m cua cac c^nh A B v a A D ; H la giao d i e m ciia

C N va D M Biet S H v u o n g goc v o i mat phang ( A B C D ) va S H = aVs T i n h the tich khoi chop S C D N M va khoang each giiia hai d u a n g thang D M va S C theo a 5 ;

Phuong phdpgidi Todit Hinh hoc theo chut/en de- Nguyen Phu Khdnh, Nguyen Tai Thu

V^y CN J_ DM tu do SC ± DM boi vay:

Thay len tren ta c6 khoang each can tinh la: 2a J —

Bdi 2.4.20 Cho hinh lang tru tarn giac deu A B C A ' B ' C c6 AB = a, goc gic,g

hai mat phang (A'BC) va (ABC) bang 60" Goi G la trong tam tarn giac A'Bc

Tinh the tich khoi lang try da cho va tinh ban kinh mat cau ngoai tiep tu dien

GQI I la tam cua tam giac ABC, suy ra GI / / AA' GI ± (ABC)

Gpi J la tam mat cau ngoai tiep hr dien GABC suy ra J la giao diem ciia G I

voi duong trung true doan G A ; M la trung diem GA, ta c6:

r^-KKi^h. / ^ T / ^ T n GM.GA GA^ 7a

GM.GA = GJ.GI x> R = GI = = = —

GI 2GI 12 Bai 2.4.21 Cho hinh chop S A B C D c6 day A B C D la hinh vuong canh a, canh

ben SA = a; hinh chieu vuong goc cua dinh S tren mSt phang ( A B C D ) la diem

AC

H thuoc doan AC, AH - — Gpi CM la du-ong cao cua tam giac SAC Chung

minh M la trung diem ciia SA va tinh the tich khoi tu dien SMBC theo a

Jiu6ng dan gidi

• Chung minh M la trung diem SA

rven M la trung diem SA

Bai 2.4.22 Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai A

va D; AB = AD = 2a,CD = a; goc giiia hai mat phang (SBC) va (ABCD) bang 60" Goi I la trung diem cua canh AD Biet hai mat phang (SBl) va (SCl) cung vuong goc voi mat phSng (ABCD), tinh the tich khoi chop S.ABCD theo a

Phucrtig phdp gidi Todn Hiuh hoc theo chm/en de- Nguyen Phu Khduh, Nguyen Td't Thu

Bai 2.4.23 Cho hinh lang tru tam giac ABC.ÁB'C c6 BB' = a, goc

giO-duong thSng BB' va mat pli5ng (ABC) bang 600; tam giac ABC vuong tai C v;,

BAC = 60*' Hinti chieu vuong goc ciia diemB' len mat phang (ABC) tmrig

voi trgng tam cua tam giac ABC Tinh the tich khoi tu dien Á.ABC theo ạ

Jiuang dan gidi

Goi D la trung diem AC, G la trong tam AABC

Bai 2.4.24 Cho hinh lang tru dung ABC.ÁB'C c6 day ABC la tam giac

vuong tai B, AB = a, A A ' 2a, ÁC = 3a Gpi M la trung diem cua doan thang

A ' C , I l a giao diem cua A M va ÁC Tinh theo a the tich khoi tu dien lABC

va khoang each tu diem A den mat phang (IBC)

JIu&ng dan gidi Á

Cty TNllU MTV DWH Khang Vij-t

pi^n tich tam giac ABC : S^^^BC = - AB.BC = ậ

The tich khoi tu dien lABC : V = - I H S ^ A B C = — • ' • " '

A K l A ' B ( K e A ' B ) Vi B C I ( A B B ' A ' ) nen AK 1 BC =^ AK 1 (IBC)

IChoang each tu A den mat phang (IBC) la AK

A K = 2S AAÁB A A ' A B 2aV5

^ ' ^ V A ' A ^ + A B ^ 5 Bai 2.4,25 Cho lang tru ABC.ÁB'C c6 do dai canh ben bang 2a, day ABC la tam giac vuong tai A, AB = a, AC = aVs va hinh chieu vuong goc cua dinh Á tren mat phang (ABC) la trung diem ciia canh BC Tinh theo a the tich khoi chop Á.ABC va tinh cosin cua goc giua hai duong thang AÁ, B ' C

Jiuang đn gidi

Gpi H la trung diem BC A ' H 1 (ABC)

vaAH = i B C = -7a2 + 3á =a

2 2

Do do A ' H ^ - A ' A ^ - A H ^ =3a

•ÁH = aN/3 Vay

1

^Á.ABC = - A ' H S ^ A B C

•(dvtt)

Trong tam giac vuong A ' B ' H c6 H B ' = V A ' B ' ^ + A ' H ^ = 2a

|en tam giac B ' B H can tai B' Dat (p la goc gii>a hai duong thMng A A ' va

|thi: (p = B ' B H Vay cos(p =

-1 4.26 Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh 2a,

fa, SB = ary3 va mat phang (SAB) vuong goc voi mat ph^ng daỵ Goi

f Ian lugt la trung diem cua cac canh AB, BC Tinh theo a the tich cua i6p S.BMDN va tinh cosin cua goc gii>a hai duong thJing SM, D N

Jiic&ng đn gidi

"^oi H la hinh chieu ciia S tren AB, suy ra S H 1 (ABCD)

Phucntg phdpgiai Toan Hinh hoc theo chuyen dc- Nguyen IVtii Kluiiih, Nguyen Tii't Tim

Do do SH la duong cao cua hinh chop S.BMDN

Ta c6: SA^ + SB^ = a^ + 3a^ = AB^

• ASAB vuong tai S SM = — = a

Do do tarn giac deu, suy ra SH =

Di^n tich lu giac BMDN la:

Dat q) la goc giCra hai duong thang S M va DN Ta c6: ( S M , M E ) = 9

Theo djnh ly ba duong vuong goc ta c6: S A 1 AE

Bdi 2,4.27 Cho lang try dung ABC.A'B'C c6 day ABC la tarn giac vu6n&

AB = BC = a, canh ben AA' - aVz Goi M la trung diem cua canh B C Tinh

theo a the tich cua khoi lang try A B C A ' B ' C va khoang each giua hai d u o n g

thang AM, B ' C '

Jiitang dan gidi

Tu gia thiet suy ra tarn giac ABC vuong can tai B

The tich khoi lang try la:

E

^ A B C A ' B ' C -^^'-^ABC ' ^ a 3 ( d v t t )

GQI E la trung diem ciia BB'

Khi do mat phang ( A M E ) / /B'C ^

nen d ( A M , B ' C ) = d ( B ' C , ( A M E ) ) = d ( C , ( A M E ) )

Nhan thay d ( C , ( A M E ) ) = d ( B , ( A M E ) ) = h

.i

Cty TNHH MTV DWH Khang Vift

potudien B A M E c6 B A , B M , B E doi mot vuong goc nen:

1 ] 1

+ ^ +

-B A ^ -B M ^ -B E ^ h =

>V7

\jay khoang each giua hai duong thang A M va B'C la —y

0 2,4.28 Cho hinh chop S.ABCD co day la hinh vuong canh a, mat ben I'T t^"^ S'^c deu va nam trong mat phang vuong goc voi day Gpi M , N , P Ian li"?' ''^ trung diem cua cac canh SB,BC,CD Chung minh A M vuong goc voi BP va tinh the tich khoi tu dien CMNP

Jiuang dan gidi

Gpi H la trung diem cua A D

Ta CO tam giac SAD deu nen SH 1 AD

Do (SAD) 1 (ABCD) => SH 1 (ABCD)

Suy ra M K / /SH ==> M K 1 ( C M N ) ; M K = - S H ^ VSa

2 4 Dien tich tam giac C M N : S^MN = - C M C N - —

2 8 'The tich khoi tu di?n C M N P : V^MNP = - M K S C M N

3

Si

96 (dvtt)

^^i 2.4,29 Cho hinh chop tu giac deu S.ABCD co day la hinh vuong c^inh a

E la diem do'i xiing cua D qua trung diem ciia SA, M la trung diem ciia

^E/ N la trung diem ciia BC Chiing minh MN vuong gck voi BD va tinh (theo khoang each giua hai duong thang MN va AC

Jk Jlu&ng dan gidi

•^pi P la trung diem ciia SA Ta co MP la duong trung binh ciia tam giac

" E A D => MP / / A D => MP / /NC va MN = ^ A D = NC Suy ra MNCP la hinh binh hanh MN / /CP => MN / /(SAC)

Phumtg phapgiai Toan HhtU hqc theo chiiycn ilc - Nguyen Phii Kltiiiilt, Nginfen Tat Tin,

Ta de chiing minh duoc

Bdi 2.4.30 Cho hinh chop S.ABCD day la hinh thang, ABC = BAD = 90°

BA = BC = a, AD 2a Canh ben SA vuong goc voi day va SA = a\/2 Goi

la hinh chieu ciia A len SB Chung minh tarn giac SCD vuong va tinh (theo

a) khoang each tu H den mp(SCD)

Ma SA 1 (ABCD) =^ SA 1 CD //

nen ta c6 CD 1SD hay ASCD vuong

Goi di, d2 Ian lugt la khoang each

tu B,H den mp(SCD)

Ta c6:

ASAB ~ ASHA: S A ^ S B S H ^ S A ^ ^ 2 , S H ^ d

SH SA SB ~ SB^ ~ 3 "^^ SB ~ d — = — - = - ma — = ^ = - 3 > d o = - d i 3 2 1 The tich khoi tu dien S.BCD: V

j _ Mdl cau ngoai tiep hinh chop:

Hinh chop S.A,A2 A„ noi tiep cau khi va chi khi day la (Ja giac noi tiep Khi do de xac (Jinh tarn mat cau ngoai tiep, ta

lam nhu sau:

• Dung duong thang A vuong goc voi day tai tam cua day ^'

• Dung mat phang trung true (p) ciia mot canh ben

• Giao diem I = A n (a) la tam

mat cau ngoai tiep hinh chop

2 Mat cau ngoai tiep hinh Idng try

Hinh lang try c6 mat cau ngoai tiep khi va chi khi do la ISng tru dung va day la da giac noi tiep Khi do tam mat cau ngoai tiep la trung diem doan noi tam ciia hai day

3 Vi tri tuang doi cua mot hinh

phdng mi mat cau

Cho mat cau S(0,R) va mot

mat phSng (P) bat ki trong khong gian Goi H la hinh chieu ciia O len (P)

• Neu OH > R thi (P) khong cat mat cau

• Neu OH = R thi (P) va (S) eo mpt diem chiing duy nhat la H Khi do ta noi: (P) tiep xiic voi mat cau va(P) goi la mat phang tiep dien, H goi la tiep diem

• Neu OH < R thi (P) cat mat cau theo mpt duong tron (C) c6 tam H ban kinh r = 7R^ - O H ^ ' ' '"''^ Neu O n3m tren (P) thi (C) goi la duong tron Ion va c6 ban kinh R ^ ^

Phuintg gh'ii Toiiti llhili hoc theo chiiyeu de - Nguyen Phii Khanh, 'Nguyen Tai Thu

4 Vi tri tuang doi cua mot duang thAng v&i mat cdu

Cho mat cau S(0,R) va mot duong d ba't ki trong khong gian Goi H 1^

hinh chieu cua O len d

• Neu OH > R thi d va mat cau khong c6 diem chung

• Neu OH = R thi d va mat cau (S) c6 mot diem chung duy nhat la H Khi (JQ

ta noi d tie'p xuc voi mat cau va d goi la tie'p tuye'n ciia mat cau, H goi la tie'p

diem

• Neu HO < R thi d va mat cau c6 dung hai diem chung Khi do ta noi d cat

mat cau tai hai dieiVi phan biet

5 i)ien tich mat cdu vd the tich khdi cdu

Dien tich hinh cau ban kfnh R : S = 47rR^

The tich khoi cau ban kinh R: V = - TIR^

3

II Mat tru, mat non tron xoay

Cong thuc tinh dien tich va the tich

• Dien tich xung quanh hinh non S^^ = :tRl

• Di^n tich toan phan cua hinh non = S^^ + = 7iR(l + R )

• The tich khoi non V = -TiR^h

3

• Dien tich mat xung quanh cua hinh tru S^^ = 27tRh

• Di^n tich toan phan ciia hinh try: Sjp = Sxq + 2S^ = 27:R(R + h)

• The tich khoi tru : V ^ TtR^h

Vi dxjL 2.5.1 Cho mat chop tam giac deu S.ABC c6 canh ben bang 2a, mat

ben tao voi day mot goc 60"

1) Tinh the tich khoi cau ngoai tie'p hinh chop S.ABC

2) Tinh the tich va dien tich xung quanh cua hinh non c6 dinh S va day la

duang tron ngoai tie'p tam giac ABC

jCgigidi

1) Dat AB = X Goi O la tam ciia day, suy ra SO i (ABC)

Goi M la trung diem doan AB, suy ra B C 1 (SMA) nen goc SMA la goc

giija mat ben va mat day ciia hinh chop, suy ra SMA = 60"

Goi F la trung diem canh SA, trong mat phang (SMA) duong trung trifC

doan SA cat SO tai I Ta c6 I la tam mat cau ngoai tie'p hinh chop S.ABC va ban

3 63V21 2) Ta CO ban kinh duong tron ngoai tie'p tam giac ABC la r = 4aV7

2 167ia^ Suy ra dien tich duang tron ngoai tie'p tam giac ABC la: Sj = 7tr = —— Vay the tich khoi non ngoai tie'p hinh chop la:

V = isoS = ^ -^^"^^ ^ 327ia^V2T

Vidu 2.5 J Trong hinh phang (P) cho nua luc giac deu ABCD noi tie'p duong

tron duong kinh AD = 2R Qua A ke duong thSng Ax vuong goc vai (P), tren

Ax lay diem S sao cho goc giiia hai mat phang (SDC) va (P) bang 60" Xac djnh tam va ban kinh hinh cau di qua nam diem S , A , B , C , D -

Gpi O la trung diem doan thang AD, ta c6 O la tam duang tron ngoai tie'p giac ABCD Ke Ox song song vai SA, ta c6 Ox la tryc duong tron ngoai tie'p ,tu giac ABCD Trong tam giac SAC tu trung diem J ciia canh SA ke Jy song Song voi AC, ta c6 Jy la duang trung true canh SA Goi I la giao diem ciia Ox vajy

; Ta CO I thuoc Ox nen lA = IB = IC = ID va I thuoc Jy nen IS=IA v^y I la tam

•hinh cau di qua nam diem S, A, B, C, D Va ban kinh hinh cau nay la r = lA

185

Phucmg phtip giiii Toiin Hinh hoc theo chuycn dc - Nguyen Phi'i Khanh, Nguyen Tai Thu

Theo gia thie't ta c6:

V i d y 2.5.3 Cho hinh chop S.ABCD c6 day la hinh thang vuong tai A , D,

A B = A D - a, CD - 2a Canh ben SD 1 (ABCD) va SD = a Goi E la trung

diem ciia DC Xac d j n h tam va tinh ban kinh mat cau ngoai tiep hinh chop

BE 1 CD nen trung diem M

ciia BC la tam d u o n g tron

ngoai tiep tam giac EBC

D y n g A la true d u o n g

tron ngoai tiep tam giac EBC

thi A song song voi SD

D u n g mat phang

trung tryc canh SC, mat

phang do cat A tai I

D i e m I la tam

m|t cau ngoai tiep

hinh chop S.BCE

K e S N / / D M c 3 t M I tai N , taco S D M N la hinh chi> nhat voi S D - a va

^ ^ , 2 D B ^ + D C ^ BC^ A B ^ + A D ^ + D C ^ EC^+EB^ Sa^

D M = = =

2 4 2 4 2

Ta CO SI^ = SN^ + N I ^ = SN^ + ( N M - IM)^ = ^a^ + (a - I M ) ^

Cty TNIIII MTV DWH Khang Vift

Ma IC^ = I M ^ + M C ^ = I M ^ + — va R = IC = IS

nen ^a^ + ( a - I M ) 2 = I M 2 + y « I M = | a

Vay ban kinh mat cau ngoai tiep hinh chop S.BEC la: R = J l M ^ + — =^!^a

Vidu 2.5.4 Cho hinh chop S.ABCD c6 day ABCD la hinh c h u nhat A B = a,

A D = - ^ M a t phang (SAB) 1 (ABCD) va SA = SB = a Xac djnh tam va

3

tinh the tich khoi cau ngoai tiep hinh chop S.ABD

JCffigidi

Vi tam giac SAB can tai S va

(SAB) 1 (ABCD) nen goi H la trung diem cua A B thi SH 1 (ABCD)

Goi O la tam hinh chu nhat ABCD thi O la tam d u o n g tron ngoai tiep tam giac A B D , true d u a n g tron ngoai tiep tam giac A B D la d u o n g thang A song song voi SH

Goi G la trong tam tam giac deu SAB ^

V i H O 1 (SAB) nen true d u o n g tron ngoai tiep tam giac SAB la d u o n g

th^ng qua G, song song voi H O cat A tai I thi I la tam mat cau ngoai tiep hinh

chop S.ABD T u giac G H O I la hinh ehi> nhat i'^ :

Vi du 2.5.5 Cho t u dien A B C D c 6 A B = 6, CD = 8, cac canh con lai bang

\f74 Hay t i m ban kinh hinh cau ngoai tiep t u dien A B C D

Phmnig plriipgiiii Toiin Iliuh hgc theo chttyen de- Nguyen Phii Khdnh, Nguyen Tat Thu

J:fflgidi

Goi O la tarn mat cau ngoni tiep fu dien

ABCD Do O each deu AB nen O thuoc

mat phang trung true canh AB Do chinh

la mp(MCD) Voi M la trung diem AB

Tuong tu O each deu CD nen O thuoc mp

trung true canh CD la mp (NAB), voi N la

trung diem CD Vay O nam tren giao

tuyen M N cua mp(MCD) va MP (NAB) ^

Vay ban kinh R = 5

Vi du 2.5.6 Cho hai tia Ax, By cheo nhau va vuong goc, nhan AB la doan

vuong goc chung, AB 2a Tren Ax, By Ian lugt lay cac diem C, D

1) Chung minh CD tiep xuc voi mat cau duong kinh AB <=> CD = AC + BD

2) Vdi dieu kien 6 cau 1, chung to rang diem tiep xuc cua CD voi mat cau

duong kinh AB thuoc mot duong tron co dinh va goc tao boi CD vdi mat

phang chua duong tron do khong ddi

3) Neu CD thda man dieu kien AC^ + BD^ = b^ (b la so cho trudc) Hay tim

lien h^ giOa a va b deCD tiep xuc vdi mat cau duong kinh AB

• Ngu(?c lai neu CD = AC + BD thi ta chung minh CD tiep xuc vdi mat can

dudng kinh AB

188

Cty TNUH MTV DWH Khang Viet

Do i n 1 CD nen ta can chung minh I H = A B = a That vay

Neu IH < a Ap dung djnh ly pitago cho hai A vuong lAC va IHC

CH = 7ci2 _ ipj2 > ^ci2 - lA^ = AC (1)

Ap dung djnh ly pitago cho hai A vuong IBD va IHD

trai vdi gia thie't M'

* Neu I H > a chung minh tuang tu ta cd CD < AC + B D trai vdi gia thie't

Vay I H - a hay CD tiep xuc vdi mat cau dudng kinh

AB

Cdch 2 Lay diem M la diem tren tia doi cua tia Ax sao cho A M = B D

Ta de chung minh duoc OM = OD va CM = CD => ACOM = ACOD

Suy ra lA = OH = a (Hai dudng cao tuong ung)

2) • Chung minh H thuoc dudng tron co djnh Ke tia Bx' song song va cung chieu vdi Ax Ggi C va H ' Ian lugt la hinh chieu cua C, H len mp( Bx'y) Ta cd BH' la phan giac ciia goc x'By Nen H ' co dinh thuoc tia phan giac Bz cua gdc

" x'By Vay H thuoc mp(ABz) cddjnh Mat khac H lai thuoc mat cau codinh Nen

H thupc dudng tron co'djnh la giao cua mp (ABz) vdi mat cau dudng kinh AB

• Chung minh gdc giCra CD va mp (ABz) khong ddi ' ' GQI K la hinh chieu cua D len Bz ta cd gdc DHK la gdc giua CD va mp (ABz) Ta cd: ABDK = AHDK ^ DHK = 45° (khong ddi) , , >

3) He thuc lien h? giija a va b

Do CD tiep xiic v a i mat cau nen theo cau 1 ta cd CD = AC + BD = c + d Suy ra cd = 23^ (1)

Ta cd AC^ + BD^ = b^ ^ CD = ^4a^ + b^ ^ c + d = V4a^ + b ^ (2)

Tir (1) va (2) ta cd c, d la nghiem cua phuong trinh: x^ - V4a^ + b^ x + 2a^ =0 Vay h^ thtfc lien giira a va b de CD tiep xiic vdi mat cau dudng kinh AE

l a b > 2 a

Phuonig phlipgilii Todtt Hinh hoc theo chuyeu ile - Nguyen P/iii Khmih, Nguyen Tat Thu

Vi du 2.5.7 Cho hinh chop tu giac deu S.ABCD c6 canh day bang a va

ASB = a

1) Xac djnh tam va ban ki'nh mat cau ngoai tiep hinh chop a ' iA'

2) Xac djnh tam va ban kinh mat cau npi tiep hinh chop ' "

3) Tim gia trj cua a detam mat cau npi tiep, ngoai tiep trung nhau

Xffi gidi

1) Tam va ban kinh mat cau ngoai tiep

Gpi H la tam ciia hinh vuong ABCD Ta c6 SH la tryc du-ong tron ngoai tiep

day Trong mat ph5ng (SHA) ke dirong trung true canh SA cat SH tai O Ta c6

O la tam mat cau ngoai tiep hinh chop S.ABCD, ban kinh R = SO

Gpi N la trung diem SA, ta c6:

ASNO ~ ASHA: SN SH SO SA SH 2SH so = SN.SA

Ap dung djnh ly sin cho tam giac can SAB ta c6 :

a

47cosa.sin —

2 2) Tam va ban kinh mat cau npi tiep

Ta CO tam I ciia mat cau npi tiep thupc duang thSng SH Gpi M la trung

diem cua AB, ta c6 AB 1 (SHM) tai M Gpi I la chan duong phan giac trong

ciia goc SMH (I e SH) Ta c6 I la tam ciia mat cau npi tiep hinh chop Ban kinh

r = I H

De tinh ban kinh r ta c6 the tinh theo hai each sau:

Cdch l.Dua vao tinh chat ciia duang phan giac ta c6

I S 2(sin~avcosa + C O S - )

2 2

^ Tam ciia hai hinh cau trung nhau <=> R + r = SH

aVcosa aVcosa Hay:

4\/cosa.sin^ 2(sin^+cos^) 2sin^

a 2 2 a

sin-<=> 1 + sina - 2sin^ — = 2cosa o sina = cosa sin-<=> a = 45°

2 Vay khi a = 45" thi tam mat cau ngoai tiep va tam mat cau npi tiep ciia hinh chop trung nhau

Vi du 2.5.9 Ben trong hinh tru c6 mot hinh vuong ABCD canh a npi tiep ma

hai dinh lien tiep A, B nam tren duang tron day thu nhat ciia hinh tru, hai dinh con lai nam tren duong tron day thu hai ciia hinh tru Mat phang hinh vuong tao voi day hinh tru mot goc 45" Tinh dien tich xung quanh va the tich ciia hinh try do

JCgigidi

Ta CO DC 1 BC,CE la hinh chieu ciia duang xien

=i>DClCE(djnh li ba duong vuong goc)

do do BCE = 45° la goc giiia hinh vuong ABCD

Va day hinh try

Trong tam giac BEC ta c6

BE - BC.sin45° = — ; BD - V2a

2

Trong tam giac vuong BED ta c6: DE^ = DB^ - BE^ =

191

Dien tich xung quanh hinh tru:S = 27trh = \l3na~

The tich khoi tru; V = nr^h = 3N/2;

16

Vi du 2.5.10 Cho tii dien ABCD c6 trong tarn G, npi tiep trong mot^rnlt^iiT

(0;R).Cac duong thclng GA,GB,GB,GC Ian lirot cat mat cau tai diem thu

hai A',B',C',D- Chimg minh rSng V^BCD ^

"^A'B'CD-Loi giai

Theo tinh chat phuong tich cua mpt diem doi v6i

mgt duong tron, ta c6

GA.GA' = GB.GB' = GC.GC = GD.GD' = - OG^

Vol mpi diem M ta c6

MA^ = MA^ = ( M G + G A ) ^ = MG^ + 2MG.GA + GA^

GB'.GC'.GD' ^ GC'.GD'.GA' ^ GD'.GA'.GB' GA'.GB'.GC

I GB.GC.GD GCGD.GA ^ GD.GA.GB ^ GA.GB.GC J

'GA' GB' GC GD'

> 4

' ' G A G B G C G D

Tu (1) va (2) suy ra dieu phai chung minh

Dau dang thuc xay ra khi G ^ O, hay ABCD la t i i di^n gan deu

(2)-192

p BAI TAP ^, ,

gdi 2.5.1 • Mot khoi tru c6 ban kinh R va chieu cao VsR *

1) Tinh dien tich ciia thiet di qua A B va song song voi true cua khoi tru

2) Tinh goc giiJa hai ban kinh day di qua A , B

3) Cho hai diem A , B Ian lugt nam tren hai day tron sao cho goc giCra A B va tryc hinh tru bang 30° Tinh khoang each giira A B va true cua hinh tru

Jiimng ddn giai

\ Tu A , B ta ke A A ' , B B ' song song voi true hinh trv- Khi do thiet di^n la hinh chii nhat A A ' B B ' , ta

CO A B B ' = 30° la goc giira A B va true hinh tru

Di?n tich thiet di^n

0 0 ' va mat phSng ( A A ' B B ' ) Ggi H la trung diem cua A B , thi O H la khoang each giiia O O ' va mat phang ( A A ' B B ' ) Ta c6 O H - ^

Bai 2.5.2 Cho hinh non dinh S, duang cao SO Ggi A,B la hai diem thuoc duong tron day cua hinh non sao cho khoang each tu O den ABbang ava SAO = 30°,SAB = 60° Tinh di?n tich xung quanh va the tich cua hinh non

Phuangphdpgiai Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

3 OA Xet tarn giac SAO, ta c6 SA = -

Di^n tich xung quanh cua hinh non la: S^^ = Ttrl = VsTia^

The tich khoi non la: V = ^ T i R^ h = ^Tt-OA^-SO = — 7 t a ^

3 3 4

Bdi 2.5.3 Cho hai mat phang (P) va (Q) vuong goc vai nhau c6 giao tuyen la

duong thang A Tren A lay hai diem A, B voi AB = a Trong mat phang (P) lay

diem C, trong mat phang (Q) lay diem D sao cho AC, BD ciing vuong goc v6i

A va AC = BD = AB Tinh ban kinh mat cau ngoai tiep t i i di^n ABCD va tinh

khoang each tu A den mat phang (BCD) theo a

(De thi tuyen sinh dai hpc khoi D nam 2003)

J{u&ng ddn giai

Vi (P) 1 (Q) va C A 1 A nen C A 1 (Q) => C A 1 AD

Tuong tu BD 1BC, nen cac diem

B, A cung nhin doan CD duai mot goc

vuong, do do mat cau ngoai tiep tu

difn ABCD c6 tarn la trung diem CD

CD

va CO ban kinh R =

A p dung djnh ly Pitago cho cac

tarn giac ABD,ACD taco

Bdi 2.5.4 Cho hinh 15ng try tam giac deu ABCA'B'C c6 AB = a, goc giiia h;

mgt phang (A'BC) va (ABC) bang 6OO Gpi G la trpng tam tam giac A'BC Tinh

the tich khoi lang try da cho va tinh ban kinh m^t cau ngoai tiep tu di?n GABC

theo a

Cty TNHHMTV DWH Khang Vi?t

Jiuang ddn giai Qoi M la trung diem cua BC

po tam giac ABC deu nen BC 1 A M

^ A'M 1 BC (djnh li ba duong vuong goc)

' •••(••mi ijf

V^y A'MA = 60° Ta c6 A M = — a nen A'A = AM.tan A'MA = - a

it ••.:••• : :

The tich khoi lang tru ABCA'B'C la V - AA'-S^BC = 2 ^" 4 " ~8~^^'

Ggi H la trong tam ciia tam giac ABC, ta c6 = i = nen GH / / A A '

=> GH 1 (ABC) Do H cung la tam duong tron ngoai Hep tam giac ABC nen

GH la true duong tron ngoai tiep tam giac ABC Gpi I la giao diem cua GH vai trung true ciia GA (qua trung diem N ciia GA) thi do la tam mat cau ngo?i tiep tu di^n GABC

Phuang phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phii Khdnh, Nguyen Tat Thu

Jiu6ng ddn gidi

Gpi O la giao diem hai duang cheo ciia hinh thoi ABCD

Theo bai ra ta c6 tam giac ABD la tam giac deu canh a => BD = a

Ma tam giac SBD vuong tai S

nen SB = SD = — a , S O - -

2 2 Goi H la hinh chieu cua S tren mat

phang day thi H la tam duang tron

ngoai tiep tam giac ABD (do cac canh

Gpi K la tam cua tam giac deu BCD thi K la trung diem cua HC, true duong

tron ngoai tiep tam giac BCD di qua K va song song vol SH nen la trung true

cua HC cat SC tai diem I Ta c6 I la trung diem cda SC nen IS = IC, do do I

chinh la tam mat cau ngoai tiep hinh t u dif n SBCD Ban kinh cua mat cau la

R = isC = : ^ a

Bdi 2.5.6 Cho t u di^n deu ABCD c6 tam mat cau ngoai tiep t u di?n la O va

H la hinh chieu ciia A len mat phang (BCD)

OA 1) Tinh ty 1? k =

OH

2) Gia su mat cau ngoai tiep tii di?n c6 ban kinh bang 1

Tinh dp dai caC;Canh ciia t u di^n

Jiu&ng ddn gidi

1) Do t u dien ABCD deu nen cac

khoi chop O.ABC, O.ACD, O.ABD,

O.BCD CO the tich bang nhau

Cty TNHH MTV DWH Khang Vi?t

Tam O la giao diem cua duang trung true canh A B ciia tam giac A B H 2) •' (

Xa CO R = O A = 1 => O H = - Tam giac B O H vuong tai H

BH = V O B 2 - O H 2 = V R ^ - O H I - j i - - = -j-g CO BH la ban kinh duang tron ngoai tiep tam giac deu BCD ' < i

V^y canh ciia tu di^n deu la BC =

3 2^6

Bdi 2.5.7. Cho hinh chop S.ABC c6 (SBC) 1 (ABC) va cac canh AB = AC = SA

aSB = a Xac djnh tam va ban kinh hinh cau ngoai tiep hinh chop khi SC = x

cliop S.ABC ta CO OS = OB = OC nen O thupc B

thang d qua I vuong goc vdi (SBC)

Ta CO (SBC) 1 (ABC) => d c (ABC) =^ O e (ABC)

=> O la tam duang tron ngoai tiep AABC

Gpi K la giao ciia AI voi duang tron ngoai tiep tam giac ABC

Ta CO : AB^ - AI.AK = AI.2R ^ R AB^

2AI

Ta co:

A l 2 = A B 2 - B l 2 a 2 - ^ ^ ' -2 = a -• SB^+SC^ 7 a^ + x^ Sa^-x^ = a 2

-Jq^2 _ 2

^Al = — — (0 < x < V3a) Vay R =

^ 2 5 8 Cho hai duong tron (O^r^) va (02,r2) cat nhau tai hai diem A,B

^"an lupt nam tren hai mat phang phan bi^t (P) va (P')

^) Chung minh c6 mat cau di qua hai duang tron do

ta c6: O j O j = OO^ + OOJ - 2C)Oi.002 « + y^ - xy = 21 (1)

D o t u giac M O j O O j n o i tiep nen

M O i OO2 + MO2 OOi = M O O j O j o 4x + y = N/2T.Z (2)

O M 2 = M O ^ + O j 0 2 = M O ^ + 0 2 0 ^ < » z 2 = 1 6 + y 2 = l + x2 (3)

Giai h | g o m ba p h u o n g t r i n h (1),(2) va (3) ta t i m duoc:

X = 3V3; y = 273; z = 2V7 OA^ = AO^ + OjO^ = 37 R = O A = N/37

Bai 2.5.9 Cho h i n h chop S.ABCD c6 day A B C D la h i n h thang can va A B // CD

D u o n g tron tarn O nOi tiep trong h i n h thang c6 ban k i n h r Biet SO 1 (ABCD)

va SO = 2r Xac d i n h tarn va tinh ban k i n h mat cau npi tiep h i n h chop S.ABCD

Jiu&ng ddn gidi

Gpi M , N , P, Q la cac tiep diem

ciia d u o n g tron n o i tiep h i n h thang

v o i cac canh cua h i n h thang

D o SO 1 (ABCD) nen cac tam giac

SOM,SON,SOP.SOQ bSng nhau va

m p i diem tren SO each deu cac mat

ben cua h i n h chop Tam mat cau n o i

tiep la giao cua phan giac trong goc

SNO v o i SO

Phuongphap giai Toan Hinh hyc theo chuyett de- Nguyen Phu Khdnh, Nguyen Tat Thu ^^^^If

2) T i m ban k i n h R cua mat cau biet = 5;r2 = VlO; A B = 6;Oj02 = ^ 2 1

Jiitong ddn gidi

Gpi d i , d2 Ian l u p t la hai d u o n g thSng d i qua O i , O2 va v u o n g goc v o i (P)

( F ) Gpi M la t r u n g d i e m A B ta c6 (MO1O2) 1 A B ^ di,d2 c (MO1O2) • Goi Q

la giao diem ciia d i va di Ta c6 O la tam mat cau chua (O^) v a (O2) Ban kinj,

^ C h u n g m i n h rang k h i do, ban k i n h mat cau ngoai tiep t i i dien A B M N la nho riha't

dupe k h i X = y = aV2, hay A M = B N = ^ A B

2) Ban k i n h m a t cau ngoai tiep t u d i ^ n la R = ^x^ + y^ +4a^ nen

R = ^x2 + y2+4a2 > i/2xy + 4a2 = ^ = 2^f2a

m i n R = ly/la k h i x = y = a y / l , hay dat dupe tuong u n g v o i ban k i n h mat

^au n p i tiep dat gia t r j Ion nha't

Phuang phap giai Todn Hinh hpc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

^htemq 3

P H U O N G P H A P T O A D O T R O N G K H O N G G I A N

A,T6MTATLYTHUY^T

I Tpa dp trong khong gian

l.Jie true toa do trong khong gian Oxyz

• H§ gom ba true Ox,Oy,Oz doi mot

vuong goc duoc goi la he true toa dp

vuong goc trong khong gian

• Diem O goi la goc cua he toa dp, true

Ox la true hoanh, Oy la true tung va Oz

• Vai vecto l i trong h^ tpa dp Oxyz luon ton tai duy nha't bp (x;y;z) thoa

2 Toa do vecta - Toa d0 diem

Cty TNHH MTV DWH Khang Vift

c) Cac ung dung cua tich c6 huang

* Tudi?n: V ^ B C D ^ ^ AB,AC AD

I Dieu kif n 3 vecta dong phang:

V

Phuang trinh mdt cdu ,

Mat cau (S) tarn I(a;b;c), ban kinh R c6 phuang trinh ' ^- ^ ^'^

201

Phuamg phap gidi Todn Hinh hQc theo chuyen de- Nguyen Phu Khanh, Nguyht Tat Thu

1 Vec taphdp tuyen:

^inh nghia: Cho mat phang ( a ) Vec to

n ^ 0 gQi la vec to phap tuyeh (VTPT)

ciia m p (a) neu gia ciia n v u o n g goc v o i

* N e u ( a ) : A x + By + Cz + D = 0 thi n - ( A ; B; C) la m o t VTPT cua (a)

* N e u A(a;0;0), B(0;b;0), C(0;0;c) ; a b c ^ O thi p h u o n g trinh ciia (ABC) c6

4, JChodng each tu m0t diem den mdt mdt phdng:

K h o a n g c a c h t u M ( x o ; y o ; z o ) d e h m p ( P ) : A x + By + Cz + D = 0 la:

A x o + B y o + C z o + D d(M,(P)) =

• /'V -'! , ,}

V A ^ + B ^ + C ^

I I I P h u o n g t r i n h d u t m g thang trong k h o n g gian

/ ^hucmg trinh thorn s6 cua duang thdng:

a) Vec to c h i p h u o n g ciia d u o n g thang:

Cho d u o n g thang A Vec to u 0 goi la vec to chi p h u o n g (VTCP) cua duong thang A neu gia o i a no song song hoac trung v o i A Ta k i hi§u u ^ la VTCP ciia A

-* N e u u la VTCP ciia A t h i k.u (k ^ 0) cung la VTCP ciia A •

* N e u d u o n g thSng A d i qua hai diem A , B thi A B la mot VTCP

* N e u A la giao tuyeh ciia hai mat ph^ng (P) va (Q) t h i n p , n Q = U A la m p t VTCP ciia A (Trong do np,nQ Ian lupt la VTPT ciia (P) va ( Q ) )

b) P h u o n g t r i n h t h a m so cua d u o n g thang

Cho d u o n g thSng A d i qua M(xo;yo;Zo) va c6 VTCP u = ( a ; b ; c ) K h i do

x = xo + a t phuong trinh d u o n g thang A c6 dang: • y = yg + bt t e R (1)

Z = Z Q + Ct

(1) gpi la p h u o n g trinh tham so ciia d u o n g thSng A, t gpi la tham so

Chil y. Cho d u o n g thang A c6 p h u o n g trinh (1)

u = (a; b;c) la m p t VTCP ciia A

* M € A < : » M ( x o + a t ; y o + b t ; Z o + c t )

2 Phuong trinh chinh tdc:

H Cho d u o n g thSng A d i q u a M ( x o ; y o ; Z o ) v a c6 VTCP u = (a;b;c) v o i abc 0

K h i do p h u o n g t r i n h d u o n g thang A c6 dang:

Phuong phdp giai Toan Hinh hoc thco chuyen de- Nguyen Phu Khdtth, Nguyen Tat Thu

X - X Q _ y - Y o _ Z - Z Q

(2)

a b c

(2) goi la phuong trinh chinh tac ciia duong thang A

3 Vi tri tuang doi giun hai duang thdng

Cho hai duong th^ng d : = = di qua M(xo;yo;zo) c6

VTCP ^ = (a;b;c) va d':^ = ljl^ = ^ di qua M'(x',;r,;z'^) e6

a O C

VTCP uj =(a';b';c')

* Neu [ud,Ujj.]MM' = 0 d va d' dong phang Khi do xay ra ba tmong hop

z) d va d' cat nhau [ u , u ' ] ^ 0 va tpa dp giao diem la nghi?m cua h$ :

X-XQ ^ y - y o ^ Z - Z Q

X - X Q ^ y - y b ^ Z - Z Q

b' ii) d//d'<:^

a D c'

[u.ir] = 0

[ u , M M ' ] ^ 0 J[u,u'] = 0

m) d = d <=>-^

[ u , M M > 0

* Neu [u,u']MM';^0=> d va d' cheonhau ,|

4 Vi tri tuang doi giOn duang thdng vd mat phdng

Cho mp(a): Ax + By + Cz + D = 0 CO n = (A; B; C) la VTPT va duong thang

^ i i Z ^ ^ y : _ y ^ ^ z ^ a-(a;b;c) laVTCPvadiqua Mo(xo;yo;zo)

• A cat (a) o n va u khong cung phuong o Aa + Bb + Cc ^ 0 Khi do tpa do

Ax + By + Cz + D = 0 (a) X-XQ _ y - y o ^ Z - Z Q

9 0 4

Cty TNHHMTV DWH Khang Viet

* A -L (a) <=> n va u ciing phuong o n = k.ii

S^Xhodngcdch

a) Khoang each tir mpt diem den mpt duong thang:

Cho duang thang A di qua MQ , c6 VTCP u va diem M « A Khi do de tinh Idioang each tu M den A ta c6 cac each sau:

[MoM,u]

CI: Su dung cong thuc: d(M, A) = C2: Lap phuong trinh mp(P) di qua M vuong goc vai A Tim giao diem H cua (P) voi A Khi do dp dai M H la khoang each can tim ^

b) Khoang each giua hai duong thang cheo nhau:

Cho hai duang thang cheo nhau A di qua Moco VTCP u va A ' di qua MQ '

CO VTCP u ' Khi do khoang each giiia hai duong thang A va A ' duoc tinh

theo cac each sau:

CI: Su dving cong thuc: d(A,A') = .MoM'o

u,u' C2: Tim doan vuong goc chung M N Khi do do dai M N la khoang each can tim

C3: Lap phuong trinh mp(P) di qua A va song song vai A ' Khi do khoang each can tim la khoang each tir mpt diem bat ki tren A ' den (P)

IV GOC

I Gdcgiua hai duang thdng:'

I Cho hai duong thing A : ^ ^ = ^ ^ = ^ ^ c6 VTCP u=(a;b;c) va

a b c X - X Q _ y - y o _ z - z o

|uongthang A':

a' b Dat a = (A,A'),khi do:

Cho mp (a): Ax + By + Cz + D - 0 C O n = (A; B; C) la VTPT va duong thSng

I a b c

la duong thang A, khi do ta c6: '

Phuang phdp giai Todn Hinlt hoc titco chuycn de- Nguyen Pliii Khdnh, Nguyen Tat Ttiu

sincp = C O S |Aa + Bb + Cc|

A ^ + B ^ + ^4

3 Goc giica hai mat phdng

Cho hai mat phSng (a): Ax + By + Cz + D = 0 c6 VTPT = (A; B; C)

va (p):A'x + B'y + C'z + D' = 0 c6 VTPT = (A';B';C')

Goi ip la goc giiia hai mat phang (0° < ip < 90*^) Khi do:

AA'+ BB'+CC'I cos 9 = cos

V A ^ + B ^ + C ^ V A ' ^ + B ' ^ + C ' ^ '

B M O T S 6 BAI T O A N THl/ONG G A P

§ 1 T f C H C O H I / O N G C U A H A I V f i C T O V A TJTNG D U N G

Cac ung dung ciia tich c6 huang ciia hai vec to

• Di?n tich tam giac: Dien tich tam giac A B C dugc tinh boi cong thiic

1

' A A B C - • A B , A C The tich:

+ ) H i n h h p p : The tich hinh hop A B C D A ' B ' C ' D ' dugc tinh boi cong thuc

V A B C D A ' B ' C ' D ' = | [ A B , A D ] A A +) T u di^n: The tich tu dien A B C D dug-c tinh boi cong thiic

V A B C D = T r A B , A C ] A D Dieu k i f n 3 vecta dong phang:

+) Ba vec to a, b, c dong phSng Ichi va chi khi a,b

+) Bon diem A, B, (t, D dong phSng khi va chi khi

.c = 0

AB,AC AD = 0

Vi du 1.1.3. Trong khong gian Oxyz cho bon diem A(4;2;l), B(-l;0;3),

C(2;-2;0), D ( - 3 ; 2 ; l )

1) Chiing minh rang A, B, C, D khong dong phang;

2) Tinh di^n tich tam giac BCD va duong cao BH cua tam giac BCD;

3) Tinh the tich t i i dien ABCD va duong cao ciia tu di^n ha tu A;

4) Tim toa dq> diem M nam tren duong thang AB sao cho tam giac MCD c6

di|n tich nho nha't

Xgi giclL

Cty TNHH MTV DWH Khang Viet

ra BC,BD - 2 - 3 2 - 2 -3 3 -2 - 2 "2 2 3 - 2 = (10; 12; 2) jsjgn BA BC, B"D = -70 0 Do do, ta c6 A, B, C, D khong dong phang ' c _ ^

2) Ta CO- JABCD - 2 BC,BD = -VlO^ +12^ +2^ = 2^62

205

161 Vay S^Mco nho nha't khi va chi khi t = hay M ' 3 _ 88 ,527

.4r205'205j

T?jdu2.i.3.Trong khong gian Oxyz cho A(4;0;0), B(xo;yo;0) vdi Xo,yo >0

thoa man AB = 2^/l0 va AOB = 45°

1) Tim toa dp diem C nam tren tia Oz sao cho the tich t u dien OABC bang 8 2) Tim tpa dp diem M thupc Ox, diem N thupc Oz sao cho tam giac AMB can tai M va tu di?n ABMN c6 the tich bing 20

JC^gidL

Taco: OA = (4;0;0), OB = (xQ;yo;0) suy ra OA.OB = 4xo Theo gia thiet bai toan ta c6 h? phuang trinh sau:

(xo-4)^ + yo=40 4xn 1 o x o + y o - 8 x o = 2 4

.V2xo = sjxo + yo

yo =4

x § - 4 X 0 - 1 2 = 0

Phuorng phdp gidi Toan Hinh hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Tim

Vi du 3.1.3. Cho hinh lang tru diing ABC.A'B'C c6 day ABC la tam giac

vuong voi A(1;0;1), B{2;0;0),C(0;1;0) The tich cua khoi lang tru bang 3

1) Xac dinh tqa do cac dinh con lai cua lang tru,

2) Xac dinh toa dp diem each deu tat ca cac dinh ciia lang tru

1) Taco AB(l;0;-l),AC(-l;l;-l),rAB, ACl = (l;2;l) Dodo S ^ A B C =

The tich ciia kho'i lang try la 3 nen AA' = ABC.A'B'C

' A B C = V6

Tu AA' = t AB, AC = (t;2t;t) suy ra |t| = 1

• Neu t = 1 thi A'(2;2;2),B'(3;2;1),C'(1;3;1)

Neu t = -l thi A'(0;-2;0),B'(l;-2;-l),C(-l;-l;-l)

2) Diem I each deu tat ca cac dinh ciia lang try la trung diem cua duang noi

tam hai day

Gpi M, M' Ian lugt la trung diem ciia BC, B'C

Cty TNHH MTV DWH Khang Vi^t

X'i du 4.1.3 Trong khong gian voi h? tpa do Oxyz cho ba diem A(2;l;0),

B(0;4;0), C(0;2;-l) va duong thang d: x-1 _ y + 1 _ z-2 1 Lap phuong

trinh duong thing A vuong goc voi mat phang (ABC) va cat duong thang d tai diem D sao cho bon diem A, B, C, D tao thanh mot tii di^n c6 the tich

Xffi gidi

Taco AB = (-5;l;-3), AC = (-4;l;-4),AD = (6;-l;2) Suy ra AB A A C - (-l;-8;-l), (AB A A C ) A D = 0

Do do A, B, C, D dong phSng va n = (1;8;1) la VTPT ciia mat phSng (ABCD)

Vi M la trung diem CD nen M(5;-l;l), suy ra phuong trinh SM:

x-5 y+1 z-1

Do do S(5 + t;-l + 8t;l + t) Taco: D B = (-ll;2;-5), D C = (-10;2;-6):::>DBADC = (-2;-16;-2)

Suy ra S A B C D - ^AABC + ^ABCD - 2 A B A A C 1

+ —

2

D C A D B

2

Phumtg phdp gidi Todn Hinh hoc theo chuyert de- Nguyen Phu Khdnh, Nguyen Td't Thu

Nen Vs ABCD = - S M S ^ g c D = = 66 ^ S M = 2>y66

Va hai diem A ( - 2 ; l ; l ) , B(-3;-l;2) Tim tpa dp diem M thupc A sao cho tam

giac ABM c6 dien tich bSng sTs

Xgigidi

Taco M ( - 2 + t ; l + 3 t ; - 5 - 2 t ) ; A B = (-1;-2;1), A M = (t;3t;-2t - 6)

Suy ra A B A A M = (t + 1 2 ; - t - 6 ; - t ) nen S^^MB =-^^/3t^T36t7l80

Ma S ^ M B "=375 nen taco: t^ +12t +60 = 60 c:>t = 0,t =-12

Tir do ta tim dupe hai diemM(-2;l;-5) va M(-14;-35;19)

Vidu 7.1.3.Trong khong gian Oxyz cho hinh chop S.OABC c6 day OABC la

hinh thang vuong taiOva A(3;0;0), AB = OA = i o C , S(0;3;4) va y c > 0

1) Tim tpa dp cac dinh con lai va tinh the tich cua hinh chop S.OABC

2) Mpt mat phSng (a) di qua O va vuong goc voi SA cat SB,SC tai M va

N Tinh the tich khoi chop SOMN

JCffigiai

1) Do ABCD la hinh thang vuong tai A va

O, dong thoi A € Ox, y^ > 0,OC = 6

Nen ta suy ra dupe C(0;6;0)

2) Ta c6: SB = (3; -3; -4), suy ra phuong trinh mat phSng (a): 3x - 3y - 4z = 0

Cty TNHH MTV DWH Khang Viet

Vi SB = (3; 0; -4), SC = (0; 3; -4) nen ta c6 phuong trinh SB:

VidvL 8.1.3 Cho lang tru deu A B C A ' B ' C c6 canh day bang a Gpi M la

trung diem CC, biet A M I B ' M Chpn he tryc Oxyz sao cho A = 0, C

thupc tia Ox, A' thupc tia Oz va B thupc mien goc xOy

1) Xac dinh tpa dp cac dinh ciia lang try, 2) Tren cac canh A'B', A'C, BB' Ian lupt lay cac diem N , P, Q thoa A'N = NB', A'P = 2C'P, B'Q = 3BQ Tinh the tich khoi da difn AMPNQ

Dat A A ' - 2 x , x > 0 1) Taco A(0;0;0), C(0;a;0), A'(0;0;2x), C'(0;a;2x) Gpi K la hinh chieu eiia B len Oy, taco: BK = A B s i n 6 0 ° = — , A K = -

2 2 Nen B

2 ' 2 ' , B'

aVs a

-2 -2 Suy ra M (O; a; x) A M = (O; a; x), B'M =

2 2

=> AM.B'M = - — x^ Ma AM.B'M = 0 nen suy ra x =

Dodo A'(0;0;a72) va B' ^ ; | ; a N / 2

Phucmg phdp gidi Todii Hinh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

1) Tim X de a A b vuong goc voi c

2) Tim X de goc giua hai vec to a A b va c bang 120"

Jiuang dan gidi

Cty TNHHMTV nVVII Khang Vict

P^i 2.1 •3- Trong khong gian vai he tpa dp Oxyz cho diem A (3;-2; 4)

f im tpa dp cac hinh chieu ciia A len cac true tpa dp va cac mat phSng toa dp

2j Tir" ^ ^ ^ *a"^ 8'^*^ AMN vuong can tai A 3^ Tim tpa dp diem E thupc mat phang (Oyz) sao cho tam giac AEB can tai E

CO dien tich bang 3^y29 voi B(-1;4;-4) ,^

Jiucang dan gidi

\ Gpi A i, A 2 , A 3 Ian lupt la hinh chieu cua A len cac true Oa, Oy, Oz

^ ' " ^ chieu cua A len cac mat phang tpa dp (Oxy),(Oyz),(Ozx)

n = •

n = •

22 + 3 N/23T

5 22-3^/23T

Phuortig phap giai Todn Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

3) Vi Ee(Oyz) nen E(0;x;y)

Suyra AE = (-3;y+ 2 ; z - 4 ) , BE = ( l ; y - 4 ; z + 4)

= (8y + 6z-8;4z + 8;10-4y) AE,BE

Nen tir gia thiet bai toan ta c6:

2) Tinh chieu cao ve tu B ciia tam giac BCD va chieu cao ciia t i i dien ABCD ve

tu A

3) G(?i M, N Ian lugt la trung diem ciia AB va CD Tinh c6 sin ciia goc giiia hai

duong thang CM va BN

4) Tim E tren duong thSng AB sao cho tam giac ECD c6 di^n tich nho nhat

Jiuang dan gidi

^ABCD 3) Ta CO M ^•1-1

2'2'

3 7]

~ ' ' 2 ' 2 j

- - - ^7 1 Suy ra CM= - ; - ; - 4

27,

f 7 ,38,107'

27' 9 ' 27 ,

Bai 4.1.3 Trong khong gian Oxyz, cho ba diem A ( l ; l ; l ) , B(5;l;-2), C(7;9;l)

1) Chung minh rang cac diem A, B, C khong thang hang

2) Tim toa do diem D sao cho ABCD la hinh binh hanh

3) Tinh cos A, sin B, tan C ciia tam giac ABC

4) Tinh do dai duong cao tu dinh A, ban kinh duong tron ngoai tiep, ban kinh duong tron noi tiep ciia tam giac ABC

5) Tim toa dg giao diem ciia phan giac trong, phan giac ngoai goc A v6i duong thSngBC

Jiuang dan giai

1) Ta CO AB(4;0;-3),AC(6;8;0) nen cac diem A, B, C thang hang khi va chi khi

[4 = 6k ton tai so thyc k sao cho AB = kAC, tiic la

V%y A, B, C khong thang hang

0 = 8k (v6 li)

-3 = O.k

Phumigphdpgidi Todit llhih hoc llico clim/eit ile- Ngni/On Phil Khi'iiih, Njjuylit Tat Thu

2) V i A , B, C khong t h i n g hang nen A B C D la h i n h b i n h hanh k h i va chi khi

= V l - c o s B = - , 5 V 77

+) tanC,cosC cung dau tanC = |—I 1 =

k o s ^ C 38 4) Dien tich tam giac ABC la S = - B A B C s i n B = - 5 N / 7 7 - j — = VisT

2 s i n B 2 s i n B = 25

77

r = ^ = 2S

481 2V48T

p A B + BC + CA 15 + V77*

Chii y: Co the tinh dipn tich bang cong thuc S = A B , A C

5) Goi E, F Ian l u g t la giao diem ciia phan giac trong, phan giac ngoai goc A voi

d u o n g thSng BC

Theo h'nh cha't phan giac, ta c6 — = — =

^ ^ EC FC A C +) V i E nam trong doan BC nen EC = -2EB

G i a s u E{x^;y^;z^) thi EC(7 - X E ; 9 - y j : ; l - ),EB(5- X E ; 1 - y ^ ; - 2 - z ^ )

A ( 2 ; 3 ; l ) , B ( - l ; 2 ; 0 ) , C ( l ; l ; - 2 )

1) Tim toa do chan d u o n g vuong goc ke txx A xuong BC

2) Tim tga do H la true tam cua tam giac ABC

3) Tim tga do I la tam d u o n g tron ngoai tie'p cua tam giac ABC

4) Gpi G la trgng tam cua tam giac ABC C h u n g m i n h rang cac diem G, H , I nam tren mot d u o n g t h i n g

Phucntgphapgiai Todn Hinli iiQC theo chuyen de N^iiijcti I'hu Khanh, Nguyen Tat Thu

15 30 nen H G = 2 G I , tiic la ba diem G,

H , I nam tren mot duong thSng

Bai 6.1.3 C h o tam giac deu A B C c6 A ( 5 ; 3 ; - l ) , B ( 2 ; 3 ; - 4 ) v a diem C nam

trong m^t phang (Oxy) c6 tung do nho hon 3

1) T i m tpa dp diem D biet A B C D la tu di?n deu

2) T i m tpa dp diem S biet S A , S B , S C doi mpt vuong goc

Jiuong dan giai

V i C e ( O x y ) nen C(x;y;0)

T a CO A B ( - 3 ; 0; - 3), A C ( x - 5;y - 3; 1),BC(x - 2;y - 3;4)

T a m giac A B C la tam giac deu nen A B = A C = B C , do do

( x - 5 ) 2 + ( y - 3 ) 2 + 1 ^ = 1 8 ( x - 5 ) 2 + ( y - 3 ) 2 +1^ = ( x - 2 ) 2 + ( y - 3 ) 2 +4^

Cty TNHl! MTV nVVH Kluuix Viet

V i C CO tung dp nho hon 3 nen C(l;2;0)

l ) G p i D('<;y;z)- ' '

K h i do A D ( x - 5;y - 3;z + l ) , B D ( x - 2 ; y - 3 ; z + 4),CD(x - 1 ; y - 2;z)

T a m giac A B C la tam giac deu nen A B C D la tu d i f n deu khi v a chi khi

= B D = C D = A B = 3 N/2 Ta c6 h§ phuong trinh [(X - 5f + (y - 3)2 + (z +1)2 = (X - 2 )2 + (y - 3)2 + (z + 4)2 ^J' (x - 5 )2 + (y - 3)2 + (z +1)2 = (X -1)2 + (y - 2)2 + z2

z = l - x

y - 1 6 - 5 x 3x2-16x +20 = 0 V ' ' •

x = 2

x = — •

10 _ 2 _ 7

3 ' 3 ' 3

Giai phuong trinh 3x2 - 16x + 20 = 0 ^j^^^^

Vay tpa dp cac diem D la D(2; 6; - 1 ) hoac D

x + y - 4 z = 12 -3x - 3z = - 3 x2 + y2 + z2 - 6x - 5y + z = -11

Giai phuong trinh 3z2 + lOz + 8 = 0 ta dupe z = -2;z = - - ' ''^^'

Phucmgphapgidi Todn limit UQC theo chtiySn de- Nguyen Phi'i Khanh, Nguyen Tat Thu

^7 13 4~

Suy ra hai diem S thoa man la S(3;1;-2),S

3' 3 Bdi 7.1.3 Trong khong gian Oxyz, cho hinh hpp chu nhat ABCD.A'B'C'D'

CO A = 0 , B € 0 x , D € 0 y , A ' G 0 z va AB = 1, A D = 2, A A ' = 3

1) Tim toa dp cac dinh ciia hinh hop

2) Tim diem E tren duong thang DD' sao cho B'E 1 A ' C

3) Tim diem M thuoc A ' C , N thupc BD sao cho M N 1 B D , M N 1 A ' C Tu

do tinh khoang each giua hai duong thang cheo nhau A ' C va BD

Jiucang ddn gidi

1 ) Taco A(0;0;0),B(1;0;0), D(0;2;0), A'(0;0;3)

Hinh chieu ciia C len (Oxy) la C, hinh chieu cua C len Oz la A nen

A N = AB + BN = AB + y.BD = ( l - y;2y;0) => N ( 1 - y;2y;0)

Theo gia thiet cua de bai, ta c6:

gai 8.1.3 Trong khong gian v6i he true toa dp Oxyz cho hinh chop S.ABCD

CO day ABCD la hinh thang vuong tai A, B voi AB = BC = a; A D = 2a ;

A = 0 , B thupc tia Ox, D thupc tia Oy va S thupc tia Oz Duong thang SC va

BD tao vai nhau mot goc a thoa cosa = - ^

1) Xacdjnhtpa dp cac dinh cua hinh chop (;••;(• 2) Chung minh rang ASCD vuong, tinh dien tich tam giac SCD va tinh c6 sin cua goc hpp bai hai mat phang (SAB) va (SCD)

3) Gpi E la trung diem canh AD Tim tpa dp tam va tinh ban kinh mat cau ngoai tiep hinh chop S.BCE

4) Tren cac canh SA, SB, BC, CD Ian lupt lay cac diem M , N , P, Q thoa SM =

MA, SN = 2NB, BP = 3PC,CQ-4QD Chung minh rang M , N , P, Q khong dong phang va tinh the tich kho'i chop MNPQ ' ' ' '

o x^ + 2a^ = 6a^ o X - 2a S(0;0;2a)

2) Ta CO CS = (-a;-a;2a),CD = (-a;a;0) CS.CD = 0 => ASCD vuong tai C

Do do: S'ASCD c r n = ^CS.CD - i.aV6.aV2 = a^v/s

Gpi l ( x ; y ; z ) la tam mat cau

^ ngo^i tiep hinh chop SBCE ' x

77i