10 trọng điểm bồi dưỡng học sinh giỏi môn toán lớp 12 phần 2

s Hipang din giai Hinh chop S.ABCD c6 cac canh bing 1 nen hinh chieu cua S len day la H each deu A, B, C, D do d6 hinh thoi ABCD la hinh vuong nen hinh ch6p la hinh chop deu.. Hay xac d

10 trQng diem bai duong hoc sinh qioi man Joan i>' ih Phi

VAMDK = VK AMD - SAMD-X

Mgt khac VAMDK = VMADK = ^SADK.ci(M, ( A ' D K ) ) = ^ (^a ^ a)a = ^

N6n SA.MD= g ^ ' - ^ " ^ " 2Vi^~2~^^

Vgy d(eK, A p ) = X = -

Bai toan 13 26: Cho hinh ch6p tu- giac c6 tat ca cac canh bing 1 Mot mat

phing qua mot canh d^y, chia hinh chop ISm 2 ph^n tuang du'ang Tinh

chu vi thi^t di$n s

Hipang din giai

Hinh chop S.ABCD c6 cac canh bing 1 nen

hinh chieu cua S len day la H each deu A, B,

C, D do d6 hinh thoi ABCD la hinh vuong

nen hinh ch6p la hinh chop deu

Mat phing qua cgnh AB cSt hinh chop

theo thi^t di?n la hinh thang can ABEF

O0t EF = X thi SE = SF = X

AF^ = BE^ = x^ + 1 - 2x.cos60° = x^ - x + 1

Theo gia thiet: Vs ABCD = 2.Vs ABEF = 2(VSABF + VSBEF)

Ma S A B F _ S F V, SBEF _ ^ SF ^ ^2

S A B D SD = ' ' V 3 3 , , SC SD

nen x^ + X - 1 = 0, chon x =

Chu vi thiet dien: C = AB + EF + 2BE= ^{1 + Vs + 2V1O - 2>/2)

Bai toan 13 27: Cho dilm M nSm trong tCc dien ABCD

Dgt Va = VMBCD- Vb = VMACD Vc = VMABD Vo = VMABC •

ChLPngminh: V3.MA + Vj,.MB +V^.MC +V^.MD = 6

TNHHMTVDWH Hhong Vi^t

Hu'O'ng din Hu-o-ng din giai:

V = Vg.MA + V^.MB + V MC +V^.MD Gpi A ' la giao diem ciia A M va ( B C D ) Ta c6:

Bai toan 13 28: Chu-ng minh ring mot ti> dien thoa hai dieu kien: n^m canh

CO dp dai nho hon 1, con canh thu sau c6 do dai tuy y thi the tich V <

-8

Hu'O'ng din giai A

Xet tic dien ABCD c6 5 canh bSng 1 va cgnh con lai AD = a tuy y

Ta Chung minh the tich cua tu- dien nay 1^ Vi < I

8 That vay, ha AH vuong goc vb'i (BCD),

AK vuong gocBC thi:

3 4 3 4 3' 4 2

^ CO ti> dien thoa d4 bai c6 the tich nho hon V, ^ dpcm

^' toan 13 29: Gpi V va S l l n lugt la thd tich va di^n tich toan phin cua mpt

dien Chung minh ring — > 288

Hu'O'ng din giai

tip dien da cho la ABCD Goi dien tich cac J^t ABC, ACD, ADB, BCD l l n luqyt la So SB

^c, SA

10 trqng diem bSi dudng hgc sinh gioi m6n Toan in

Gpi B' la hinh chi§u vuong goc cua B len m0t phing (ACD) va C la hinsl

chidu vuong goc cua C len mat phing (ABD) , y

2 2 Lgp luan tu'ang ty ta du'gc < -SCSBSA, < -SQSASB va <

y y -SASBSC Do do V® < ( - ) V (SASBSCSD)^ Vi d i n g thCcc khong d6ng tho

dien SABC, V la t h I tich ti> dien SAMN Chung m i n h l < 9 V 2 < —,

Hiro'ng din giai

Gpi A' la trpng tam ASBC, I la trung dilm BC

Ta CP A,G, A' thing hang, S, A', I thing hang

- xy — - — => y

Cti^ TNHHMTVDWH Hhang Vi0t

K^t ho'p ta CO di^u ki^n ^ < x < 1 , ^ < y < 1

Ta c6: ^ = XV = • Xet f(x) = , I < x < 1 rtn,,-^:ini?

f'(x) = BBT:

(3x - ^f

4 V, 1 Vay - < — ^ - •

MA SBCD + MB SACD + MC.SABD + MD.SABC ^ 9V

A

Hiro'ng din giai

Ha AAi, MA2 vuong goc vai mp(BCD)

Ta c6: AM + MA2 > AA2 > AAi

=^ A M> A A i - M A 2

Dau bing trong xay ra khi M thupc du-an^ca^

AA, cua tir dien

Do do AM.SBCD > AAi ,SBCD - MA2.SBCD > SV - 3\

Ly luan tu'ang ti/ ta c6:

BM.SACD > 3V - 3VMACD; MC.SABD > 3V - 3VMABD ' MD.SABC > 3 V - 3 V M ABC

Ta CO MA.SBCD + MB.SACD + MC.SABD + MD.SABC

> 12V - 3(VMBCD + VMACD + VMABD + VMABC) = 9V DIu "=" xay ra khi M d6ng thai thupc 4 duang cap cua tCf dien ABCD nen M tam H cua tu" dien ABCD

^' toan 13 32: Cho hinh chop cut c6 chilu cao h, di$n tich cua thilt di#n

^ong song va each ddu 2 day la S Chung minh the tich V thoa man: Sh < V

4-Hifang din giai

Si, S2 la dien tfch 2 day hinh chop cut

Ta Chung minh : S - 2 1

N2

1 + 2k

B.i toan 13 3 3 ^ h o hinh ch6p ^ABCD eg d ^ - a Nnh^b^hJ^nK G K ^

trung d i l m cua SC Mat phang qua AK cat SB.SD tai M, N u a i vi

3 V 8 Hipo-ng din giai

s

va V = VsABCD ChCrng minh:

SM SN

Ta CO Vi = VsAMK + VsANK

u-ang ti^ Vs ANK = ^

3 V 8 Bai toan 13 34: Cho goc vuong xOy Tren cac tia Ox va Oy, Ian lu'p't lay hai

di§m M va N sao cho MN = a, vai a la mot do dai cho tru-ac

Tim tap hap trung diem I cua doan MN

|) Tren duang thing vuong goc vai mat phing (Oxy) tai O, lay mot dilm A ' djnh Hay xac djnh vj tri cua M va N sao cho dien tich tam giac AMN dat

"r$p hgp cac diem I la phan cua du'ang tron

l^m o ban kinh ^ nlm trong goc xOy

"^i^ng AH 1 MN thi theo djnh li ba du'ang vuong g6c OH 1 MN

Ta c6: SXAMN = - AH.MN = -a.AH

2 2 Dien tich tarn giac AMN I6n nhit khi va chi khi

AH Ian nhil D'iku nay xay ra khi va chi khi OH

Ian nhlt

Trong tarn giac vuong OHI ta luon luon c6:

OH < Ol: 0 H <

-2

Gia tri Ian nhat cua OH la - gia tri nay dat du'gc khi va chi khi H trung v6i |

khi do OMN la tarn giac vuong can '"^

Vay dien tich Ian nhat khi: OM = ON =

Bai toan 13 35: Cho tu- dien ABCD trong do goc giu'a hai du-ang thing AB va

CD bing u Gpi M la dilm b i t ky thupc canh AC, dat AM = x (0 < x < AC)

Xet mat phlng (P) di qua dilm M va song song vai AB, CD Xac dinh vj tri

dilm M d l dien tich thilt dien cua hinh tu- dien ABCD khi c i t bai mp(P) dat

gia trilan nhat

Hu'O'ng din giai

Thilt dien la hinh binh hanh MNQR

SMNQR = NM.NQ.sinMNQ

Do MN // AB, NO // CD nen goc giOa MN va NO

bing goc giOa AB va CD nen sin M NO = sina

MN: AB

AC ( A C - x )

AC MR = ^ x AC nen SMNQR = ' '"'^^ (AC - x)x.sina <—AB.CD sin a

AC^ 4 Tie do SMNQR max ce>AC-x = x o x = AC 01

Vay khi M la trung dilm cua AC thi dien tich Ian nhlt

Bai toan 13 36: Cho hinh chop S.ABC day la tam giac deu canh a Hinh chie^J

cua S len mat day trung vai tam O cua du'ang tron ngoai tilp day, SO = ^

Mot lang tru tam giac d i u c6 day du-ai n i m tren day hinh chop, ba dinh cu3

day n i m tren ba canh ben hinh chop

a) Tinh canh day lang tru khi mat ben la hinh vuong

b) Tinh t h I tich lan nhat cua lang tru khi a, h khong d6i

Hu-o-ng din giai

Gpi MNP.M'N'P' la lang tru, x la chilu dai ' c9nh day

I trung d i l m cua AB, SI n M'N' = 1'

, 3 c 6 C I = ^ , P T = i ^ I'P' _ SO' ^ SO' X

Ap dung b i t ding thu-c BCS:

Bai toan 13 37: Cho tam giac ABC, AB = AC Mot d i l m M thay d l i tren duang

I (ilhing vuong goc vai mat phing (ABC) tai A

a) Tim quy tich trong tam G va tryc tam H cua tam giac MBC

b) Gpi O la tri/c tam cua tam giac ABC, hay xac dinh vi tri cua M d l t h I tich tCf dien OHBC dat gia tri Ian nhlt

Hu'6'ng dan giai

^'GpiDla trung dilm cua BC

Ta c6: MB = MC Do do

1 BC va trpng tam G

^ua tam giac MBC n i m tren

'^D thoa m§n h^ thccc

DM Vay G la anh cua

^ trong phep vi tu- tam D, ti s6 vi tu- - ^

3 f ^ quy tich cac trpng tam G cua tam giac MBC la du-ang thing d' vuong vai mat phIng (ABC) tgi trpng tam G' cua tam giac ABC

f V

7 7 7 mpnp TTT^m 7 7 7

Hg CD ± AB, CF ± MB ta c6 H = DM n CF la tri/c tarn cua tarn giac MBC, Q %

DA o CE la try-c tarn cua tarn giac ABC Do CE ± AB va CE ± MA nen Cg ^

(MAB) Vi CF 1 MB nen EF 1 MB Do do MB 1 (CEF), ta suy ra MB i 0|~|

Chu-ng minh tu'ang tu ta c6 MC 1 OH Ti> do ta suy ra OH 1 (MBC)

DHO = 90° Vay quy tich true tarn H cua tarn giac MBC la duang tron duorig

kinh DO nSm trong mat phlng (D, d)

b) Goi HH' la chieu cao cua ti> dien OHBC, ta c6 H' thupc DO

Hinh chop nay c6 day OBC c6 dinh nen V Q H B C Ic^n n h i t khi va chi khi Hhf

Ian nhat Oi4m H chay tren du-ang tron duang kinh OD nen HH' Ian nhat khi

HH' = ^ D O nghTa la DHH' la tam giac vuong can tgi H', suy ra tam giac

DMA luc do vuong can tgi A

Vay tu- dien OHBC c6 t h i tich dat gia tri Ian nhit, c i n chon M tren d (v§ hai

phia cua A) sao cho AM = AD

Bai toan 13 38: Cho ba tia Ox, Oy, Oz vuong goc vai nhau tung doi mot tao

tam dien Oxyz D i l m M c6 dinh nam trong goc tam dien Mot mat phing

qua M c^t Ox, Oy, Oz l^n lu'p't tai A, B, C Goi khoang each iix M d i n cac

mat phing (OBC), (OCA), (OAB) l i n lu'p-t la a, b, c Tinh OA, OB, OC theo

a, b, c d§ tu- dien OABC c6 the tich nho nhit

Hu'O'ng din giai:

OA.OB.OC nho nhat

OA OB OC V§y; V nho n h i t « OA = 3a, OB = 3b, OC = 3c

Bai toan 13 39: Cho tu di^n ABCD c6 the tich V Mpt mSt phIng di qua trpf^^

tam M cua tu di§n c i t DA, DB, DC tgi A', B', C Tim gia tri nho n h i t cua: T '

V A A ' B C ' V B A ' B ' C "*• VcA'B'C'

Oy TNHHMTVDWH Hhong Vl$t

HiKO-ng d i n giai

Q0\i = (DAM) n (DBC), DBi = (DBM) n (DAC)

p(3^ = (DCM) n (DAB) DM n (ABC) = H 1^ trpng tSm AABC n^n

^ _ DA DB

Ma 4 = f DA'

AA'

DB' D C BB' C C

D A B C ,

V$yminT= | ^ V D A B C khi-^"^' BB' C C 1

64 " " " " ^ D A ' — w Bai toan 13 40: Cho kh6i chop t u gi^e d i u S.ABCD ma khoang c^ch t u dinh

A din mp(SBC) b i n g 2a Vai gia tri nao cua goc giua mat ben va m0t day cua khii chop thi t h i tich cua k h l i chop nho nhlt

Hu'O'ng din giai

Hg SO 1 (ABCD) thi O la tam hinh vuong

ABCD Gpi EH 1^ duang trung binh cua hinh vu6ng ABCD

AD // BC ^ AD // (SBC) d(A, (SBC)) = d(E, (SBC))

^9 E K i S H t a c6: EK 1 (SBC)

EK = d(A, (SBC)) = 2a ^

^ SHO 1^ goc giua mat ben (SBC) v^ m^t phIng dSy

^^t-SHo = x Khi do: * •

-a cosx

10 trpng diSm bSi dudng h<?c sinh gioi m on Toan 1£ icH.^o^ihPho

1 4a^

Vay SsABCD = — SABCD-SO = - — ^ •

.1 g ScosxsitT^x

Do (56 Vs ABCD nho nhit <=> y = cosx.sin^x dat gia trj Ian nhlt

Ta c6: y' = -sin^x + 2sinxcos^x = sinx(2cos^x - sin^x)

= sinx(2- 3sin^ x) = 3sinx

-y

V$y Ss ABC dat gia tri Ian nhJit khi x = a

Bai toan 13 41: Tren canh AD cua hinh vuong ABCD c6 dp d^i canh la a, |l

dilm M sao cho: AM = x (0 < x < a) Tren niia du'ang thing Az vuong goc vj

m|it phIng chica hinh vuong tai dilm A, l^y dilm S sao cho SA = y (y > 0)

a) ChLPng minh ring (SAB) 1 (SBC) va tinh khoang each ti> dilm M din

- mp(SAC).Tinh thi tich khii chop S.ABCM theo a, y va x

b) Bilt ring x^ + y^ = a^ Tim gia tri Ian nhit cua thI tich khoi ch6p S.ABCMj

Hipang dan giai a) Ta CO BC 1AB, SA nen BC 1 (SAB)

Do do (SAB) 1 (SBC)

Vi (SAC) 1 (ABCD) theo giao tuyin AC

nen ha MH 1 AC thi MH 1 (SAC)

Vay MH la khoang each tu" M tai mgt

The tich khoi chop S.ABCM la:V= - y -a(a + x)= ;Jya(a + x)

Vay f(x) dat gi^ trj I6n nhit tai x = | , khi do thI tich cua khoi chop S.ABCM

dat gia trj Ian nhit la: V = 7maxf(x) =

8

Bai toan 13 42: Cho hinh chop S.ABCD c6 bay cgnh bing 1 va cgnh ben SC

HiFang din giai

D ^y ABCD c6 4 canh bIng 1 nen la 1 hinh thoi

=>AC1BC

Ba tam gi^c ABD, CBD, BSD c6 chung canh BD, cac canh c6n lai bIng nhau va bIng 1 nen bIng nhau, c^c trung tuyIn

Suy ra tam giac ASC vuong tai S ta du-ac AC = Vx^+1

Gpi H la hinh chilu dinh S tren d^y (ABCD)

Do SA = SB = SD = 1 nen HA = HB = HD => H la tam du'ang tron ngoai tilp tam giac ABD => H e AC => SH la du-ang cao cua tam giac vuong ASC

Tac6SH.AC = SA.SC=>SH= , ^

4 2

^'^u ki^n X ^ < 3 O 0 < X < N / 3

lOtrpng c/u^m bSi duang hQc sinh gini mon Taan ly U- H~ , ;';

Ta CO SABCD = A C O B = = ^V(x^ +1)(3 - x ^ )

1

; V|y VsABCD = — SABCD SH = - X N / S - X ^ 6

Ta CO the dung dgo ham hay blit ding thu-c Cosi:

1 fTTr 2T 1 x ^ + 3 - x ^ 1

D l u "=" khi x^ = 3 - x^ » 2x^ = 3 » X = —

Bai toan 13 43: Cho diem IVI trong tu- dien ABCD C^c duo-ng thing MA

MB, IVIC, MD c i t mat doi dien tai A', B'.C D' tuang Ceng Tim GTNN cua

thing hang Goi V, Vi, V2, V3, V 4 ISn

iLfot la the tich cua t i i dien ABCD va 4

hinh chop dinh M voi cac day la cac

tam giac BCD, ACD, ABD, ABC Ta c6:

V | MC V

M C V, - 1

MD V MD' V, - 1

eai tlP 13 1: Tam giac ABC c6 BC = 2a va dudyng cao AD = a Tren dub-ng

thing vuong goc vai (ABC) tai A, lay di§m S sao cho

SA = a N/2 • Goi E va F Ian luol la trung di^m cua SB va SC

a) Gpi H la hinh chi§u cua A tren EF Chung minh AH n i m tren (SAD) Hay cho biet vi tri cua diem H d6i vai hai dilm S va D

[3) Tinh dien tich cua tam giac AEF

Htpo'ng d i n

a) Chung minh BC vuong goc vai (SAD)

K^t qua H Id trung dilm cua SD ,^ , ,,

b) K e t q u a S A E F =

Bai tap 13 2: Cho hinh chop S.ABCD c6 day Id hinh vuong canh a va SA 1

(ABCD), SA = X Xac dinh x d§ hai mat phIng (SBC) va (SDC) tao vai nhau

Khoang cdch giua cdc cSp canh d6i dien cua tu dien 6hu Id do dai doan noi

2 trung dilm.K§t qua — va VABCD = - ^ - ^ ' ^ ' ^

2 12

Bai t|p 13 4: Cho khdi tu dien ABCD c6 t h i tich V Tinh th§ tich kh6i da dien

CO 6 dinh la 6 trung diem cua 6 canh cua \(y dien ABCD

Hu-ang d i n Sosanhthltich K § t q u a - V

2

^3' tap 13 5: Trong mat phIng (P) cho tam giac ABC vuong tai A, AB = c, AB

b Tren duang thing vuong goc vai mat phIng (P) tai A, l l y dilm S sao cho SA = h (h > 0) M la mot dilm di dong tren canh SB Gpi I, J l l n lugt la

cac trung diem cua BC va AB

3) Tinh dp dai doan vuong goc chung cua hai duang thing SI va AB

Tinh ti s6 giOa t h i tich cac hinh chop BMIJ va BSCA khi dp dai dogn

^Uong goc Chung cua hai duang AC va MJ dat gia tri Ian nhat

^ung AC song song vai (SIJ) Ket qua ,

V b' + 4 2

W trQng diem bSl dUOng HQC s/nh gioi mon To6n 12 - LS Ho> i - •,• • \hd_

b) K^t qua

'BSCA

Bai t$p 13 6: Cho hinh chop ti> giac a§u S.ABCD Bi4t trung doan bing d

goc giOa cgnh ben d^y bing cp, tInh the tich cua kh6i chop

H i r i n g d i n Tinh cgnh day a bang each lap phuang trinh

, 4N/2d^tan(p ,

Ket qua —p ^ • >'i y

3V(2tan^(p + l f

Bai tap 13 7: Cho lang tru ABC.A'B'C Hay tinh the tich tu- dien ACA'B' bidt

tarn giac ABC la tarn gi^c d§u cgnh bSng a, AA' = b AA' tgo vai mat

phing (ABC) mot goc bing 60°

Hu-ang d i n j j j ; ,

Xac dinh hinh chieu cua A' len mp(ABC) Kk qua VACAB' = ^ ^ ^ ^

o

Bai tap 13 8: Cho hinh chop tarn giac SABC c6 SA = x, BC = y, cac canh con

lai"d§u b^ng LTinh thi tich hinh chop theo x, y Vai x, y nao thi thi tich

hinh chop Ian nhat?

HiFang d i n Gpi M trung di6m BC thi th§ tich hinh chop chia doi b^ng nhau bai mp(SAM)

K§t qua V = y ^1 - ^ the tich hinh chop Ian nhlt khi x = y =

Bai tap 13 9: Cho hinh chop S.ABC c6 day ABC la tarn giac vu6ng can tai

dmh B, AB = a, SA = 2a va SA vuong goc vai mat phing day Mat phing

qua A vuong goc vb-i SC cSt SB, SC lln lu 'O't tai H, K Tinh theo a thi tich

khoi ti> di^n SAHK

Hu-ang d i n 8a^

Dung ti s6 thi tich Kit qua VSAHK = ^ •

Bai tap 13 10: Cho tCf dien ABCD c6 BAD = 90°, CAD = ACB = 60°, va AB =

AC = AD = a Tinh thi tich tip dien ABCD va tinh khoang each giOa hai duong

thing AC va BD

Hu'O'ng d i n Xac dinh dgng tarn giac BCD suy ra hinh chieu len (BCD)

42

Kit qua VABCD = va d(AC; BD) = |

Cfy TNHHMTVDWH Hhong Vi$t

ifit C

jjI^N THliC TRQNG TAM ' |\/l?t clu va kh6i clu

Cho mat cau S(0; R) duac xac dinh khi bilt jgfii va ban kinh R hoac bilt mot duang kinh cua no

Qj^n tich mgt clu: S = 47rR^

, Thi tich khii clu ( hinh cau): V = -TTR^

3

Vj tri tu-cng d6i giOa mat clu va mat phing

Cho mat clu S(0; R) va mp(P) Gpi OH = d la khoang each tCr O din (P) thi:

, NIU d < R: mp(P) cit mat clu theo du-ang trbn giao tuyin c6 tam H, ban kinh r = VR^ - d^ Dat biet, khi d = 0 thi mp(P) di qua tarn O cua mat clu, m|t phing do goi la mat phing kinh; giao tuyIn cua m^t phing kinh v6'i m|t clu la du'ang tron c6 ban kinh R, goi la du-ang trbn Ian cua mat clu

- Nlu d = R, mp(P) va mat clu S(0; R) c6 dilm chung duy nhat la H Khi d6 m^t phing (P) tilp xue vai mat clu tai dilm H hoac mp(P) la tilp dien cua m$t clu tai tilp dilm H

- Neu d > R: mp(P) khong eo dilm chung vai mat clu

tri tiFcng d6i gifra mat clu va du-o-ng thing:

Cho mat clu S(0; R) va du-ang thing ^ Gpi H la hinh chilu cua O tren A

d = OH la khpang each ti> O tai A

jjlu d < R: duang thing A cit mat clu tai hai dilm '

d = R, (ju^ang thing A va mat clu S(0;R) c6 dilm chung duy nhit la H

h' do, du-ang thing A tilp xuc vai m^t clu tai dilm H hoac A la tilp tuyIn [.'^^mat clu tai tilp dilm H

^ U d > R: du-ang thing khong c6 dilm chung vai mat clu

(J^h ly: Nlu dilm A nim ngoai mat clu S(0; R) thi qua A c6 v6 so tilp vai mat clu Khi do

, d^j cac doan thing nli A v6-i e^e tilp dilm deu bing nhau

•'^ hgp cac tilp dilm la mot du-ang tron nIm tren m$t clu

10 tr<?ng diS'm bSl dUOng hgc sInh gioi mdn Todn 12 / , „ /; Phd

P h u ' c n g tich:

C h o mat c a u S ( 0 ; R ) va didm M Qua

diem M, ve 2 cat t u y ^ n cSt mat c i u tgi

A, B C, D thi

M?t c l u ngoai ti4p khdi da d i ? n : Mat c a u di qua moi dinh cue hinh da cjj^

gpi la mat c l u ngoai ti§p hinh da dien va hinh da dien gpi la npi ti§p mat c^u CJQ ^

- Dieu ki§n c i n va du d ^ mot hinh chop c6 mat cku ngoai t i l p la day cua hint,

' chop do CO du-ang tron ngoai ti§p

- Dieu kien c4n va du d l mot hinh lang tru c6 m^t c ^ u ngoai tiep la lang ^

d i f n g va day cua hinh lang tru do c6 du-ang tron ngogi t i l p

Xac djnh tarn O cua mat c l u ngoai ti6p

- Hinh chop S.AiA2 An c6 day la da giac npi ti4p du'ang tron (C), gpi A la true

cua du-d-ng trpn dp va gpi O la giao d i l m cua A v a i mat p h i n g trung triic

cua mpt canh ben, chSng han canh S^^ thi O S = OAi = OA2 = = OAn nen

O la tam mat c^u ngoai ti§p

- Hinh ISng tru di>ng c6 day la da giac npi ti§p du'ang tron G p i I, 1' la hai tam

cua du'o-ng tron ngoai tiSp 2 day thi II' la true cua 2 du-b-ng tron G o i 0 la

trung d i l m cua H' thi O each d i u cac dinh nen O la tam mat cau ngoai ti§p

Mat cilu noi ti4p hinh d a dien: Mat c^u t i l p xuc v a i mpi m§t cua hinh da

dien gpi la m a t c l u npi t i l p hinh da di?n va hinh da di$n gpi la ngoai tigp

m#t c l u do

Xac djnh t a m I c u a mat cku npi ti6p;

T i m diem I each d ^ u t^t ca cac mat cua kh6i da dien V a i 2 mSt song song

thi I thupc mSt p h i n g spng spng each d ^ u , v o l 2 mat p h I n g c§t nhau thi

thu0e mat phan giac (ehCea giac tuyen va qua mpt du'ang p h ^ n giSc cua goc

tap bP-i 2 d u a n g t h i n g Ian lu-pl thupc 2 mat p h i n g , vucng gpc vai giao tuyen)

Mgt tru, hinh tru, khdi tru

, MSt tru la t a p hp'p t i t ca cac d i ^ m M each du'ang t h i n g A c6 dinh mot

- Dipn tich xung q u a n h : S^q = 2nRI

- Jhk tich khdi tru: V = TtR^h

- Thiet d i ^ n song song v a i true hinh tru la mpt hinh chu- nhat, t g c b d i 2 du^"

sinh song spng va b i n g nhau D ^ c biet, thidt d i ^ n qua true hinh tru la ^ '

hinh chQ' nhat c6 2 kich thu-ac la d u d n g kinh day va chi§u cao hinh try

^

k

cr<V TNHH MTVDWH Hhang Vi$t

''Oft.'

K/||t n o n , hinh n o n , khdi non

M^t non sinh ra khi quay du-dng t h i n g / c I t A c6 dinh va h a p v 6 i A goc a I<h6ng d6i, quanh A Mat non c6 true A va goc a dinh la 2 a

Hjnh non, khdi n o n : Tryc SO Du-dng sinh S M = /

Goc a dinh la 2 a Ban kinh d a y R va chidu cap h t h i :

^2 = h ' + R '

, Dipn tich xung q u a n h : Sxq = TiRC / ^ ^ rf;:)f/

_ T h i tich khdi non: V = - r t R ^ h

3 _ T h i l t dien c I t b a i mpt mgt p h i n g di qua dinh hinh ndn thi c I t m^t nen theo

2 du-dng sinh SA, SB b I n g nhau tao thanh t a m giac can S A B D a c biet,

thidt dien di qua true hinh non thi la tam giac c § n S A B v d i SA, S B 1^ 2

du-dng sinh b I n g nhau va A B la du-dng kinh day

Chu y : Phu-ang phap du-dng sinh

2. C A C B A I T O A N

Bai toan 1 4 1 : T i m tap h a p t a m cac mat cau

a) D i qua ba didm khdng t h i n g hang A, B, C cho tru-dc b) Tidp xuc v d i ba canh cua mpt tam giac A B C chp t r u d c

Hu-d-ng d i n giai

a) 11^ tam cua mat c l u di qua ba d i l m phan bi?t A, B, C chp t r u d e khi va chi khi lA = IB = 10 V a y tap h a p cac diem I la true cua du'dng trdn ngpai tidp tam giac A B C

b) Mat cau t a m O tiep xuc v d i ba canh A B ,

BC, CA cua tam giac A B C Ian lu'p't tai c^c d i l m I J, K khi va chi khi 01 1 AB,

OJ 1 BC, OK 1 CA, 01 = OJ = OK Gpi

0 ' la hinh chidu v u d n g goc cua didm O tren m p ( A B C ) thi cac didu kien la: O'l 1

AB, O'J 1 BC, O'K 1 CA, O'l = O'J =

O'K, hay O' la t a m du'dng trdn ndi t i l p tam giac A B C

yay tap hp'p cac t a m O la true cua du'dng trpn npi tiep t a m giac A B C

^ ' t o a n 14 2: T i m tap h a p eac d i l m M sap cho tdng binh phu'ang cac

•^hoang each tu- M t d i 8 dinh cua mpt hinh hdp chp tru-dc b I n g k^ chp tru'de

Hu-d-ng d i n giai

^ ' a su- ba kich t h u d e cua hinh hop la A B = a, BC = b, C C = c t h i : '

AC'^ + BD'2 + CA'2 + DB'^ = 4(a^ + b^ + e^)

1 »\

M \

' \

c

Gpi O la tarn cua hinh hop, ta c6:

Bai toan 14 3: Cho tip dien ABCD Tu' mot dilm M ve 4 cat tuyen MAA',

MBB', MCC, MDD' vai mat cau noi ti^p Tim tap hgp cac di^m M sao cho:

MA MB MC MD

= = + = + = + = = 4

MA' MB' MC MD'

Hu-ang din giai

Gpi G la trong tarn tip dien ABCD

Gpi mat c^u ngoai ti§p S(0; R) Ta c6:

sj^y H c6 dinh nen t^p hgp cac diem M la m§t phing vuong goc vai OG tai H

toan 14 4: Cho P la mot di§m c6 dinh nkm ben trong mpt mSt clu cho

trii'd'C- Ba day PA, PB, PC vuong goc nhau tCrng doi mpt Gpi Q Id diu mut

tHi> hai cua du-ang cheo PQ cua hinh hop chu nh^t ma cdc canh Id PA, PB,

pC Tim quy tich cac di^m Q khi ba dilm A, B, C chay tren m^t cku

Hu-ang din giai ,.,5 ^g^ ,

Theo gia thiet ta c6: PQ PA + PB + PC f Qpi G la trong tam cua tam giac A B C thi: PA + PB + PC = 3 P G - -

Do do PA = 3PG nen quy tich cua Q la anh cua quy tich cua G qua phep vj ti^ tam P ti s6 bSng 3 Ta c6:

Ciipn di§m I CO dinh: Pi = -PO thi 2IP +10 = 6 Khi do:

'^o do diem G chay tren mat cku tam I ban kinh r =

^^uyra quy tich cua Q

^' |oan 14 5 : Cho 2 du-ang tron (O; r), (C, r') cit nhau tai A, B va Ian lu-gt

"^^m tren 2 mat phing phan biet (P), (P')

^^ij-ng minh mat cku (S) di qua 2 du-ang tron d6

^ho r = 5 , r' = N/IO , 0 0 ' = V 2 I , AB = 6 Tinh ban kinh cua (S)

HiPO'ng d i n giai:

a) Gpi M Id trung d i i m cue AB thi OM 1 AB, O'M 1 AB

Tu- do m p ( O M O ' ) la mp trung tru-c cua A B

Gpi A va A' l l n \ua\a trgc cua du-ang tron C ( 0 ;

" r) va C ' ( 0 ' ; r') thi A va A' cung vuong goc vai AB

n6n A, A' cung n i m trong mp(OMO') va cSt

Nen = IB^ = IM^ + MB^ = 37 Vay R = V37

Bai toan 14 6: C h o mot tu- dien deu A B C D c6 canh bSng a Mot mat c§u (S)

tiep xuc v a i ba du'ang t h i n g A B , A C , A D l l n lu-gt tai B, C va D Mot mat cku

(S") CO ban kinh R' < R, ti§p xuc vai mgt cau (S) v^ cung nhgn cac du-ang

t h i n g A D , A B , A C lam cac t i l p tuySn

a) Tinh ban kinh R cua mat c l u (S)

b) Tinh t h i tich kh6i c l u (S')

Hu'O'ng d i n giai a) Gpi O la tam cua mat c l u (S) thi OB = OC = O D

= R va OBA, OCA, O D A la nhu-ng tam giac

vuong tai cac dinh B, C, D Gpi H la giao d i l m

cua A O va mp(BCD) thi H la tam cua tam giac

deu B C D

Ta CO A H = 1N/6 DH =

Do do R = O D = a V 2

b) Gpi O' la tam mat cau (S') vS D' IS d i l m tiSp

xuc cua (S') v a i A D , c i t ca hai mat c l u bai mat

p h i n g (ADO) ta du-ac hinh g6m hai d u a n g trbn

t a m O, tam O' t i l p xuc v a i nhau va cung t i l p

Bai t o a n 14 7: C h i j n g minh c6 mat c l u t i l p xuc v a i sau cgnh cua hinh

ABCD khi va chi khi: A B + C D = AC + BD = A D + BC

H u - a n g d i n g i a i Gia su- c6 mat c l u tiep xuc vai sau canh cua tu- dien A B C D tai cac d i l m M, N, P, Q,

R, S nhu- hinh ve thi A M = AP = AR = x,

2 = ^ [CA + CB + C D - ^ (AB + BD + DA)]

t = ^ [DA + DB + DC - ^ (AB + BC + CA)]

^ h a n xet r i n g true cua cac du-ang tron npi tilip cac mat day (M, N, P, Q, R,

^ cung la cac t i l p d i l m ) d6ng qui tai d i l m J Mgt c l u tam J di qua cac d i l m

•vl, N, P, Q, R, s la mat c l u c i n tim. ,n ± r J^oan 14 8: Cho 4 hinh c l u c6 cung ban kinh r va chung du-gc s i p x i p sao

doi mpt tiep xuc v a i nhau Ta d i / n g 4 mat p h I n g sao cho moi mat

Phang deu tidp xuc v a i ba hinh c l u va khong c l t hinh c l u con Igi Bon mat Pnang do tao nen m p t tu- dien ddu Hay tinh th§ tich cua kh6i tii- di^n do

tneo r

lO tr<?ng aiem bOI dUOng hgc sinn gi6t iHM l&Cih I a-LB HC>0/)n hVO ^

Hipo-ng din giai

Gpi Mi, Ma, Ms M4 1^ tSm cua 4 hinh clu d§ cho thl d6 Id 4 dinh cua m^,

tu" dien d§u CO canh b^ng 2r va do do c6 chi§u cao hi = —XSIQ va Vnk tich u

3 '9

3 ^-^.i- na 'Vii

Gpi O la tarn cua ttp dien deu M1M2M3M4. Ta c6 tii' di^n d§ chp d6ng

v6i tCr dien M1M2M3M4 han nOa O chinh la tarn dong dgng Ta gpi cac ^\^\

cua tLf dien da cho la Ai, A2, A3 va A4 sao cho trong phep d6ng dgng t[ u

1 biln M, thdnh A, (i = 1, 2, 3, 4)

^' Ta CP hai m$t phIng (M2M3M4) vS (A2A3A4) spng spng v6i nhau va c6

khoang cdch dung bing ban kInh r. Gpi G Id tarn cua mSt (M1M2M3) va G' |^

tarn cua m$t (A1A2A3) thi:

Bai toan 14 9: Canh day va du'ang cac cua hinh ISng try lyc giac deu

ABCDEF A'B'C'D'E'F' \kn \uai blng a va h Chii-ng minh ring s^u m|l

phIng (AB'F'), (CD'B'), (EF'D'), (D'EC), (F'AE), (B'CA) cung tiep xuc vai mOI

m$t clu, xac djnh tarn va ban kinh

Hu'O'ng din giai

Gpi O la tarn hinh lang try Mat phIng (AB'F') ti§p xuc vai m$t c^u tarn 0

mat clu (S) nay du-gc xac dinh duy nhJit Sau mat phIng d6u each d^u 0

suy ra rSng ca sau mat phlng d§u ti^p xuc vai m§t clu (S) tam O

Gpi P la trung diem canh AE, P' la trung diem cgnh A'E'; Q la trung diim

canh PF', va gpi R la hinh chilu cua 0 len du'ong thing PF', thi c^c di4m ^<

P', Q, R, O, F' cung nim tren mot mat phing

Ta CP F'P' = - va QO = — Vi QO // F'P' nen RQO - P P ' F ' Ngpai ^\

ORQ = P ^ ' = - 90° nen suy ra hai tam giac ORQ va PP'F' dpng d

nhau DP dp, ban kinh cua (S) la:

gai toan 14 10: Tic dien ABCD c6 AB = 6, CD = 8, cdc canh c6n Igi diu blng

7 / 4 Djnh tam va tinh dien tich hinh clu ngoai ti^p tu- dien

Hu'O'ng din giai

Gpi M, F thCr ty la trung diem cua AB, CD va K Id t§rn duang tron ngoai tiep AABC Khi do K thupc CM Ha K G 1 FM thi O la tam mdt cau

ngogi tilp tip dien ABCD, R = CD

Ta CP CM = DM = V74-9 = N/65

Vd MF = 765-16 = 7 Gpi R la ban kinh duang tron ngoai tilp AABC

Ta c6 R =

suy ra

abc 4S ^

Vgy dien tich mat cdu S = 47:R^ = IOOTT

Bai toan 14 11: Cho hinh chop S.ABC cc SA = SB = SC = a, ASB = 60°,

BSC = 90° va CSA = 120° Xac djnh tam va tinh ban kinh mat ciu ngpai tiep

Hu'O'ng d i n giai:

Ta CP AB = a, BC = a N/2 va AC = a v's

ndn tam giac ABC vuong a B Gpi SH la du-d^ng cao cua hinh chop, do SA = SB = SC nen HA =

HB = HC suy ra H la trung di§m cua canh AC

Tdm m$t clu thupc true SH Vi g6c HSA = 60°

n§n gpi O la dilm d6i xu-ng vai S quadiemHthi:OS = OA = OC = OB = a

• P u y ra mat clu ngoai tiep hinh chop ^

S.ABC CP tam O va c6 ban kinh R = a 11'/ toan 14 12: Cho hinh chop tam giac deu SABC c6 dudyng cao

-So = 1 va canh day bing 2 V6 Diem M, N la trung diem cua canh AC, AB

W n g trng Tinh t h l tich hinh chop SAMN va ban kinh hinh clu npi ti§p hinh chop do

10 tTQng diSm bSl dUCing HQC smh gioi mdn Toon 1£

HiFang din giai:

Do ABC 1^ tam gi^c d4u n6n:

AM = MN = NA =

S M M N = ^AM.AN.sin60°= ^

2 sin

1 3V3 , V3 Dodo: VsAMN= - • 2 ^

Vi SABC la hinh chop d4u nen O trung

vai tarn du-ang tron npi tiep tam giac

ABC

Do do OM 1 AC, ON 1 AB va do SO 1 (ABC) n6n ta suy ra SM 1 AC, SN

AB va SM = SN Xet tam giac vuong AOM; SOM:

OM = ATtan30° = V6 ^ = V2 = ON

SM^ = OM^ + SO^ = 2 + 1 = 3 SM = V3 , nen: » '•'

SsAM = AM.SM = ; SsAN = AN.SN = y

-Gpi K la trung diem cua MN thi S K 1 MN

SK' = S M ' - KM' = 3 - ^ = - => SK = ^ nen:

SsMN = ^ MN.SK = I ; SAMN = ^ MN.AK = ^

Do d6 b^n kinh hinh c l u nOi ti^p: r = 3V

s,„ 1+2V2 + V3

Bai toan 14.13: Goc tam di^n Sxyz dinh c6 xSy = 120°, ySz = 60°, zSx = 90°

Tren cac tia Sx, Sy, Sz lay tuong ung cac dilm A, B, C sao cho SA = SB =

SC = a

a) X^c djnh hinh chieu vuong goc H

cua dinh S len mp(ABC)

b) Tinh ban kinh hinh c l u npi ti4p ti> di#n SA

Hu>6ng din giai

a) Do BSC = 60°, nen SBC 1^ tam giac d4u,

po A S C Id tam gidc vu6ng cSn n§n A C = a V2

f !> tam gidc cSn A S B c6 g6c dinh Id 120° n§n A B = a V3

AD = BC = a

a) Tinh ban kfnh mdt cau ngoai tiep tLP diSn R

b) Tinh ban kinh mdt cdu nOi tidp r

Hu'O'ng din giai

Xem tu- dipn ABCD la mot ph^n cua hinh hpp chu' nh|t vai 3 kich thuc^c m,

m ' + n ' = c ' m' + p2 = a ' =:>

n' + p 2 = b '

m'=l(a'+c'-b') n2=:I(b2+c'-a')

= V(a + b + c)(a + b-c)(b + c-a)(a + c-b)

\^$y r = ^ = : l /2(b^ + C -a')(a^ + b^ -c^Xa^ - b ' T ^

S,p 4V (a + b + c)(a + b-c)(b + c-a)(a-b + c) '

I

Bai toan 14 15: Gpi (P) Id m#t phIng <3i qua A va ti§p xuc vai mSt c l u ngog

ti§p tu dipn ABCD Cac mgt phIng (ABC), (ACD), (ABD) cSt rngt phIng (pj

l^n lu'p't theo cdc giac tuy§n d, b, c Bi^t d, b, c tao v6i nhau thdnh 6 QQ

bing nhau Chung minh ring: AB CD = AC BD = AD BC ^

,j Hu'O'ng din giai o ,

' Tren AB, AC, AD ta l l y Ian lu'p't cac dilm B', C,

D' thoa m§n:

AB' = ACAD, AC = AB.AD,

AD' = A B A C Ta CP — = — = AD

AC AB nen 2 tam giac ABC va ACB' d6ng dang

Vi d Id tilp tuy§n cua du-ang tron (ABC)

Suy ra dAC = ABC (chdn cung AC)

Do do dAC = AC' B' d // B'C Tuang ty b // CD', c//B'D'

Vi b, c, d tao thanh cac goc bdng nhau, suy ra tam giac B'C'D' deu

B'C

Ta lai c6

BC, = AD suy ra B'C = BCAD

Tuang ti/ CD' = CD.AB, D'B' = DB.AC ^ dpcm I

Bai toan 14 16: Cho tip dien ABCD c6 tinh chit: Mat c l u npi ti4p cua tip dien

ti§p xuc vai mat (ABC) tai tam duang tron npi ti§p I tam giac ABC, tiep xuc

vai mat (BCD) tgi true tam H cua tam giac BCD va ti§p xuc vai mat (ACD)

tai trpng tam G cua tam giac ACD Chung minh ABCD la tu dien d4u

Hu-ang din giai

Vi I la tam cua duang tran npi tiep AABC nen

a = lAB = lAC; p = IBC = I B A , y = ICA = ICB

Theo tinh chdt tiep tuy§n ta cc:

MAC - AGAC; AIBC = AHBC

suy ra (x = GAC, p = HBC, y = GCA = HCB

Trong tam giac ABC ta c6:

a + p + y = 90°, suy ra trong ABCD ta c6:

a = HBD = HCb , p = hIDC , y = HDB ,

Mat khac AHCD = AGCD suy ra a = GCD , p = GDC

Gpi P la trung di6m cua AC ta cp P G C = a + p suy ra G P C = 180° - (a + PI

+ y) ^ 90° Do do DP la dub-ng cap vua la duang trung tuy§n nen ADA^

can dinh D suy ra AGAC can dinh G Tu d6 a = y nghTa la H C D = HC^^

nen CH la phan giac cua goc DCB Tu do ADCB can a C, vay CB =

Mat khac \ABC can a B n6n BA = BC Vay DA = DC = BC = BA M0t KH^'^

JO a = Y GCfK = GCD , v$y ACAD cdn a C, ngodi ra ACAD c6n cdn a

f^§n Id tam gidc d6u suy ra p = 30° ChCpng minh tuang tu a = y = 30° nen

^gC va ABCD d4u suy ra 6 canh cua tup dien bdng nhau Vay ABCD la tip

.-toan 14 17: Cho t u di^n ABCD c6 dp dai cac cap cgnh d6i Idn lup't la a,

' b, b', c, c' Gpi V vd R Id t h i tich va ban kinh mdt cdu ngoai ti§p tip di^n

a) ChLfnO "^'"'^ ^ "^9t tam giac c6 dp ddi 3 canh Id a.a', b.b', cc'., J,) Gpi S Id di^n tich tam giac do ChCpng minh rdng: S = 6.V.R

jHifdng din giai

Gia s"^ = a, CD = a', AC = b, BD = b',

AD = c BC = C

£)^tk = a.b.c -fren AB, AC, AD ta Idy Idn lup't cac diem Bi, Ci, Di thpa

ABi.AB = ACi.AC =ADi.AD = k AAB1C1 dong dang AABC • - b

^^'""'"-^KQ ' ab

\/$y B1C1 = cc'

Tipang tu d D i = aa'; B1D1 = bb' nen A1B1C1 Id tam giac cdn tim

b) Gpi I Id tdm mdt cdu ngoai tilp tip di$n ABCD, O Id tam duang tron ngoai

tiep tam giac ACD, M Id trung diem AD

Ta c6: AOM = A C D = A D C , => Tu giac OED1M npi ti^p

^ O A l C D , = Al ± C i D i Tuang tu Al 1 B1C1 =:> A l l (B1C1D1) Al cdt (AiBid) t?! H TacpAH.AI =AE.AO = ADi.AM= - A D i A D = - a b c

W trQng dii^^Tboi duanq HQC sinh qlOl mon lOUii - ce , lucrnn

^ Hipo-ng din giai:

[cpi Si 1^ di^n tich m|it d6i di^n dinh Ai, ta c6:

Bai toein 14, 19: Trong m^t phing (P) cho du'6'ng trdn (O; R) va diem A sao

cho OA = 2R, Tren du'6'ng thSng d vu6ng g6c (P) tai A liy mot di^m s ci5

djnh Cho M e (O; R), gpi I, J la trung diem SM, AM Chirng minh r^ng khi M

chuyen dpng tren (C) thi dogn IJ sinh ra m$t xung quanh cua mpt hinh tru

HiPO'ng din giai

Gpi H Id trung dilm cua OA thi HJ = — suy ra J

thupc du'6'ng tron (tH; — ) nim trong (P)

Vi IJ Id du'ang trung binh cua tam gid

SMA n§n: ^

IJ = - y vd IJ 1 (P)

Do d6 khi M chuy4n dOng tren du'6'ng trdn (O) thi dogn IJ sinh ra mpt

trg c6 trgc Ht 1 (P) vd bdn kinh y

Bai toan 14 20: MOt hinh trg c6 ban kinh R vd chieu cao R

a) Tinh di$n tich xung quanh, di^n tich todn phin vd the tich

b) Cho hai dilm A vd B lln lu-pt nlim tr§n du-dng tr6n ddy sao cho goc

AB vd trgc cua hinh trg blng 30° Tinh khoang cdch giu-a AB va trgc

A-H \ - - - T 0 '

B

.J

-^pj o vd O' Id tdm cua hai du'b'ng trPn ddy Gpi

f) 1^ duang sinh cua hinh try thi O'A' = R, AA'^, _

^ R V3 va g6c BAA' bdng 30° Vi 00'//mp(ABA) nen khoang cdch giOa 00" vd bdng khoang each giu-a OO' vd mp(ABA')

Qpj H Id trung dilm BA' thi khoang cdch d6 bdng O'H

Tam gi^c BA'A vuong tgi A' nen:

= AA'tan30° = R V3 = R

V3

DO d6 BA'O' la tam gidc deu, vdy O'H =

Bai toan 14 21: Tren hai ddy cua hinh try c6 du-b-ng cao gap doi bdn kinh

cf^y, ta lay hai ban kinh ch6o nhau, dong thd-i tgo vb-i nhau mpt goc la 30°

Biet rdng dogn thing noi hai diu mut cua hai bdn kinh khong di qua tdm duang tron c6 dp ddi Id a

a) Tinh tang cua g6c hp-p tryc vd doan thing qua 2 mut d6

b) Tinh thi tich cua khii try, ,

Hirang din giai

a) Gpi bdn kfnh cua hinh try Id R, hai bdn kinh ch6o nhau Id OA' vd O'D Ve du-dyng sinh DA thi: 9(00', A'D) = g(AD, A'D) = ADA' = a Trong tam goc AOA' cdn tgi O:

AA'^ = 2R2 - 2R^cos30° = (2 - VB )R^

^ AA' = V 2 - V 3 R Tam gidc ADA' vuong tgi A nSn: DA'^ = AD^ + AA'^

6 - V 3 6 - V 3 AA' V2-N/3

.h = 2 a

6-V3

V6-^/3 _ 27iV6 - Vsa^

giSi m6n To6n 12 - LS Hodnh Phd

V6i

Bai toan 14 22: Cho hinh try c6 b^n l<inh R du-d-ng cao R V2 Gpi

CD la hai duong kinh thay doi cua hai dudng trbn day AB vu6nQ

vai CD ^

a) ChLPng minh r i n g ABCD Id ttp di^n d4u

b) Chu-ng minh r i n g cac duang t h i n g AC, AD, BC, BD luon t i l p xuc

mpt mat tru c6 dinh

Hifo-ng d i n giai

a) Gpi A', B' Ian lu'p't la hinh chi4u cua A, B tren

mat phing chtpa du-ong tron day c6 du-ong

kinh CD, thi A, B thupc duang tron ndy Khi

do A'B' 1 CD nen A'CB'D la hinh vuong c6

duang ch6o CD = 2R, do d6 A'C = R V 2 ,

ngoai ra AA' = R V2 nen ta suy ra AC = 2R

Tu-ong t y ta c6 AD = BC = BD = 2R

Vay ABCD la tip dien deu

b) Gpi O, O' l i n lu-gt la trung d i l m cua AB va CD, H trung d i l m A'C Ta c6:

d ( 0 0 ' ; AC) = d ( 0 0 ' , (AA'C)) = OH' =

Tu-ong ti^, khoang each giOa m6i du-ang thing AD, BC, BD vd 0 0 ' d§u

b i n g — ^ Tu- d6 suy ra cdc duang thing AC, AD, BC, BD deu tilp xuc

C

voi mat try c6 tryc Id 0 0 ' vd c6 bdn kinh R V 2

Bai toan 14 23: Tren duong tron ddy cua mpt hinh try, ta lay hai diem xuyen

tam A va B, tren duong tron ddy thCf hai ta l l y diem C khong n i m tren

phIng (AOB), voi O la trung diem cua tryc hinh try Chu-ng minh ring t^iS

cac goc nhj di?n cua goc tam di$n vai dinh O vd cac cgnh OA, OB, 0^

bIng 360°

Hipvng d i n giai

Gpi C Id d i l m doi xu-ng cua C qua tam O Khi

do, n l u b goc tam di^n OABC c6 cac goc nhj

di^n voi cac canh OA, OB, 0 0 l l n lu-gt bIng

a, p, y thi g goc tam di?n OABC c6 cac goc

nhj dien vgi cdc cgnh OA, OB, 0 0 Ian lu-gt

bIng: 1 8 0 ° - a , 180° - p y

Gpi I la tdm du-6-ng tr6n du-b-ng kinh AB, trong tii- dien OABC, cac go'^

di^n vgi canh OA vd O C bIng nhau (vi tii- di^n ndy nhgn m0t phan gia'^

g6c nhj dien canh 01 lam m|t phIng d6i xu-ng); ngodi ra, trong ti> ^

OIBC, cdc goc nhj di$n vb-i c?nh OB vd O C bIng nhau

Cty TNHHMTVDWHHhang Vi^t

00 do: (180° - a) + (180° - p) = y => a + p + y = 180°

pai toan 14 24: Cho hai d i l m A, B c6 djnh Tim tdp hgp nhu-ng du-o-ng thing

d qua A va each B mpt dogn khong doi bIng d

Bai toan 14 25: Cho hinh non dinh S du-o-ng cao SO, A vd B Id hai d i l m thupc

duo-ng tron day hinh non sao cho khoang each tu- O d i n AB bIng a va SAo

= 30°, SAB = 60° Tinh t h i tich, dien tich xung quanh hinh non

• sinlAO =

A O V3 V6 a

^' toan 14 26: Cho hinh non S, goc giu-a du-gng sinh d vd mdt day Id a Mpt

f^St phIng (P) qua dinh S, hgp v6i mdt day g6c 60° Tinh di$n tich thilt

"^'en vd khoang each tu- O d i n mp(P)

Hu'O'ng d i n giai: / V'!^

•"hilt dien Id tam giac SAB edn tgi S.Gpi I Id trung d i l m AB , ' '

10 tr<?ng dlSm bSl dUOng HQC sinh gl6i mdn To6n 12 - LS Ho6nh Phd

AOHI Id nu-a tarn gidc d4u n§n : d(0,(P))= OH = =

Bai toan 14 27: Cho hinh n6n (N) c6 bdn kinh ddy bing R, dird-ng cao SO

Mpt m0t phIng (P) c6 djnh vuong g6c vai SO tgi O' cSt non (N) theo duong

tr6n c6 bdn kinh R' M0t phIng (Q) thay d6i, vuong g6c vb-i SO tgi dilm 0,

(O1 ndm giOa O vd O') cit hinh trdn theo thilt di^n Id hinh trdn c6 ban kinh

X H§y tinh x theo R vd R' de (Q) chia phin hinh n6n ndm giu-a (P) va day

hinh n6n thanh hai phin c6 thfe tich bdng nhau

Hu-ang din giSi:

Gpi Vi Id the tich phdn hinh non gi&a dinh S vd mp(P) V2 Id t h i tich ph^n

hinh n6n giOa hai m$t phdng (P) vd (0)

V3 Id t h i tich phin hinh nbn giu-a m$t

phSng (Q) vd ddy hinh n6n d§ cho

Cty TNHHMTVDWHHhang Vi$t

(jAj toan 14 28: MOt hinh n6n c6 chi^u cao bing h vd bdn kinh ddy bdng r

a) Tinh bdn kinh m0t c4u npi tiep hinh n6n d6

Tinh ban kinh m^t c^u ngogi ti^p hinh non

Hipo^ng din giai

, Gia su' hinh non c6 dinh S vd c6 ddy Id du-^ng t'"^" C(0; r)

Lgy diem A c6 djnh tr§n du-b-ng tron ddy

ya gpi I Id diem ndm tren SO sao cho Al Id phan gidc cua goc SAO thi I tarn cua mdt clu nOi tiep hinh n6n, bdn kinh R = 10

Ta c6 SA = V O S ' + O A ' = V h ' + r ' Theo tinh chdt du'ang phan gidc, ta c6:

r + Vh

b) Gia sO hinh n6n c6 dinh S vd lly dilm M c6

dinh tren du-o-ng tron day (O; r) thi tam gidc SOM vuong 6- O Tdm I cua mdt cdu ngogi tiep hinh non Id giao dilm cua SO vd mdt phdng trung tri^j-c cua SM, ban kinh R = IS

Gpi SS' la duang kinh cua mdt cdu ngoai tiep hinh non (SS" > h) Tam gidc SMS' vuong tgi ?

M, CO du-dyng cao MO nSn:

2

MO^ = OS.OS' => = h(SS' - h) => SS' = ^ + h =

\^dy ban kinh mdt cau ngoai tilp hinh n6n Id: R =

toan 14 29: Mpt hinh non tron xoay c6 chieu cao bdng 3, c6 day la hinh

trbn CO ban kinh 1 Mpt hinh l|p phu-ang npi tiep trong d6 sao cho mpt mdt

*hi ndm tren mat phIng day, 4 dinh cua mdt doi dien cua hinh Igp phuang

W thupc mdt non Tinh thi tich hinh Igp phu-ang

Hu'O'ng din giai

xet mdt phdng chua true hinh ndn vd hai dinh

^^i di$n cua day hinh lap phuo-ng Mdt phdng '^^y se cdt hinh Idp phu-ang theo thilt di^n Id

^ifih chu' nhdt MNPQ c6 mpt cgnh bdng MQ = s,

^9nh kia bdng MN = s 72 , vai s la dp ddi canh hinh lap phuang

10 trpng diem bSl dU<Sng HQC sinh gioi m6n To6n 12 - LS Hodnh Ph6

M$t p h i n g noi tren cung cSt hinh non theo thi§t di^n la tam giac SAB

tarn gi^c dong dang AQM va ASO cho ta:

'2' I = ^ suy ra s = ^ V v V - ^ 3 ( 9 ^ ^

•

• Vay V = $ 3 =

i^i > j r " 7 ' n n i u q - • 3 4 3 ' ' ' "

B a i t o a n 14 3 0 : C h o tu- d i ^ n vu6ng OABC dinh O G p i R, r l l n lu-p't la

kinh mat cau ngogi, npi ti§p tCp dipn

B a i t o a n 14 3 1 : Cho r, R l l n lu-pt la ban kinh m§t cau npi tidp, ngogi tiep cua m

tCp di$n CO t h ^ tich la V Chu-ng minh rSng: 8R^r > 3 Vs V Suy ra V <

Hu'O'ng d i n g i a i

Gpi O, G l l n lu'p't IS tarn mSt cSu ngoai ti4p vS trpng tSm tup di$n A B C D GPi

BC = a' A D = a', CA = b', B D = b', A B = c, C D = c' Gpi Sa, Sb, Sc, Sd, S,p

lu'p't Id dien tich cdc m $ t d6i d i ^ n v&\c dinh A, B, C, D vS di^n tich ^o3<^

r-Vi R' > r => R > 3r => d p c m Bai t o a n 14 3 2 : Cho \\Je di^n ABCD c6 cdc du-dng cao AA', BB', C C , DD' dong

quy tai rnpt d i l m H thupc mi§n trong cua tie di$n C a c du'6'ng t h i n g AA',

BB', C C DD' lai c i t mat cau ngogi ti6p tLc dipn A B C D theo thu- tu' tai Ai, Bi,

tu' di$n A B C D thi A'l = IJ

Do H la t r y c tam tam giac ABI nen:

A'H.A'A = A'B.A'I = - A'B.A'J = - A'Ai.A'A •

W trgng diem bSi dUOng hqcsinh gl6i mon Toan 1£ Lc Hoanh '^hd

Theo bit ding thCpc B C S :

Bai toan 14 33: Cho ti> di^n A1A2A3A4 c6 G Id trpng tarn, gpi (S) Id m|t cly

ngoai ti§p tu- di^n tren Cac du-ang thing GAi, GA2, GA3, GA4 cit (S) tai

' A"i,A '2,A'3,A'4.Chu'ngminh: ^ _ 1 _ < ^ ^

Hifang din giai

Gpi O va R Id tdm vd bdn kinh cua m$t c^u (S)

Bai toan 14 34: Cho hinh hpp chu- nhgt ABCD.A'B'C'D' Gpi R, r, h, V Ian lu^'

Id ban kinh mdt cdu ngogi tiep, npi tiep, ducyng cao ke tu- A' vd the tich cua

tLK di$n A'AB'D'.Chung minh; X^^zll < i

R2.r.h 3 Hipang din giai

D$t AA' = a; AB' = b; A'D' = c Ta c6

L i t / iivnn mi v uvvn nnang vi^

Ti> dien A'AB'D' vuong tai A' nen R = Va^Tb^Tc^

Bai toan 14 35: Ti> dien ABCD npi tiep trong mdt clu (O, R) Gpi ma, nib, nic,

md la dp ddi cdc trpng tuyen ve tu- A, B, C, D

3

Chu-ng minh R > — (ma + mt + mc+ md)

16

Himng din giai

Gpi G Id trpng tdm tip dien GA + GB + GC + GD = 0

^ai toan 14 36: Cho tCf di^n OABC trpng d6 OA, OB, OC dpi mOt vu6ng g6c

vai nhau, c6 dirdng cap OH = h Gpi r Id bdn kinh mdt clu npi ti6p tCr di$n Tim gid tril6n nhltcua -

TDtri,ina diem hfli JUcmq hoc '^inh qiol man JOOm^ - ••- "nnnil riiu

Vay gia th Ian nhit cua - 1^ 1 + Vs khi OA = OB = 0 0

Bai toan 14 37: Cho hinh ch6p tii- gi^c d&u, gpi R, r Ian lu'p't la ban kinh mst

cau ngogi tiep va mat cau npi ti4p cua hinh ch6p 66 Tim gi^ tn \6fn nhat cQa

t f s 6 - ^ • •

R

HiPO'ng din giai

Xet hinh ch6p ILF gi^c deu S.ABCD c6 #

canh day a, duang cao h Gpi a Id g6c / / 11

hp'p bai mat ben vai day Gpi O, I l^n

* luat tarn mat c l u ngogi tiep vd nOi ti§p

cua hinh chop thi O, I e SH

Ta c6: OS.^ = OB^ = OH^ + BH^ ^

gai toan 14 38: Trong cdc tii- dipn npi tiep hinh cku c6 ban kinh R = 1, tim tip

di^n CO dien tich toan phin Ian nhlt

Trong mpi tam gidc a, b, c, dipn tich S thi:a^ + b^ + c^ > 4 V3 S

Ap dung l l n lu'at vao cac mSt tCp di^n A B C D r6i cpng lai thi du-ac:

2(AB^ + AC^ + AD^ + BC^ + BD^ + CD^) > 4 V3 S,p Gpi O, G l l n lu-gt la tam va trpng tam tu- di?n A B C D , ta c6:

-Do do ABCD la tLP di^n d^u

^ai toan 14 39: TIP di$n ABCD c6 cdc cgnh AB, BC, CA d4u nho han DA,

DB, DC Tim gia trj Ian nhat vd nho nhat cua PD, trong do P la dilm thoa (Ji4u ki^n PD^ = PA^+ PB^ + PC^

HiPO'ng din giai

Qpi O la diem sao cho O A + OB + OC - OD = 6 (1) «

"•"a CO PD^ = PA^+ PB^ + PC^

^ ( 0 A 0 P ) 2 +(OBOP)^ + ( 0 C 0 P ) 2 (ODOP)^ 0

-^20P2 - 2 0 P ( 0 A + 0 B + 0 C - 0 D ) = 0D2 -(OA^ + 0 6 2 + 0 0 ^ )

^ 20p2 = OD^ - (OA^ + OB^ + OC^) (2)

^'fih phuang 2 ve cua (1), ta suy ra 20D^ - 2(0A^ + OB^ + OC^)

4 '-t.,'"

* DA^ + DB^+ DC^ - (AB^ + BC^ + CA^) (3)

D0t DA^ + DB^+ DC^ = X, AB^ + BC^ + CA^ = y ^'

Tu- (2) (3) suy ra OP^ = (x - y)/ 4 > 0 do gia thi§t

Do d6 P thupc m0t ciu (O) tarn O b^n kinh (yjx-y)/2 (

Tu' (1) ta CO OD^ = (3x-y) / 4 suy ra D nlm ngoai (O) Duang thing OD cj*

(0)taiPi,P2(DPi < D P 2 )

D P > D O - P O = D O - P 1 O = DPi, dau bing khi P = Pi

DP < DO + PO = DO + P2O = DP2, dau bing khi P = Pj

Vay minPD = DPi, max PD = DP2

Bai toan 14 40: TIP di^n ABCD gkn deu Tim dilm M sao cho

f(M) = MA'°°' + MB'°°' + MC'°°' + MD^""" min

Hiro-ng din giai

Gpi G la trpng tam cua tip dien, vi t(y di^n gan deu nen G cung Id tSm mat

cau ngoai tilpiGA = GB = GC = GD

Ta CO bSt ding thCcc vai n nguyen duang

V$y f(M) nho nh^t khi M trung trpng tam G^

Bai toan 14 41: Chung minh trong cdc tii- di$n ABCD nOi tilp m$t clu (0,

cho trudc thi hinh c6 g6c tam dien dinh A vu6ng khi va chi khi:AB' + AC

AD' - BC' - CD' - D B ' min

Huxyng din Himng din giai:

Khai triln (OB + 0C+ OD-OAy >Oia c6:

O B ' + O C ' + O D ' + OA'

+ 2 ( 0 B.0C+0B.0D-0B.0A+0C.0D-0C.0A-0D.0A);

md 20B.OC = OB' + OC' - BC' = 2 R' - BC'

tu-ang ty v6i OB OD , OB OA OD OA

Ctj/ TNHHMTVDWH Hhong Vl^t

' ^ f i l vdo khai triln tr6n ta du'P'c: A B ' + AC^ + AD^ - B C ' - C D ' - D B ' > -4R'

ping thuKC xay ra <=> O B + O C + OD = OA

^ A B + A C + AD = 2AO - AA' (vdi AA' la du-dyng kinh) ^' Tfi

^ hinh hpp ABD'C DC'A'B' npi ti§p mSt clu (O, R)

^ g6c tam dien dinh A Id tam dien vuong

- j toan 14 42: Cho hinh try npi tilp hinh cdu S(0; R)

a) Hinh tru nao c6 dipn tich xung quanh S Id'n nhlt

Hinh tru nao c6 the tich Ian nhlt <

Hu'O'ng din giai:

a) Gpi X la khoang each tCp tam hinh cau O

L$p BBT thi V dat gid tri I6n nhlt khi x =

^ac/j /(h^c; dung bit ding thtpc BCS

^^"'toan 14 43: Cho tCc di$n diu ABCD c6 canh bing a Gpi O Id tdm cua tam

9i^c BCD, dyng mp(P) vucng gpc vai AO tai mpt dilm I thupc dpgn AO, (P)

AS, AC, AD lln luat tai M, N va P Chp mpt hinh try c6 mpt day la hinh

^^^n (I) npi tilp tam giac MNP vd ddy kia nim tren (BCD) Xdc dinh vj tri I

'^^n AO d l khii try CP t h i tich Idn nhlt

Hu-ang din giSI

IK = X, vi AMNP Id tip dipn dIu nen KM = KA = 3IK = 3x

^ 0 = VAB'BO' = a '

Bai toan 14 44: Trong cac hinh n6n ngoai tiep hinh cau bdn l^inh r hay

dinh hinh n6n c6 t h i tich nho nhSt s

Hu-ang d i n glai Thilt di$n qua true Id tarn gidc can SAB ngoai tiep

du'ang tron (O; R) Gpi chieu cao cua hinh non la

h, ban kinh day R Ta c6 SSAB = - AB.SI = p.R

l^p BBT thi maxV = V(4r): chieu cao h = 4r

•j toan 14 45: Cho hinh non trdn xoay (H) dinh S, ddy Id hinh tr6n bdn kinh

p, chieu cao bdng h Gpi (H') Id hinh try tr6n xoay c6 ddy Id hinh trdn bdn

Kinh r (0 < r < R) noi tilp (H)

a) Tinh ti s6 thI tich cua (H') vd (H)

b) Xac dinh r d l (H') c6 thI tich \&n nhk

Hirang d i n gidi

g) Goi ' '' 9'3° ^^"^ duang cao liinh n6n (H) vd hai ddy cua hinh try (H')

^ SI' Khido - - ^

Do do nen 1'! =

BBT thi maxf = f( ^ ) Vgy V(H , Ian nhdt khi r = —

^^'toan 14 46: Cho goc vuong xOy va hai d i l m M, N Ian lupl di dpng tren Ox

Oy sao cho MN = 2a khong d6i Gpi A, B, C Ian lupt Id trung dilm cdc

^09n OM, MN, ON Dat OA = x (0 < x < 2a) l-iai canh MN, MO cua tarn gidc UJON va cac doan CB, BA, AO quay quanh NO sinh ra mot hinh non vd mot

^ try npi tiep hinh non c6 chung true NO

' "^'fih dien tich xung quanh Si cua hinh non vd di?n tich todn phan S2 cua

^'f^h try theo a va x

dinh OM d l ti s6 dien tich dgt gia trj Ian nhdt

2R

10 trqng diS'm bSi dUOng h<pc sinh gidi m6n To6n 12 - LS Ho

Himng din giai

a) Ta c6 ABCD Id hinh chQ- nh|t

AB = V M B ' - M A ' - V a ' - x '

11 Di^n tich xung quanh cua hinh n6n id

51 = TtOM MN = Tt2x2a = 47:ax

Di^n tich to^n phin cua hinh trg Id:

52 = 27tOA.AB + 27:OA^ = 27:xVa' - x * + 27iX

Bai toan 14 47: Cho mpt hinh cSu npi tilp trong mpt hinh non trbn xoay M^'

hinh try ngogi tiep hinh clu do c6 day du-ai n§m trong mSt phing day ciia

hinh non Gpi Vi, V2 lln lu-p't la thi tich cua hinh n6n va cua hinh tru Tim

gia trj nho nhSt cua tf so — B ^

Hirang din giai

Gia su- hinh non c6 du-ang cao BH = h, ban

kinh d^y la DC = a, g6c giua duang sinh v^ \

tryc IS a; ban kinh hinh cau npi tiep hinh non

a) Vai ti> di0n ABCD ; MA^ + MB^ + MC^ + MD^ = k^ k cho tru-^c

b) Vb'i n dilm Ai(i = 1 n): ^aiA//l'=/t'(ai, k Id hing s6)

Hirang din

a) Dung binh phu-ang v6 hu-ang va chen trpng tam G cua tu" di?n ABC b) Dung binh phu-ang v6 hu-ang vd ch§n tSm tf cy I cua h$ dilm

Bai t|ip 14 2: Cho dilm A a ngoai m|t cau S(0; R).Mpt mgt phing bit ki di

qua AO, cit mat clu theo mOt dud-ng trbn (C) Gpi AH la mOt tilp tuyin cua (Ju-ang tron do tai H

a) Chu-ng minh ring AH cung tilp xuc v&[ mSt clu tai dilm H

b) Tim quy tfch cac tilp dilm H

Hirvng din

3) DCing AH Id mpt tilp tuyIn cua du-dyng tron (C) tgi H

^) Kit qua du-b-ng tron giao tuyIn cua m^t clu va mp(P)

^ ' tap 14 3: Cho tam gidc can ABC c6 goc BAC = 120° va du-ang cao

'*^H = a V2 Tren du-ang thing A vuong g6c vb-i m^t phIng (ABC) tai A lly

^ai dilm I vd J a v l hai phia cua dilm A sao cho IBC Id tam gidc diu vd

"^^C la tam giac vuong can

^) ChCeng minh ring BIJ, CIJ la tam gidc vuong

1^) Xac djnh tam va tinh theo a ban kinh cua m|t clu ngogi tilp tu- di$n

iJBc - , :

nOtrQng diem boi dUcrng hQC Sim giOl man roon TX - L B nuunii rinj ng Vi^t

a) Xac dinh tam va ban kinh hinh c l u ngoai tiep tif dien SABC

b) Tinh tan cua goc a = ASB d l hai mat p h l n g (SCA), (SCB) ho'p nha^

goc 60°

Hu-ang d i n , , ; A ^

a) Tam I cua hinh cau ngoai ti§p tu- dien SABC c^ch d^u S,A,B,C,

K e t q u a R = a ^

b) Ket qua tan a = — n < K •"

Bai tap 14 5: Cho tCp dien ABCD c6 mat c l u npi tiep (I, r) Cac mat p h i n g ti^p

xuc vai m$t cau do v^ song song voi cac mat t i i dien, chia tip dien ABCD

thanh 4 t u dien c6 4 mat c l u npi ti§p ban kinh ri, r2, rs, Xi,

C h u n g minh n + r2 + ra + r4 = 2r

Hu'O'ng d i n

Dung ti s6 dien tich, ti s6 t h i tich cua cac hinh dong dang

Bai tap 14 6: Cho tam giac vuong can ABC c6 cgnh huyen A B = 2a Tren

du'ang t h i n g d di qua A va vuong goc vai mat p h l n g (ABC), l l y mot dilm S

k h a c A

a) C h u n g minh t u dien SABC chf c6 mot cap d6i dien vuong goc vai nhau

b) Xac dinh tam mat c l u ngoai ti§p t u dien SABC Tinh the tich mat cau

(SBC) tai vai (ABC) mot goc b i n g 60°

Hu'O'ng d i n

t|P 14 8: M$t p h l n g di qua tryc cua mot hinh tru, c i t hinh tru theo t h i l t

^ j j ^ n hinh vuong canh 2R

Tinh dien tich xung quanh, dien tich toan p h i n va th§ tich

Tinh t h I tich khoi ISng tru t u giac d§u npi t i l p hinh trg ,r,fr;-

Hipang d i n

Hinh tru c6 thi§t dien qua true la hinh vuong canh 2R nen h = 2 R, ban kfnh day R- K i t qua Sxq = 47tR2; S,p = STIR^; V = 2%R^

gai t?P l^- 9: Cho hinh non c6 goc dinh 2a Tinh ti so ban kinh mat c l u ngoai

tjlp va ban kinh m$t c l u npi t i l p hinh non

Bai tap 14 7: Cho mot hinh tru c6 ban kinh day R va chieu cao 2R Tren caC

duang tron day (O) va (O') lln lupt lly hai d i l m M, N Mot mat phang (^^J

qua MN va song song vai true hinh tru cIt hinh tru theo t h i l t dien la ttr gi^ j

MPNQ

^' a) Xac dinh khoang each tu GO' din (P) d l thilt dien c6 dien tich bIng 21^

\) Xac djnh vi tri M, N tren (O) va (O') d l k h i i tu dien MONO' c6 thI tich

t nhlt

Hu'O'ng d i n

R.-\f3

a) Dung hai duang sinh MP va NO K i t qua OH =

b) K i t qua OM hp'p vai ON goc 90°

Kit qua

-sin2a

Bai tap 14 10: Cho tam giac ABC npi t i l p d u a n g t r 6 n ( 0;R) Tfnh t h I tich t u

dien ABCD biet r i n g DA =BC, DB = CA, DC = A B va ban kinh m$t c l u npi tilp trong t u dien A B C D dat gia tri nho n h l t

Hu'O'ng d i n

TLf dien g i n d i u K i t qua V = ' '^^^"^^

4

Bai tap 1 4 1 1 : Trong cac hinh hop npi t i l p mat c l u ban kinh R, hay xac dinh

hinh hop CO dien tich toan p h I n Ian nhat • f "n

Hu'O'ng d i n

Dung b i t d i n g thuc A M - GM

[^^t qua hinh hop la hinh lap p h u a n g

t?p 14 12: Cho hinh ch6p n - giac deu, gpi R, r l l n lugt la ban kinh mat cau ngoai tiep va mat c l u npi t i l p cua hinh chop do Tim gia trj Ian n h l t cua

tls6 ^ '

Hu'O'ng d i n

dinh trub-c tam mat c l u ngoai t i l p va tam mSt c l u npi t i l p cua hinh

op n - giac deu chinh la giao d i l m cua tryc SO vai m$t trung t r y c va mgt f^fian giac t u a n g Crng , '

tqua

1 +

c o s ^

« A [ O A :

W trqng diSn* b_oi_ duang HQC sinh giSTmSn loQn I a - le noann rno

cnur^n as i3: TOII D O K H O N G G I A N

- 3 vectc a , b, c khong dong p h i n g : [ a , b ] c 0

Di^n tich v a t h i tich

Di$n tich tarn giac ABC: S = ^ I [ A B , AC ]

The tich ti> di^n ABCD: V = I [ A B , AC ] A D I

6 The tich hinh hop ABCD.A'B'C'D': V = I [ A B A D ] A A '

f T h i tich hinh ISng trg ABC.A'B'C: V = - | [ A B , A D ] A A ' |

G o c gifra 2 m|it p h i n g : m$t p h i n g (P) c6 vectc ph^p t u y l n n vS m^t

p h i n g (Q) c6 vectc phSp tuyen n ' thi

cos((P), (Q)) = I cos( n , n ' ) I

G o c giira 2 diro'ng t h i n g :

d c6 VTCP u va d' c6 VTCP v thi cos(d, d') = I cos( u, v ) I

Goc giOa du-o-ng t h i n g v a m|t p h i n g :

- 4

Khoang e a c h tCr Mo(xo, yo, Zo) d i n m|t p h i n g :

- ( O x y ) l a Izol; (Oyz) 1^ |xol; (Ozx) 1^ |yo

- (P): Ax + By + Cz + D = 0 la:

d(Mo, P) = ' A X o ^ y o ^ C ^ o ^ D I

V A ^ + B ^ + C ^

Khoang e a c h tu- mot d i l m d i n 1 dipo-ng t h i n g :

,Cho Mo(xo, yo, Zo) va d i r c n g t h i n g d qua A vd

[AMo

u c6 VTCP u = A B thi d(IVIo, d) =

Khoang e a c h giira hai dipo'ng t h i n g c h e o nhau:

di qua Ml v ^ c6 VTCP u i ;d2 qua M2 va c6 VTCP U2 thi

d(d,,d2) =

, , ::/V cr iV'

Phipo-ng trinh t i n g quat c u a m|it p h i n g :

•^It p h i n g qua Mo(xo,yo) v ^ vectc phap tuyen n = (A,B,C)

Ax + By + Cz + D = 0, A^ + B^ + C^;^0 i : U , 0 ; t K , '

j^ay A(x - xo) + B(y - yo) + C(z - Zo) = 0

fhu-o-ng trinh c u a du-ang t h i n g : di qua Mo(xo,yo,Zo) v ^ c6 vectc chi

Phucng u =(a,b,c), a^ + b^ +c^ =^ 0

Phuang trinh tham so: d:

Phuang trinh chinh tSc khi a, b, c ^ 0: ^

Phu-ang trinh mat c'au: ' '

Mat cku (S) tarn l(a, b, c) b^n kinh R: ' "^r

' •• ( X - af + (y - b)^ + (z - c)^ = R^ hay:

• + y^ + z^ + 2Ax + 2By + 2Cz + D = 0, + + - D > 0

CO tarn l(-A, -B, -C) va ban kinh R = VA^ +6^ - D

Vj tri tipo-ng ddi ctia 2 mat phing:

Vj tri tifcng d6i cue 2 diforig thing:

f qi 3 A(x , yA ZA) va c6 vecta chi phu-ang u (a,b,c)

c qua B(> „ y t , ZB ) va c6 vecto' chi phu'ang v (a'.b'.c') ^.^ ^^.i ,

Chec nhau: [ u, v ] AB 0

cat nhau: [ u , v ] AB = 0 va a : b : c a' : b': c'

Trung nhau: a : b : c = a': b': c' = (XB - XA) : (ys - YA) : (ZB - ZA)

Song song: a : b : c = a': b': c' ^ (XB - XA) : (YB - YA) : (ZB - ZA)

' Hai dilm Mi(xi; y i ; z^) v^ M2(X2; ^2', Z2) nSm vk hai phia cua mSt ph^ng

(P): Ax + By + Cz + D = 0 khi va chi khi:

(Ax, + By, + Cz, + D). (AX2 + By2 + Cz2 + D) < 0

Vj tri tirang d6i cua 1 diro'ng thing va 1 mat phing:

Du-dyrig thing d qua A va c6 vecta chi phuang u va mat phing (P) qua Mo

va CO vecta phap tuy^n n

cat nh£u: u n 0

Song song: u n = 0 va A g (P)

Ou-ang thing thupc mat phing: u n = 0 v^ Ae (P)

Vj tri tirang ddi giCra mat ciu va mat phing: 3''

Cho mgt ciu S(l; R) va mp(P) Gpi IH = d la khoang each ti> tam I den (P) thi:

c) N I U d > R: mp(P) khong c6 diem chung vai mat ciu , ^„^->- ^; ,

• O'ng dung glai bai toan khdng gian:

Oua tpa dp Oxyz vao bai toan hinh hpc khcng gian thuin tuy, bing each chpn he true thugn Ip'i d l giai toan

2.CACBAIT0AN

Bai toan 15.1: Cho hinh binh hanh ABCD v6i A(-3; -2; 0), B(3; -3; 1), C(5; 0; 2)

Tim toa dp dfnh D va tinh goc giua hai vecta AC va BD

Hu'O'ng dan giai

Ta CO BA = (-6; 1; -1), BC = (2; 3; 1) Vi toa dp cua hai vecta do khong ti 1$ nen ba dilm A, B, C, khong thing hang

Gpi D(x; y; z) Tii- giac ABCD la hinh binh hanh khi va chi khi ^^^j^

b) Tinh dp dai du'ang phan giac trong BD , ,r + i) ^ HA

Hu'O'ng din giai

10 trQng diSm bSi dUSng fiQC sinh gioi / le rroanii rinj

<te , HiFcyng din giai

AB = (-1; 1; 7), AC = (-6; -6; 16), hai vecta ndy khong

cung phu-ang vi toa dp khong ti I? suy ra A, B, C khong

thing hang va c6 :

DC = (-2; 2; 14) = 2 AB =^ AB // CD

V$y ABCD Id hinh thang n§n

SAJCD = SABC "*" SADC

= |[AB, AC]| + ^ i[AD, AC]| = 3Vl046

Bai toan 15 4: Cho ti> dipn ABCD c6: A(-1; 2; 0), B(0; 0; 1), C(0; 3; 0), D(2; 1; 0)

a) Tinh di?n tich tarn giac ABC va the tich ti> dien ABCD

b) Tim hinh chieu cua D len m$t phing (ABC)

Hiro'ng din giai

a) d <Ji qua dilm Mi(1; - 1 ; 1), c6 vectc chi phu'ong u^ = (1; - 1 ; 0) d' di qua

dilm M2(2; -2; 3) c6 vecta chi phu-ang Ug = (-1; 1; 0) *^ ''^

Vi u, vdUg cung phu-ang nhu-ng u^ Ug khong cung phu-ang v6i M^Mg = (1; - 1 ; 2) n6n hai duang thing do song song

TM^Mg.Ugl

Vdy d(d, CI') = d(Mi, d') = ^ ^ = 2

b)dqua M(0; 4, -1) c6 VTCP u = (-1; 1; -2) d' qua M'(0; 2; 0) c6 VTCP u' = (-1; 3; 3)

nhcu Oc dc d(d, d') = u,u' MM' - 1 0 - 2 1

12 u,u' \/81 + 25 + 4 V l 1 0

Hu-ang din giai

M thupc (Oxz) tren M(x; 0; z) Ta c6; MA = MB = MC

A M ^ = B M 2 ^ j ( x - 1 ) 2 + 1 + (z-i)2=(x + i)2 + u 2 ^

A M 2 = C M 2 • [(x-1)2 + 1 + (2-1)2 = (x-3)2 + U ( 2 + i)2

4x - 2z = 1 I ^ ^ / 4J< + 4Z - 8

^oan 15 7: Cho hai dilm A(2; 0; -1), B(0; -2; 3)

^) Tim toa dp (Jilm C e Oy tarn giac ABC c6 dien tich bing Vri va thoa Tim dilm D e (Oxz) d l ABCD Id hinh thang c6 cgnh day AB

W tr<?ng diSm b6i dU6ng hoc sinh gidi m6n loan I a - L& Moonn i-^o

Hirang d i n

a) Gpi C(0; y; 0) => AB = (-2; -2; 4), AC = (-2; y; 1)

Ta c6: SABC =

« 1 | [ A B , AC]| = >/ri«|V(2 + 4 y ) ' + 3 6 + (2y + 4 f = Vvl

<=> 20y^ + 32y + 12 = 0 o y = -1 ho$c y = - - (loai)

VayC(0;-1;0) o rr :!~ ,r)f!V1 rr4:r>

b) Gpi I3(x; U; z) e (Oxz) DC = (-x; - 1 ; -z) , <j

ABCD la hinh thang khi va chi khi AB , DC cung hu-ang

- X -1 -z >0<=>x= 1,z = -2 VayD(1; 0;-2)

-2 -2 4

Bai toan 15 8: Tim toa dp diem H la hinh chiSu cua

a) A(-2; 1; 0) tren duang thing BC vai B(0; 3; - 1 ) , C(-1; 0; 2)

b) D(1; 1; 1)len mat phIng (ABC) vai A(4; 1;4), B(3; 3; 1), C(1;5; 5)

Hu-o-ng din giai ,^ : ^ V - £ /q> \

a) Gpi D(0; y; 0) thupc tryc Oy Ta CP: " '

<:> I - 4y + 2 I = 30 » y = -7; y = 8 ipij , sV S * fiA 6o e': •

V^y CO 2 dilm D tr§n tryc Oy; (0; -7; 0) vd (0; 8; 0)

b) Ta coAC = (3; -3; -3),BC = (2; 1; -3) nen lap du'ac phu-ang trinh m^t phIng (ABC): 3x + y +2z - 6 = 0

Gpi H(x; y; z) la tri/c tam tam gi^c ABC

^ AH = (x; y-4; z-1), BH = (x-1; y; z-1), ta c6:

v = 25

AHBC = 0 [2x + y - 3 z - 1 = 0 BH.AC = 0 o ] x - y - z = 0 o H : H€(ABC) 3x + y + 2 z - 6 = 0

\3 13 26^

toan 15 10: Cho hai dilm A(0; 0; -3), B(2; 0; -1) va m$t phing (P)

3x - 8y + 7z - 1 =0

^) Tim giao dilm I cua du-ang thing AB vai mat phIng (P)

^) Tim diem C n i m tren mp(P) sao cho ABC l i tam giac diu

Hu'O'ng din giai Qoi l(x; y; z) ^ AB = (2; 0; 2), AJ = (x; y; z + 13)

10trong cJic.m boi duong ho<: 'iinh gi\-^i m6n loc .- —

VI Al va AB cung phuang nen c6 mOt s6 k sao cho Al - k AB hay

x = 2k

y = 0

z + 3 = 2k

y = 0 Mat khdc I e (P) 3x - 8y + 7z - 1 = 0 V$y ta c6 he:

Giai ra c6 hai dilm: C(2; -2; - 3 ) , C 2 2 1

3' 3' 3

Bai toan 15 11: Cho tarn giac ABC c6 C(3; 2; 3), duong cao AH nlm tren

dL^ong thing (d.): ^ = = ^ , di^ang phan giac trong BM cua goc

1

x - 1 y - 4 z - 3

B nim tren duong thing (d^): ^ - ^ = ^ • 'Jfnh A va B

HifO'ng din giai

Mat phing (P) qua C, 1 (di) la:

K d6i xi>ng vai C qua (62) thi K nim tren duo- thang chua cgnh AB vi

trung di§m cua CK nen K(1; 2; 5)

'1;

Dudyng thing (A) di qua KB la : y = 2 + 2t

[z = 5 - 2t Dod6 :(A)cit(di)taiA(1;2;5)

PtoAn 15 12: : Cho A(1; 0; 0), 8(0; 1; 2) Tim C e Oz d4 m0t phSng (ABC) hp-p

vo'i ni^* phing (a): 2x - 2y - z +5 = 0 mOt g6c bing 60°

Hu^ng din gidi

Gpi C(0; 0; m) e Oz Ta c6: AB = (-1; 1; 2), AC = (-1; 0; m)

^ u = rAB,AC = (m; m-2; 1) Id vecto phdp tuy^n cua (ABC)

I\/I$t phing (a) c6 vecta phdp tuy4n n = (2; -2; -1)

p(ABC) vd (a) hgp nhau g6c 60° n§n:

cos60° = cos(u,n) 2m + 4 - 2m -1 1 „ 2 ± V 2

, • = — «t:> m =

3Vm2 + 1.Km-2)2 2 2

V$y c6 hai dilm C(0; 0; - ^ ^ ^ ) , C'(0; 0; ^ — ^ )

Bai toan 15.13: Cho diem A(1; 0; -1), B(2; 3; -1), C(1; 3; 1) vd dudyng thing d Id

giao tuyen cua hai m$t phing c6 phu-ong trinh: x - y + 1 =0,x + y + z - 4 = 0

Tim tog dp di4m D thupc duong thing d sao cho th§ tich cua khoi tu di0n ABCD bing 1

Hipang din giai

Tac6 A"B =(1;3;0) AC = (0; 3; 2) nen d c6 VTCP u =[AB, AC ] = (6;-2; 3)

x = t huong trinh cua dudyng thing d Id: y=1+t

^'^y c6 hai diem D thoa mdn bdi todn Id D(-1; 0; 5) vd D(5; 6; -7)

todn 15.15: Cho hai dudyng thing: d,: ^ J - ~ = ^ vd dz Id giao tuy4n hai m^t phing c6 phuong trinh: 5x - 6y - 6z + 13 = 0, x - 6y + 6z - 7 = 0

^) Chung minh ring di vd dj clt nhau tgi dilm I

Tim tea dO cdc dilm A, B lln lup't thuOc di, dz sao cho tam gidc lAB cdn '5' I vd c6 di^n tich blng

#

741

42

401

b) Vecta chi phu'cng cua di la u i = (2; 2; 1) ^"'^

Vecta chi phu-ang cua dz la u 2 = [ n, n'] = (-72; -18; -12) hay (6; 3; 2)

Gpi (p la g6c giu'a di va 62 ta c6: coscp - U1U2 20 V41 21 s i n c p — -21

do hai mat phing c6 phuang trinh la

Ctjy TNHHMTVDWH Hhang Vi$t

_ y + 3z - 5 = 0 va 4x - 2y + 6z - 10 = 0 n§n chiing trung nhau Vly:

(<h6ng CO gia trj m nao d§ hai mat phlng do song song

l^f^j rn = 1, hai mat phing do triing nhau .y.^, ,5;^ ^.^ „ , I^HJ m 5^ 1 hai mat phing do c i t nhau

j^gi fTiat phing do vuong goc vai nhau khi va chi khi n, ng = 0

9

^ 2 ( m + 3) + 2m + 3(5m + 1) = 0 c:>19m + 9 = 0 < » m =

19 flfli toan 15 17: Xac djnh cac gia trj p va m d l ba mat phlng sau day di qua

^ ni6t du-ang thing:

31 9 Cho z = 0 X = — ,y = — suy ra B 31 9

10 10

)

; 0

Ba mat phing cung di qua mot du-ang thing khi m^t phing:

5x + py + 4z + m = 0 di qua hai diem A va B:

^^ij'ng minh (di), (dj) va A cung thupc mOt m0t phing

Hu'O'ng din giai

•^2) qua B(0; 1; 4) va c6 VTCP u = (1; 2; 5)

qua B va c6 VTPT n = [ u , A B ] = (-J; - 8 ; -4) hay (1; 2; -1) nen .^PWng trinh: x + 2 y - z + 2 = 0

'^'^^ (di), (d2) cung thupc mpt mat phing

^1 ^ (di) qua IVI(0; - 1 ; 0) va N(-1; 1; 3)

•^N thupc mp(A, d2) nen di thupc mp(A, d2

Wtr^ng ^^-^^^hAi HuOno hoc sinh g,o, mon loan ,^ - n^.j^^^

Bai to^n 15 19: Cho b6n diim A(-3; 5; 15) B(0; 0; 7), C(2; -1,4) D(4; ^

ChO^ng minh hai du-drng thing AB va CD cSt nhau, tim tog dO g-ao d.em ^'

Ta c6: AB = (3; -5; -8), AC = (5; -6; -11)

AD =(7;-8;-15), CD = ( 2 ; - 2 ; ^ )

Do do [AB A C ] = (7; -7; 7) ^ [AB A C ] AD = 0 nen AB, CD (j^

phing, han nO-a AB, CD khfing cung phu-ang, do d6 2 du-b-ng thing AB

Chung minh t6n tai mpt due^ng thing (d) c i t ca b6n due^ng thing <56

phuang trinh chinh t i c cua (d)

Hipang din giai

Cty TNHHMTVDWH Hhong Vi$t

<r^^n 15 21: Cho s^u di4m A(a; 0; 0) B(0; b; 0) C(0; 0; c) A'(a'; 0; 0),

^ g'(0; b'; 0), C'(0; 0; c') vdi aa' = bb' = cc" ^ 0; a ^ a', b ^ b', c ^ c"

g) Chung minh c6 mpt m$t cau di qua s^u diem n6i tren

ChLPng minh du-b-ng thing di qua g6c tog dO O trpng tSm tam gi^c

^gC, vuong g6c voi m|t phing (A'B'C) ^

Hipang din gidi

fa xac djnh tSm vd bdn kinh R cua m^t c i u qua 4 dilm A, A', B, C \ Qpj |(x; y; z) Id tdm m^t c i u d6, ta c6: lA^ = lA'^ = IB^ = IC^

-2ax + a^ = - 2 a ' x + a'^

-2ax + a^ =-2by + b^ ' ' -2ax + a^ =-2cz + c^

Do do X = a + a'

y = b^ + a a ' ^ ^ ^ _ c^ + a a '

ram 1 a + a' b^ + a a ' c^ - - ' ^

2b ' +aa

c6; A'F' = (-a'; b'; 0) A ' C = (-a'; 0; c") ngn O G A ' B ' = - — + — + 0 = 0 ;

" H Hiring dan giai J ^ ^

(dO qua A(1; 2; 0), A (d.) (d,) vd (da) cung^cb vecta phuang u = (1: ^A'C' - ^ , f , o - 0

' ^OdoOG l A ' B ' , A ' C ' ^ O G l m p ( A ' B ' C ' )

^' toan 15 22: Chung minh cdc m^t phing (Pm): (2 + m)x + (1 + m)y + (1 + m)z

" ^ - 1 = 0 luon di qua mpt dubng thing c6 djnh u i j

Hu'O'ng din giai

.i6n (d.) // (da), (da) qua B(2; 2; 0), AB =(1,0,0)

Gpi (P) Id m0t phing qua (di), (d^) Id: PT cua (P) Id y + z - 2 - 0

(da) n (P) = E l l ; ± \(d4)n(P) = F(4;2;0)

' 2 2

x - 1 ^ " 2 - ! _ _ 2 c 6 vecta chi P^^^' Dubng t h i n g (d) qua E, F Id _1

Ill

W trQn<3 ciidrn hni r/(/'V?o hQC sinh gioi m6n To6n 12 - Le hPho

Cho z = 0 thi X = 2, y = - 3 : B(2; - 3 ; 0)

Vay c^c m0t p h i n g (Pm) «3i qua (Jirang thing c6 djnh la glao tuy^n cua 2

phing: 2x + y + z - 1 = 0 , x + y + z + 1 = 0 t C p c l a du-ang thing AB c6 djnh ^'

Bai toan 15 23: Trong khong gian Oxyz, cho hinh hop ch&

ABCD.A'B'C'D' c6 A trung vai g6c O, B(a; 0; 0), D(0; a; 0), A'(0; 0; b), {^l

b > 0) Gpi M 1^ trung diem canh C C a) Tinh t h ^ tich kh6i tu- dien BDA'M

b) Xac dinh i y so - de m§t p h i n g (A'BD) 1 (MBD)

b

Hu'6ng d i n giai

a) TCf gia thietta c6: C(a; a; 0), C'(a; a; b) ^ M(a; a; •^)

Nen B D = (-a; a; 0), B M = (0; a; - ) , B A ' = (-a; 0; b)

Z

A D BD,BM

Do d6: VBDAM =

-b BD.BM BA'

a^b (dvtt)

ab; ab; a"

Hu-ang d i n giai

a) Hinh chop S.OMAN c6 chi§u cao SO = 1

khong doi, tii' giac day n i m trong mSt

p h i n g Oxy c6 dien tich:

CtiJ TNHHMTVDWH Hhang Vi$t

S = SAOM + SAON = IO M A H + | O N A K = 1 (m + n) = I : khong d6i

2 ' 2 phii-cnQ trinh mat p h i n g (SMN) la

V$y ( S M N ) tigp xuc vai mat c l u tSm A , b^n kinh R = 1

gai toan 15 25: Trong khong gian Oxyz, cho hinh hop S A B C D c6 day A B C D

la hinh thoi, A C c i t B D tai g6c O Bi4t A(2; 0; 0), B(0; 1; 0), S(0; 0; 2N/2)

Gpi M la trung d i l m cua canh S C a) Tinh goc va khoang each giiVa hai du-ang t h i n g S A , B M

b) Gia sir mat p h i n g ( A B M ) c I t du-ang t h i n g S D tai d i l m N Tinh t h I tich

JO trpng diS'm bSi dUOng HQC sinh gioi mon Toan 12 i e / kxinh Phd

Bai toan 15 26: Gpi G \h trpng tSm cua tup di$n ABCD

ChLPng minh ring dud-ng thing di qua G mOt dinh cua tu- di$n curig

GA

qua trpng tarn cua m$t do! di$n vdi dinh do Gpi A' la trpng tSm tam gjg

BCD ChLPng minh rSng = 3

Hiring din giai

Ta giai bing phu-ang ph^p tog dO.Trong khong gian toa dO Oxyz, gja

A(xi; yi; Zi), B(X2; 72; Z2), C(X3; ys, Z3), D(X4; y4; Z4) thi trpng t§m A' cua ta^

giac BCD, trpng tarn tu- di$n G:

AC = (ci; C2; -a), OD = (di; dj; -z)

-rneo gia thi§t OA = OB = OC = OD » OA^ = OB^ = OC^ = OD^

^ ( a - z ) 2 = b^ + z^= c f + c ^ + z 2 = d f + d | + z 2

^ a ' - 2 a z = b^= c ^ + c ^ = d f + d (1)

Suy ra: GA = -3 GA' ^ G, A, A' thing h^ng ^ = 3

GA Tu-ong ty thi c6 dpcm

Bai toan 15 27: Cho tu- di$n npi ti^p trpng m§t clu t§m O va c6

AB = AC = AD Gpi G 1^ trpng tSm AACD, E, F Id trung dilm BG, AE

Chu-ng minh: OF 1 BG o OD 1 AC ^

Hu'6ng din giai l A

AB = AC = AD va OB = OC = OD

=> OA 1 (BCD) tai chdn du-dng cao H vdi

HB = HC = HD

Chpn H lam g6c tpg dp, vai h^ tryc Hx, Hy, ^""^. G)

Hz sao cho HA la tryc Hz, HB Id trgc Hy,

HD la tryc Hx

^c, + d, C g + d g a

3 ' 3 '3

•rac6:0F.BG =0<=>(Ci+dif+ (c2 + d2)^-9b^ + 7a^-12az = 0(2)

Khai tri4n (2) vd thay the (1) ta du-p-c:

(2) o 32 + cidi + C2d2 = 0 ci> OD AC = 0 ; dpcm

Bil toan 15 28: Cho hinh Igp phu-ang ABCD.A'B'C'D' c6 c?nh bIng a Gpi I, J

lln lu'P't Id trung diem cua A'D' vd B'B

a) ChLPng minh ring IJ 1 AC Tinh dp ddi dpgn thing IJ t ,,

b) ChLcng minh ring D'B 1 mp(A'C'D), mp(ACB') Tinh g6c giCra hai du'b'ng

thing IJ vd A'D

Hipang din giai

a) Chpn he tog dp Oxyz sao cho A(0; 0; 0), D(a; 0; 0), B(0; a; 0), A'(0; 0; a) Ta c6 C (a; a; a),

^^^^ ChLPng minh D'B 1 mp(A'C'D), ta chu-ng minh

•^'B 1 A'C', D'B 1 A'D D'B.A'C' = 0, D'B.>VD = 0

"Ta CO D'B = (-a; a; -a), A~^ = (a; a; 0); AT) = (a; 0; -a)

^ d o D'BA'C' = 0, D'B.A'D = 0 Tu'ong ti^, D'B 1 mp(ACB')

W trono diS'm bSi dUOng h<?csinh gi • ' •^^nJ2-_L£ Hodnh Phd

AT) = (a; 0; -a) GQi cp la goc glQ-a hal duo-ng thing IJ va A'D thj:

Bai toan 15 29: Cho hinh l§p phu-ang ABCD A1B1C1D1 canh a, tr§n BC, i^y

dilm M sao cho D,M,DA^,AB, ddng phlng Tinh di^n tich S cua AMAB1

Hipo'ng din giai

Chpn h0 Oxyz sao cho B = 0, Bi(a; 0; 0), Ci(a; a; 0), C(0; a; 0), A(0; 0; a),

, Ai(a; 0; a), Di(a; a; a), D(0; a; a)

' Vi M e BCi nen gpi M(x; x; 0)

Bai toan 15 30: Lang try tir giac d4u ABCD.A1B1C1D1 c6 chi^u cao bing ri^a

canh day D i l m M thay d6i tren canh AB Tim gia trj Ian nhat cua goc

A,MC,

Hu-o-ng din giai

Chpn h? tryc nhu- hinh ve (Aixyz)

Do do a < 90° Vay g6c a = A,MC, I6n nhat khi x = 1 tipc M trung diim AB

gai toan 15 31: Cho hinh chop S.ABC c6 du-ang cao SA = h, day la tarn giac

ABC vuong tai C, AC = b, BC = a Gpi M la trung dilm cua AC va N la dilm

sao cho SN = - S B

3 a) Tinh dp dai doan thing MN

b) Tim sy lien he gii>a a, b, h d l MN vuong goc vai SB

HiFO-ng din giai

Ta chpn h$ tryc toa dp Oxyz c6 goc

O trung vai A, tia Ox trung vai tia AC, tia Oz trung vai tia AS sao cho dilm

8 nim trong goc xOy Khi do: A(0; 0; 0), C(b; 0; 0), B(b; a; 0), S(0; 0; h), M(|; 0; 0)

:0 ; S V « '

SB = (b; a; -h) Gpi N(x; y; z) thi SN = (x; y; z - h) Tu- dilu kien SN = - SB nen

^) IVIN vuong goc vai SB khi va chi khi MN.SB = 0 ' ,2 ^2

10 trqng diSm bSl dUOng tiQC sinh gioi mon Toan IP ieti<uhi/iPhd

Hirang din giai

a) Ta chpn h$ trgc Oxyz sao cho s

V$y MN ngln nh^t bing — khit = 2a

b) Khi MN ngin nh^tthi: IVI

Ta CO MN.SA = 0

• dpcm

MN.BC = 0

Bai toan 15 33: Cho hinh chop tii- gi^c deu S.ABCD cgnh d^y

vol day goc a Tim tana d l SA vuong g6c SC

Hu-o-ng din giai

Chon he tryc Oxyz c6 O la tarn d^y

ABCD, tia Ox chua A, toa Oy chCfa B,

Qai toan 15 34: Cho hinh hOp chu' nh§t ABCD.A'B'C'D' Gpi M, N, P l l n lu-o-t

la c^c diem chia cac dogn thing AB, D'D B'C theo cung ti so !< 0, 1

Chu-ng minh ring mp(MNP) Iu6n Iu6n song song vd-i mp(AB'D')

Hu'O'ng din giai

DSt A'B' = a, A'D' = b, A'A' = c Ta dung phu-o-ng ph^p toa dO blng c^ch chpn he true tog dO voi g6c 1^: A'(0; 0; 0) sao cho B'(a; 0; 0), D'(0; b; 0) va A(0; 0; c)

Ta CO C'(a; b; 0), B(a; 0; c), D(0; b; c) va C(a; b; c) Cac dilm M, N, P chia

c^c doan thing AB, D'D', B'C theo cung ty s6 V n6n:

M _ka_

1-l< ;0;c 0;b;

-l<c 1-k , P

^ai toan 15 35: Cho hinh chop tu- giac d§u S.ABCD c6 canh day bing a vd

chi^u cao bIng h Gpi I Id trung di^m canh b6n SC Tinh khoang edeh tCf S den mat phIng (ABI)

413

Htpang din giai

Ta chpn he tryc tog dO Oxyz sao cho

goc toa do 1^ tarn O cua day, true Ox

chLPa OA, tryc Oy chu-a OB, tryc Oz

Ta c6 giao diem U cua SO Al 1^ trong tam tam giac SAC nen

M Mgt phing di qua A, B, Ml cung chinh la mSt phSng (ABM)

Bai toan 15 36: Cho hinh chop S.ABCD c6 d^y la hinh chu nhat AB = a

AD = a^/2 SA = a, SA vuong goc (ABCD) Gpi M, N la trung dilm AD, SO,

gpi I 1^ giap dilm BM AC Chu-ng minh (SAC) 1 (SMB) tinh the tich

kh6i ANIB

Hu-ang din giai

Chpn he trgc tea dp nhu- hinh ve S(0; 0; a), A(0; 0; 0) B(a; 0; 0), C(a; a

Hiring din giai

Ta kilm tra dupe ring k§t luan dung cho truang hp'p tu di^n AQBOCQDO CO 4 dinh la Ao(0; 0; 1)

cho d i n khi (T) trung vai tip dipn AOBQCQDO da noi b tren => dpem

^ai toan 15 38: Cho hai dilm A(3; 1; 0), B(-9; 4; 9) va mp(a): 2x - y + z + 1 = 0

Tim toa dp dilm M tren (a) sao cho | MA - MB | dgt gia trj Ian nhat

IHiro'ng din giai

f(x; y; z) = 2x - y + z + 1 thi f(xA; yA, ZA)-f(xB; yB, ZB) < 0 nen hai dilm A,

^ 6' khac phia doi vai m^t phIng (a)

"^Pi A' la dilm d6i xCrng cua dilm A qua m^t phlng (a),

"'"a co: I MA - MB I = | MA' - MB I < A'B (Khong dli)

^'H: x = 3 + 2t, y = 1 - 1 , z = t nen H(3 + 2t; 1-t; t) thupe (a) suy ra t = 2

^ H ( 1 ; 2 ; - 1 ) D o d 6 A X - 1 ; 3 ; - 2 ) ,

DLfdyng thing A'B c6 phu-ang trinh

Bai toan 15 39: Cho 4 diem A(1; 0; 3), B(-3; 1; 3), C(1; 5; 1) M(x; y; QJ

Tim gi^ trj nho nhit T = 2 MA + MA + MC

Hu-ang din giai:

Gpi I la trung dilm cua BC:

^ l(-1; 3; 2) IVTB + MC = 2MI ^ T = 2(MA + Ml)

ZA = 3 > 0 Zi = 2 > 0 => A I nim ve cung 1 phia d6i vb-i mp(Oxy) va

M(x; y; 0) thupc mp(Oxy) nen lly doi xi>ng l(-1; 3; 2) qua mp(Oxy) thanh

Ta tim hinh chilu A', B' cua A, B Idn d

Ta CO M bit ky thuOc d thi M(1 + 2t; 2 - 1 ; 1 + t)

" BM b6 nhat khi t = 1, khi 66 M Id hinh chilu B'(-1; 3; 0)

Tren mp(A, d) lly dilm Bi sao cho Bi vd A khdc phia d6i v6>\, BiB' -L

i BiB'=BB'

A 1 a

mrVDWH Hhang Vm

yjcfi mpi M thupc d: MA + MB = MA + MBi > AB,: khong d6l, do d6 MA + MB

nhat khi M la giao dilm cua ABi v6\ ,^.f

Ta c6 AA' / / BiB' nen M chia dogn A'B' theo ti so:

^ ' U - 1 ^ M ( 1 ; 2 ; 1 ) , r > , p

^='B,B'="Vn

^^itoan 15 41: Tim gia trj be nhdt cua: M

f(x, y)= V(x - ^f + (y + 3)2 + 9 +V(x - 2f + (y + 4f + 25

rong khong gian Oxyz, xet M(x; y; 0) va 2 dilm c6 dinh A(1; - 3 ; 3),

g^2;-4;-5) a khac phia vai mp(Oxy) ' <

) Ipl c6: f(x; y) = MA + MB > AB = N/66

(31a tri be nhdt cua f(x; y) = \/66 khi M Id giao dilm cua doan AB vai mdt phing Oxy

gaitoan 15 42: Cho 9 s6 thuc bit ki a^^, bi, Ci; 32', b2; C2; 83; ba; Cathoa man:

ai + 32 + 33 = 3, bi + b2 + b3 = 4; Ci + C2 + C3 = 12 Chifng minh bit ding thipc:

Jaf + b ^ cf + ^al ^hl^cl ^^al + b ^ > 13

hHu 'O'ng din giai

Trong khong gian vai h$ tog dp Oxyz, chpn 3 dilm:

A(ai; bi; Ci), B(ai + 32; bi + b2; Ci + C2), C(ai + 82 + 83; bi + bs + b3; Ci + C2 + C3)

hayC(3;4;12) thi c6:

OA = J a ? + b f + c f ; AB = ^a^ + b^ + c^ ; BC = ^a^+b^ + c^

N#n ta c6: ^ a f + b f + c f + ^a^+b^+c^ + ^^a+b^+c^

= 0A +AB + BC > 0 C = 13

^ BAl LUYfiN TAP

^^'%15 1: Cho l u - 2 , V I = 5, goc giu-a hai vecta u va v bing 2n

'"^kde vecta p = ku + 17v vuong goc vai vecta q = 3 u - v

Hu-o-ng din

g.^'^u ki^n tich v6 huang bIng 0 Kit qua k = 40

^^.*^P 15 2: Cho tarn giac ABC c6 A(1; 0; 0), B(0; 0; 1), C(2; 1; 1) Tinh chu

^"en tich va dp dai du'ang C30 AH

l-lu'6'ng din

^^^Q Cong thi>c Kit qua V2 + V3 + ^ / 5 ; ^ • A H = ^

Bai t|p 16 3: Cho hinh hpp ABCD.A'B'C'D' c6 cdc dilm A(1; 0; 1), B(2;

D(1; - 1 ; 1) vd C'(4; 5; -5) Tim cSc diem c6n i^i

HiPffngdln

Vi hinh hOp ABCD.A'B'C'D' nen ABCD Id hinh binh hdnh

K^t qua C(2; 0; 2) ;A'(3; 5; -6), B'(4; 6; 5), D'(3; 4; -6)

Bai t|p 15 4: Cho tip dipn ABCD c6 A(1; 0; 0), B(0; 1; 0), C(0; 0; 1) va D(-2; i;

a) Tinh g6c giu'a cac du-b-ng thing chu-a cdc cgnh d6i cua tu- di0n d6

b) Tinh the tich tec di^n ABCD va dp dai du-bng cao AH cua tu- di^n

Bdl tip 15 5: Cho 4 dilm A(2; -A; 2), B(0; 2; -2), C(4; 8; 0), D(6; 2; 4) Chiing

minh ABCD Id hinh thoi, tinh dien tich va bdn kinh r dubng trdn n0i tiep hinh thoi,

Hipvng din

Chung minh ABCD la hinh binh hdnh c6 2 cgnh lien tiep bing nhau

Ket qua SABCD = ^ 2 7 3 6 , r = 171

1 4

Bdi t?p 15 6: Chu-ng to ring cdc m^t phing (a), (P), (y) (8) sau d§y Id cac

phing chii-a b6n mdt cua mot hinh hpp chu- nhat: ,

(a): 7x + 4y - 4z + 30 = 0 , (P): 36x - 51y + 12z + 17 = 0

(7):7x + 4 y - 4 z - 6 = 0 , (5): 12x - 17y + 4z - 3 = 0

Hu'O'ng din

Chi>ng minh : (a) // (y), (p) // (5) va (a) 1 (p)

Bai tgp 15 7: Chu-ng minh cdc du-o-ng thing dk Id giao tuyen cua 2 mat ph^nj|

X + kz - k = 0, (1 - k)x - ky = 0, k 5^ 0 luon nim tren m0t phing c6 dinh

Hu'6ng din

Khu- tham so k giu-a hai phu-ang trinh mdt phing

'"Kdtqua (P): x + y+ Z - 1 =0

Bdi tap 15 8: Tim dilm M tren true Oz trong moi tru-b-ng hp-p sau:

a) M each d^u diem A(2; 3;4) vd m$t phing 2x + 3y + z-17 = 0

b) M each d4u hai mgt phing x + y - z + 1= 0 v d x - y + z + 5 = 0

Hu'6ng din

a) Dilm M tren true Oz nen M{0 ;0 ;z) Kk qua IVI(0; 0; 3)

b) Dilm M tren true Oz nen M(0 ;0 ;z) K§t qua M(0; 0; -2)

Bai t?p 15 9: Cho hinh lap phu-ang ABCD.A'B'C'D' cgnh bIng a Tr^f^

cgnh BB', CD, AD' Idn lu-p-t lay cdc dilm M, N, P sao cho:

B'M = CN = DP = k a ( 0 < k < 1)

Tinh dien tich tarn gidc MNR theo k vd a

f,) Xde djnh vj tri M tr6n BB' 66 di$n tich l\/1NP c6 gid trj b6 nhlt

HiTO'ng din

Chpn he true toa dO Axyz Kit qua SMNP = (k^ - k + 1) j ' ' Kit ^ *''^"9 ^^"^ '' " '* '

0 tiP 15- 10: Cho hinh Idp phu-ang ABCD .A1B1C1D1 Gpi M Id trung dilm

cua AD, N Id tdm hinh vu6ng CC1D1D Tim bdn kinh mdt clu di qua cdc

dilm B, Ci, IVI, N

Hira^ngdin _ _ aVis

a) MA^ + MB^ + MC^ be nhlt

b) MA^ + 1975.MB^ + 2015 MC^ b6 nhat

Hu'O'ng din

3) DCing trpng tdm G cua tam gide ABC Kit qua M(4; - 1 ; 0)

^) Dung tam ti ey-1 cua h? dilm : lA +1975IB + 2015IC = 6

4 1 9

cnuy^^n ae ic: PHVONG TAiNH DVONG VA M^t

1 K I £ N T H U C T R O N G T A M

PhiFcng trinh t6ng quat cua mat phlng: "

Mat phing qua Mo(xo,yo) va vecta phap tuyen n = (A,B,C)

Ax + By + Cz + D = 0, + + ^ 0

Phipcng trinh mat phing theo doan chin

- + ^ + - = 1 khi cit 3 true Ox, Oy, Oz tgi 3 diem khac g6c O la A(a;0

Lfng tog dp cua 2 dilm thupc giao tuyen

- Duong vuong goc chung cua 2 dudyng thing ch6o nhau:

Du'O'ng thing di qua Mi va c6 VTCP u i

Duong thing da qua M2 va c6 VTCP U 2

1: Duang vuong goc chung d c6 VTCP u = u^u^

Lap phuang trinh m^t phIng (P) chua d v^ 62

Tim giao dilm A cua di va (P) thi d di qua A va c6 VTCP u

d,

4 9 0

theo

Oy TNHHMTVDWHHhong Vi$t

^^ch 2: Gpi dogn vuong goc chung la AB, A e d, va B e dz dgng tham s6

fheo t v^ t' Tim t va t' bing he dieu ki^n:

AB.u, = 0

AB.U2 = 0 Duo-ng vuong goc chung d la du-ang thing AB

phu-o-ng trinh mat cdu: - ! ,

^/lat clu (S) tam l(a, b, c) ban kinh R:

Giao tuyin cua m^t clu (S) tSm I b^n

kinh R va mgt phIng (P) la du-o-ng tron giao tuy§n (C) c6 tam H Id hinh chieu tam mat clu I len mat phIng (P) va b^n kinh r - -d^(l;(P))

("i»>r.;.i

a 2 C A C B A I T O A N > ^ '

Baitoan 16.1: Lap phuong trinh m|t phIng:

a) Di qua hai dilm A(1; 1; -1), B(5; 2; 1) va song song vo-i tryc Oz b) Chua giao tuyin cua 2 mat phIng x - y + z - 4 = 0, 3 x - y + z - 1 = Ova (Jiqua K(2; 1; -1)

Hu'O'ng din giai

3) M§t phIng (P) song song vai Oz nen c6 phuo-ng trinh: A'x + B'y + D' = 0

10 tnpng diSm bSl dUOng HQC sinh gl6l mdn To6n 12 - LS Hodnh Phd

Cho y = 0 thi •

Cho z = 0 thi

X + z = 4 3x + z = 1

Ta lap du-gc phu-o-ng trinh (MNK): 15x - 7y + 7z - 16 = 0 ;

Bai toan 16.2: Lap phu-ang trinh m|t phlng

a) Di qua diem G(1; 2; 3) va cat cac true toa dp tai cdc diem A, B, C sao cho

V G la trpng tam cua tam giSc ABC

b) Di qua diem H(2; 1; 1) va cSt cac tryc toa dp tai cac dilm A, B, C sao cho

H la tru'c tam cua tam gidc ABC

Hu-o-ng din giai

a) Gia su- A(a; 0; 0), B(0; b; 0) va C(0; 0; c) Vi G(1; 2; 3) Id trpng tam tam giac

a + 0 + 0 , 0 + b + O _ 0 + 0 + c

ABC nen: = 1; = 2 ; = 3

3 3 o Suy ra a = 3, b = 6, c = 9

Vay phuang trinh theo doan chSn: - + + T = 1

o D y

b) N4U mat phing di qua H(2; 1; 1) va cSt cac true toa dp tai A, B, C thi tCf dien

OABC CO cac canh OA, OB, OC doi mpt vuong goc, do do H la true tam cua

tam giac ABC thi OH 1 mp(ABC)

Vay mp(ABC) di qua H va c6 vecta phap tuy§n OH = (2; 1; 1) nen co

phu'ang trinh: 2(x - 2) + (y - 1) + (z - 1) = 0 hay 2x + y + z - 6 = 0

Bai toan 16.3: Viet phu'ang trinh cua m$t phing qua dilm M(5; 4; 3) va cit ba

trge tog dp a ba d i l m khae O, cdeh d§u gdc to? dp

Hu 'O'ng din giai

Mat phing can tim co dang doan chin:

N/5|2x-y + 4z + 5| = V3|3x + 5 y - z - 1

o V 5 ( 2 x - y + 4z + 5) = ±x/3(3x + 5 y - z - 1 )

Vay tap hp'p cac d i l m M la hai mat phing phan giac:

(2V5-3^/3)x-(^/5 +5N/3)y + (4V5 + V3)z + 5V5+>/3 =0, (2N/5 + 3 N / 3 ) X - ( V 5 - 5 N / 3 ) y + (4N/5-N/3)Z + 5V5 - V S = 0

I b)Oilm M(x;y; z) each dIu hai mat phing: V^'^1^^-\x + 2y + z + 5

-^' toan 16.5: Lap phu'O'ng trinh ting quat cua mat phing (P) di qua cac dilm

^(0; 0; 1), N(3; 0; 0) va tao vai mat phing Oxy goc -

3

Vin

Hii-dng din giai

vecUy phap tuyin cua (P) la = (1; a; b) Ta c6 MN = (3; 0; - 1 )

fi MN = 3.1 + 0 - b = 0 n e n b = 3 Dod6 ri = ( 1 ; a ; 3 ) Phing Oxy c6 vecta phap tuyIn l< = (0; 0; 1)