BAI TAP TONG HOP Bai tap 1: Trong mat phang (P) cho mot diem O c6 dinh. Xet cac duong thang /

thay doi, qua O sao cho goc giua / va mp(P) bang a khong doi. Chung minh / luon luon nam tren mot mat non tron xoay.

HDDS True la duang thang qua O vuong goc vai (P).

Bai tap 2: Cho hinh non c6 the tich la 96:1, ti so giua duong cao va duong sinh la 4 - . Tinh dien tich xung quanh ciia hinh non.

HDDS Ket qua Sxq = TiRd = 6071

Bai tap 3: Cho hinh non dinh S, ban kinh day R, goc a dinh hinh non la 2a, 45^ < a < 90^ a) Tinh dien tich xung quanh va the tich hinh non. b) Tinh dien tich thiet dien do mp(P) cat hinh non theo hai duong sinh vuong

goc vol nhau. HD-DS

a) Ket qua Sxq =

b) K^t qua S =

smor ,V = TTR .cota

2sin^a Bai tap 4: Cho hinh non S, goc giua duong sinh d va mat day la a. Mot mat

phang (P) qua dinh S, hop voi mat day goc 60^. Tinh dien tich thiet dien va khoang each tu O den mp(P)

HDDS

Ket qua SSAB = ^ V4cos a- \, d(0,(P))= —^—

Bai tap 5: Cho tarn giac deu ABC canh a va (P) la mat phang qua BC va vuong goc vai mat phang (ABC). Goi (C) la duang tron duang kinh BC va nam trong mp(P) a) Tinh ban kinh mat cau di qua duang tron (C) va diem A. b) Xet hinh non ngoai tiep mat cau noi tren sao cho cac tiep diem giua hinh

non va mat cau la duong tron (C). Tinh the tich cua khoi non. HD-DS

a) Ket qua R .V3

b) Ket qua V = m

134

Bai tap 6: Cho hinh non dinh S duong cao SO, A va B la hai diem thuoc duong

tron day hinh non sao cho khoang each tir O den A B bang a va SAO = 30°,

SAB = 60* . Tinh the tich. dien tich xung quanh hinh tron.

Ket qua ,, Ml^yfl , r-a V = ,Sxq = 7ia v3

HD-DS

Bai tap 7: Cho luc giac d§u ABCDEF canh a. Tinh t h i tich hinh tron xoay sinh boi luc giac do khi quay quanh: a) Duong thang A D . b) Duong thang di qua trung diem cua cac canh A B va DE.

HD-DS

a) Ket qua V = TIO^ b) Ket qua V = 7V3: Tta

12 Bai tap 8: Cho hinh non c6 dinh S day la hinh tron (O). Tren duong tron day lay

mot diem A c6 dinh va diem M di dpng. Biet A O M = a, (SAM) tao v6i mat day hinh non mot goc P va khoang each tu O den (SAM) bang a. Tinh the tich khoi non va chung minh cac hinh chieu cua O tren mat phang (SAM) nam tren mot duong tron c6 djnh.

HDDS

Ket qua V = - . . 3 , a . . _ _

COS' ^ .sm" p.cosp

O C H U D E V I I I

TOfiN CCTC TR! KHONG GIfiN DANG TOAN

1. BAT DANG THUfC VE KHOI LANG TRU VA KHOI CHOP

Phuang phdp chung minh hat dang thuc: - Nhom hinh phuang va so sdnh - Dimg hat dang ihicc Cosi: vai cac so a,b,c khong dm thi

a + h

2 a + b + c

> -JaJ), ddu hang xdy ra khi a = h

> \fahc, ddu hang xdy ra khi a = h = c.

- Diing dao ham, dim vdo tinh chdt dan dieu hay lap Bdng hien thien de ddnh gid,...

135

CItuy: 1) Lang tru dieng khi canh hen vuong gdc vai day. Lang tru deu la lang tru dirng va c6 day la da gidc deu. 2) The tich cua mot khoi hop chit nhgt hang tich so ciia ba kich thtrac. 3) The tich cua khoi lang tru bang tich so ciia dien tich mat day va chiiu cao

ciia khoi lang tru do. 4) Ti the tich: Cho khoi chop tam gidc S.ABC. Tren ba duang thdng SA, SB, SC

V SA' SR' <sr' lay ha diem A', B', C khdc vai S thi c6 ti: -IM^ _ ^ ^

'S ABC SA SB SC

Bai toan 1: Cho tu dien ABCD thoa hai dieu kien: nam canh c6 do dai nho hom 1, con canh thu sau c6 do dai tuy y.

Chung minh the tich V < -. 8

Giai Xet tu dien ABCD c6 5 canh bang 1 va canh con lai AD = a tuy y.

Ta chung minh the tich cua tu dien nay la Vi < —. A 8

That vay, ha AH vuong goc vai (BCD), AK vuong goc BC thi:

V,.i.^.AH<i.^.AK 3 4 3 4

3' 4 • 2 8 Ta CO tu dien thoa dS bai c6 the tich nho hon Vj => dpcm.

Bai toan 2: Cho tu dien ABCD c6 mot canh lom hom a, cac canh con lai dSu

khong lom hom a. Chung minh r^ng xhk. tich V < 8

Giai Khong giam tinh tong quat, gia su AB > a khi do:

a > Max {AC, AD, BC, BD, CD} Ha cac duang cao BK, AE, AH cua tam giac

BCD, ACD va cua tu dien

Ta CO V = - AH.BK.CD 6

Dat CD = X va goi M la trung diSm cua CD.

M\ f~+~\

M

136

Xet tarn giac BCD.

„,,2 2 B C - + 2 B D ' - C D ^ ^ 4 a - - x ' r n T 4 4 2

Tuong tu d AACD ta c6: AE < ^ .-^J^a' - x^

Ma AH < AE ^ AH < - .V4a ' - x ' 2

Do do V < — (4a^ - x^)x = — (4a^x - x^) 24 24

Xet ham s6 y = 4a^x - 3x^ tren [0, a] Ta CO y' = 4a - 3x^ => y' > 0 voi moi x e [0, a]

Do do max y = 3a xay ra khi x = a nen V < 8

Dau xay ra khi • BC = BD = AC = AD = CD = a A M 1 mp(BCD)

Bai toan 3: Cho tu dien SABC va G la trpng tam ciia tu dien. Mot mp(a) quay quanh AG cat cac canh SB, SC \hn lugt tai M va N . Goi V la thk tich tu dien SABC, V, la the tich tu dien SAMN.

4 V 1 Chung minh — < — < —

9 V 2 Gidi

Goi A' la trong tam ASBC, I la trung diem BC Ta CO A,G, A' thSng hang, S, A', I thSng hang ^ . SM SN Dat — = X, — = y, von 0 < X, y < 1. SB SC , , V, SM SN laco: ' - = xy

Mat khac:

V SB SC SsMA. SM SA-

-"SIB

Tuong tu: ^ ^ ^ y ^ • S

SB SI

S

Hay ^SCB

'see

SM SN

^SCB

SMA '+SsNA ^ X + y

*SCB ^

SB SC x + y

= xy = — = ^ y = 3 x - l

Ket hop ta c6 dieu kien ^ < x < l , ^ < y ^ l

137

Fa co: — = xy = . V 3x-l

x' 1 Xet f(x) = . " < X < 1

3x-l 2 - 9x 1

f(x)= 4,f'(x) = 0<^x= - . (3x-l)- 3

BBT: X 1/2 2/3 1

f(x) 0 +

f(x) 1/2 1/2

4/9 -"^^

4 V 1 9 V 2

Bai toan 4: Cho tu dien ABCD. Hai diem M, N chuyen dong tren hai doan thang

BC va BD saocho 2~ + 3—= 10. BM BN

a) Chung minh duang thang MN luon di qua 1 diem c6 dinh

b) Chung minh — ^ ^ - . 25 ^ABCD ^

Giai ^ , BC -BD 1 BC 3 BD ,

a) 2 + 3 = 10 <=> + = 1 BM BN 5BM 10 BN 1 Goi K la diem xac dinh bai BK = - BC + — BD thi K c6 dinh. 5 10

Dat BC = x.BM; BD = y.BN. 1 , 1

Ta c6: BK = -xBM + — yBN, vi -x + —y = lnen M, N, K thang hang. 5 10 5 10

Vay MN luon qua diem c6 dinh K.

b) V ABMN BM BN 1 V MK'D BC BD xy x(10-2x)

Vi 1 < X va 1 < y va 2x + 3 y = 10 nen 1 < x < —

v. N 3 rr ^ T CM ^ 12x-30 Xetf(x)= ^ e [1; -], f^x) = ———y

x(10-2x) 2 X (10-2x) ; f'(x) = 0«x= -

2

138

B B T : X 1 5/2 7/2

f ' (x ) 0 +

f ( x ) 3/8 ^ 2/7

6/25

Vay — < f ( X ) < ^- hay — ^ < ~ 25 8 25 V ABCD 8

Bai toan 5: Cho tu dien A B C D c6 cac canh doi doi mot vuong goc voi nhau, c6 thd tich V.

Chung minh r4ng vai moi diem M nam trong tu dien ta c6 bat dang thuc sau:

M A . S,K-,) + M B . SACD + M C . S A B D + M D . S A I J C > 9V.

Giai Ha A A i , M A 2 vuong goc vai mp(BCD). Ta c6: A M + M A 2 ^ A A 2 ^ A A , => A M ^ A A i - M A 2

Dau bang xay ra khi M thuoc duang cao A A i cua tudien.

Do do A M . S i K o > A A I . S B C D - M A 2 . S » c o > 3V - SVM.BCD-

Ly luan tuomg tu ta c6:

B M . S A C I ) > 3V - 3VM ACD; M C . S A B D > 3V - 3VM.ABD

M D . S A I K - > 3V - 3VM ABC

Ta CO M A . S i j c D + M B . S A C D + M C . S A B D + M D . S A B C

^ 12V - 3(VM.BCD + VM.ACD + VM.ABD + VM.ABC) = 9V

Dau "=" xay ra khi M d6ng thai thuoc 4 duang cao cua t u dien A B C D nen M s true tam H cua tu dien A B C D . Bai toan 6: Cho hinh chop S.ABCD c6 day la hinh binh hanh. Goi K la trung • diam cua SC. Mat p h k g qua A K ck SB,SD tai M , N .

Dat V | = VsAMNK va V - VSABCD- ^

1 V 3 Chung minh: - < — < - .

3 V 8 Giai

SM SN Dat x = ; y = —

SB SD Ta CO V | = VSAMK + VSANK

V , , , , , _ SM SK

V, SAliC SB SC

xV SANK

139

yV V Tuomg tu Vs.ANK = ^ =>V| = — (x + y)

4 4

MaV, =VsAMN + VsMNK =Z^ + ^^1^ 2 4 4

X 1 Do do X + y = 3xy => y = , x > -

3x -1 3

SD X 1 1

< 1 => X ^ — nen — < x ^ 1. 3x-l 2 2

, V, 3 3x-Ta CO — = — xy =

V 4 4(3x-l)

Xet ham so f(x) -3x^

4(3x-l) vai — < X < 1

2

Tac6f'(x) = 3x(3x-2) 4(3x-l)-

,f(x) = 0<=>x = 2 3

BBT: X 1/2 2/3 1

f -0 +

f 3/8

^1/3

3 V 8 Bai toan 7: Cho hinh chop cut c6 chi6u cao h, dien tich cua thiet dien

va each deu 2 day la S.

Chung minh th^ tich V thoa man: S.h < V < - S.h.

Giai Goi Si, S2 la dien tich 2 day hinh chop cut.

Ta chumg minh: S A+Vs;

Goi S la dinh hinh chop va k la chiSu cao cua hinh chop nho, ta c6 ti dien tich:

s, rk+hV '

140

Do do - 1 = 2 VS2

- 1 4 s=(Vs;+Vs;) '

Ta CO the tich hinh chop cut: 1 v = - ( s , +Vs;s;+S2)h

(Vs;+Vs7)-- V p ; ] h < ^ (Vs7+Vs^)-h=I sh

V a V = -3

i (Vs;+Vs;)^+i(s,+s,)

2s+i(Vs;+Vs7)' h = -.3S.h = Sh 3

Vay Sh ^ V < ^ Sh.

DANG TOAN >

2. GIA TRI LQfN NHAT, BE NHAT VE KHOI LANG TRU

VA KHOICHOP

Phifong phdp tim gid tri Ian nhdt, nho nhdt: - Nhom hinh phuang vd so sdnh - Ditng hat dang thuc Cosi: vai cdc so a,b,c khong dm thi

a + h 2

a + h + c

> -ia^b . ddu bang xdy ra khi a ^ b

> \jabc, ddu bang xdy ra khi a = b = c.

- Dung tam thuc bdc hai - Diing dgo ham, dua vao tinh chdt dan dieu hay lap Bdng bien thien de ddnh gid,... Chuy: 1) Hlnh chop deu la chop c6 cdc canh ben bdng nhau vd c6 ddy la da gidc deu.

hUnh chop deu thi hinh chieu ciia dinh chop Id tam cua ddy. 2) The tich cua mot khoi chop bdng mot phdn ba tich .so cua dien tich mat ddy

va chieu cao ciia khoi chop do. 3) Tu dien hay hinh chop tam gidc cd 4 each chon dinh chop.

141

Bai toan 1: Cho tu dien SABC c6 cac goc phang a dinh S vuong. Biet rang SA = a, SB + SC = k. Dat SB = x. Tinh thk tich tu dien SABC theo a, k, x va xac dinh ' SB. SC da ihk tich tu dien SABC Ion nhk.

Giai The tich tu dien:

VsABc = - SA.SB.SC = - ax(k - x) 6 6

1 < - a

6

x + k-xV ak" 24

Dau = khix = k- x<=>x= — 2

k Vay the tich hinh tu dien SABC Ion nhat khi: SB = SC = -.

• 2 Bai toan 2: Cho tam giac ABC. AB = AC. Mot diem M thay doi tren duang thang

vuong goc yai mat phang (ABC) tai A. a) Tim quy tich trong tam G cua tam giac MBC. b) Gpi O la true tam cua tam giac ABC, hay xac djnh vi tri cua M de the tich tu

dien OHBC dat gia tri Ian nhat. Gidi

a) Gpi D la trung diem cua BC. Ta c6: MB = MC. Do do MD 1 BC va trong tam G cua tam giac MBC

nam tren MD thoa man he thuc DG = — DM. 2

Do do G la anh cua M trong phep vi tu tam D, ti

so vi tu ^. Vay quy tich cac trpng tam G cua tam giac MBC la ducmg thang d' vuong goc

vai mat phang (ABC) tai trpng tam G' cua tam giac ABC. b) Ha CE 1 AB. CI- 1 MB ta c6 H DM n CF la true tam ciia tam giac MBC,

O DA n CE la true tam cua tam giac ABC. (jpi HH' la chieu cao cua tu dien OHBC, ta c6 H' thupc DO. Hinh chop nay c6 day OBC c6 dinh nen VQUHC Ian nhat khi va chi khi HH'

Ian nhat. Diem H chay tren duang tron duang kinh OD nen HH' Ian nhat khi

HH' = ~ DO nghTa la DHH' la tam giac vuong can tai H', suy ra tam giac DMA

luc do vuong can tai A.

142

Vay tu dien OHBC c6 the tich dat gia tri Ion nhat, can chon M tren d (ve hai phia cua A) sao cho A M = AD. Bai toan 3: Cho ba tia Ox. Oy. Oz vuong goc vai nhau timg doi mot tao tam dien

Oxyz. Diem M c6 dinh nam trong goc tam dien. Mot mat phang qua M cat Ox. Oy, Oz Ian lugt tai A, B, C. Goi khoang each tir M den cac mat phang (OBC), (OCA), (OAB) lan lugt la a, b, c.

^ I3 c a) Chung minh: + + = 1.

OA OB OC b) Tinh OA. OB. OC theo a, b, c d l tu dien OABC c6 th i tich nho nhSt.

Giai: a) Ta c6: VOAHC = VMOAB + VMOBC + VVKXTA

nen - OA.OB.OC = - OA.OB.c + - OB.OC.a + - OC.OA.b 6 6 6 6

Do do: 1 a b c

OA OB OC b) Diem M c6 dinh tuc la cac so a, b, c khong doi.

Taco: V = - OA.OB.OC. 6

Do do V nho nhk OA.OB.OC nho nhat. Ap dung bat dang thuc Co si:

1 = abc

OAOBOC OA OB OC o OA.OB.OC ^ 27abc.

OA.OB.OC nho nhk ^ —=-^=--OA OB OC

Vay: V nho nhk <» OA = 3a. OB - 3b, OC = 3c. Bai toan 4: Khoi chop tam giac S.ABC c6 day ABC la tam giac vuong can dinh C

( C - 90") va SA 1 (ABC). SC = a. Hay tim goc giua hai mat phang (SCB) va (ABC) de the tich khoi chop Ian nhat.

Gidi s Ta CO BC 1 AC BC 1 SC Mat khac BC _L AC. suy ra goc SCA la goc giiJa

hai mat phang (SCB) va (ABC). Dat SCA = x Khi do SA = asinx, AC = Acosx

3 Ak

Vs asm X a" cos" x

S.ABC = —sm x.cos" X

143

Xet ham so y = sinxcos x, vai 0 < x < —

Ta CO y' = cosx - 2cosxsin^x = cosxCcos'x - 2 + 2cos^x)

= cosx(3cos X - 2) = 3cosx

y' = 0 <=> cosx

cosx • ; V3

COSX + , —

— = cosa vai 0 < a < —. 3 2

BBT: X n 0 a -

y' + 0

y

Vay Ss ABC dat gia tri Ion nhat khi x = a, 0<a<^va cosa = .

Cach khac: Dat t = sinx, 0 < t < 1 hoac diing bat dang thuc Cosi. Bai toan 5: Cho khoi chop tu giac deu S.ABCD ma khoang each tu dinh A den

mp(SBC) bang 2a. Vai gia tri nao cua goc giua mat ben va mat day cua khoi chop thi the tich ciia khoi chop nho nhat.

Gidi Ha SO 1 (ABCD) thi O la tam hinh vuong ABCD. Goi FH la duong trung binh cua hinh vuong ABCD. Vi AD // BC ^ AD // (SBC)

d(A, (SBC)) = d(E, (SBC)). Ha EK 1 SH, ta c6: EK 1 (SBC) => EK = d(A, (SBC)) = 2a.

Ta CO BC 1 SH, SB 1 OH

SHO la goc giua mat ben (SBC) va mat phang day.

Dat SHO = X. Khi do: EH 2a

,0H smx sm X

,S0 •tanx sm X cosx

1 4a' Vay Vs.ABCD = - SABCD-SO = —5— •

3 3cosxsm"x Do do Vs ABCD nho nhat <=> y = cosx.sin^x dat gia tri Ion nhat. Ta c6:

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

144

= sinx(2 - 3 sin" x) = 3 sin X 2 • — - s i n x — + s i n x

13

y' = 0 <=> sinx = V 3

Xet gia tri a sao cho: sina = , 0 < a < ^ .

BBT: X 71

0 a 2

y' + 0

y ^ ^ ^ ^ Vay Vs AHCD dat gia tr i nho nhat <^ x = a,0 < a < ^ va sina = .

' i toan 6: Tren canh A D cua hinh vuong A B C D c6 do dai canh la a, lay diem M sao cho: A M = x (0 < x < a). Tren nua duong thang Az vuong goc vai mat phang chua hinh vuong tai diem A, lay diem S sao cho SA = y (y > 0).

a) Chung minh rang (SAB) 1 (SBC) va tinh khoang each t u diem M d i n mat •ng(SAC). b) Tinh the tich khoi chop S .ABCM theo a, y va x. Gia su x^ + y' = a , t im gia

• Ion nhat cua the tich kh6i chop S.ABCM. Gidi s

a) Ta CO BC 1 A B , SA nen BC 1 (SAB). Do do (SAB) 1 (SBC).

Vi (SAC) 1 (ABCD) theo giao tuyen AC nen ha Ml 1 1 AC thi M H l ( S A C ) .

Vay M H la khoang each tir M tai mat phang (SAC). Trong tam giac vuong A M H c6:

M H = x.sin45^' = ^ 2

b) Hinh chop S.ABCM c6 duong cao SA = y va c6 day la hinh thang vuong

nen dien tich day la S = ^ a(a + x)

Do do ihk tich kh6i chop S .ABCM la:

V = | y . ^a(a + x) = ^ya(a + x) . J z o

145

Theo gia thiSt + = a => = - nen

= — a(a^ - x)(x + a)^ = — a^(a - x)(a + xf. 36 ' 36

Dat f(x) = vSai 0 < X < a, ta c6: a\3 a'(a + x)'(2a-4x)

f '(x) = (a + x)' + — 3(a - x)(a + x)' = ^ 36 36

f'(x) = 0«x= -2

36

BBT: X a

0 2

f + 0

f ^^^^

Vay f(x) dat gia tri Ion nhat tai x = —, khi do the tich ciia khoi chop S.ABCM

rV3 dat gia tri Ion nhat la: V = -JUx)^ =

8 Bai toan 7: Cho hinh chop S.ABCD c6 bay canh bang 1 va canh ben SC = x.

Dinh X de the tich khoi chop la Ion nhat. Gidi

Day ABCD c6 4 canh bang 1 nen la 1 hinh thoi =^ AC 1BC. Ba tarn giac ABD, CBD, BSD c6 chung canh BD, cac

canh con lai bang nhau va bang 1 nen bang nhau, cac trung tuyen AO,SO va CO bang nhau.

Suy ra tarn giac ASC vuong tai S ta dugc AC = Vx^ +1. Goi H la hinh chieu dinh S tren day (ABCD). Do SA = SB = SD = 1 nen HA = HB = HD ^ H la tarn duong tron ngoai tiep

tarn giac ABD =^ H e AC =i> SH la duong cao ciia tarn giac vuong ASC. X

Ta CO SH.AC = SA.SC =^ SH =

OB^ = AB^-OA^= 1-=^=>oB=Av^ 4 2

Dihu kien x^<3oO<x<V3

146

Ta CO SABCD = AC.OB = ^Vx'+l.V3-x' -^^(x'+l)(3-x')

Vay VsAHCD = |SABCD- S H = ^ x V S - x ' 3 6

Ta CO the dung dao ham hay bat dang thuc Cosi:

1 r r r z — . l x ' + 3 - x ' \4

D4U khi x^ = 3 - x^ « 2x^ = 3 « x = 2

Bai toan 8: Cho hinh chop S.ABC c6 day la tam giac ABC vuong can tai dinh B, BA = BC = 2a, hinh chieu vuong goc cua S tren mat phSng day (ABC) la trung diem E cua AB va SE = 2a. Goi I , J l in luot la trung di^m cua EC, SC; M la

diem di dong tren tia d6i cua tia BA sao cho ECM = a (a < 90°) va H la hinh chieu vuong goc cua S tren MC.

Tinh the tich cua khoi tu dien EHIJ theo a, a va tim a d8 thd tich do Ion nhat.

Giai

Vi SE 1 mp(ABC), SH ± CM nen EH 1 CM.

CE = V B C ' + B E ' = V4a' +a- = aVs

Ma IJ la duomg trung binh trong tam giac SCE nen IJ

Hon nua IJ // SE => IJ X (ABC). ^ S

Trong tam giac vuong CEH vai goc ECH = a va trung tuyen HI ta c6:

SE • a.

Si iMi= ^ s , x i i = J E H . C H 2 4

= — CE.sina.CE.cosa = - a^sin2a. \

4 8 The tich cua kh6i tu dien EHIJ la:

1 a 5a' Vi-iiij = - IJ.Si.iii = —. .sin2a = —a"* sin2a

3 3 8 24 The tich tu dien EHIJ Ion nhat khi va chi khi:

sin2a = 1 «> a = 45°.

147

DANG TOAN BAT DA NG THUfC VE KHOI TRU, KROI NON, 3. KHOICAU

Phmmg phdp chung minh bat dang thuc: - Nhom binh phuang vd so sdnh - Dung bat dang thicc Cost: vai cdc so a,b,c khong dm thi

a + b

2 a + b + c

yfa~b, ddu hang xdy ra khi a = b

> yfabc, ddu bang xdy ra khi a = b = c.

- DUng dao ham, dua vao tinh chat dan dieu hay lap Bdng bien thien de ddnh gid,...

Chuy:

1) Dien tich xung quanh cua khoi tru: Sxq = 2nRh,

The tich cua khoi tru: V = TtR^h

2) Dien tich mat cdu: S = 47rR^

The tich khoi cdu: V = — nR\ 3

Bai toan 1: Cho tu dien vuong OABC dinh O. Goi R, r \kn lugrt la ban kinh mat cau ngoai, noi tiep tu dien

Chung minh: — > "^'^"'"^ . Khi nao dang thuc xay ra. r 2

Gidi Dat OA - a, OB = b, OC = c

=> R= - Va" +b" 2

1

Ta CO r = 3V abc

•p (ab + be + ca) + Va'^b' + b^c' +c-a^ 2 2

R _ ab + be + ca + Va'b^ +b^c-

r ca"

R

2abc

3VaW + ^/3^/?bV l^/3VaV

Va" +b

3V3+3 r 2abc

Dang thuc xay ra khi a = b = c.

148

Bai toan 2 : Cho tu dien ABCD c6 cac duong cao A A', BB', CC, DD' dong quy tai mot diem H thuoc mien trong cua tu dien. Cac duong thang AA', BB', CC. DD' lai cat mat cau ngoai tiep tu dien ABCD theo thii tu tai A i , Bj , Ci, Dj .

. ^ AA' BB' CC DD' ^ 8 Chung mmh: + + + > - .

AA, BB, CC, DD, 3 Giai

Tir dien ABCD da la tu dien true tam nen A' la true tam tam giac BCD. Goi J la giao diem cua BI vol mat cdu ngoai tiep tu dien ABCD thi A ' l = IJ.

Do H la true tam tam giac ABI nen: A'H.A'A = A'B.A'I

= - A ' B . A ' J = - A ' A i . A ' A 2 2

= ^ A ' H = - A ' . A , 2

Tuong tu: B'H = ^ B'B,; C H = ^ CC,, D'H = ^ D'Di

Tir ViiBCD + ViicDA + VnDAB + VHABC = VABCD V, V V V

HDAB _|_ '^HABC _ | V V V V *ABCD ^ABCD ^ABCD * ABCD

HA HB HC HD , AA, B B CC DD, ^ + + + = 1 = > — ^ + - - = J - + — 1 + ^=2

AA BB CC DD AA' BB CC DD AA, BB, CC, DD

+ • + • I _

AA' BB' CC DD' Theo bat dang thuc Cosi:

= 6

AA' BB' CC DD' • + + + AA, BB, CC, DD, ' ,

AA, BB, CC, DD, • + -

AA' BB' CC DD' >16

AA' BB' CC DD' 8 • + + + AA, BB, CC, DD, .

Bai toan 3 : Cho hinh hop chur nhat ABCD.A'B'C'D'. Goi R, r, h, V Ian lugt la ban kinh mat cau ngoai tiep, noi tiep, duong cao ke tir A' va \hk tich cua tit dien

A'AB'D'. Chimg minh: Xi^Jlll < 1. R-.r.h 3

Giai Dat AA' = a; AB' = b; A'D' = c.

1 4 9

Ta c6: V(h-r) R'.r.h

3.V S

3.V s >p J •^ip "-"(AE'D') _ '-'xq(A.A'B'D')

3.V_ 3V

^(ABD) '-'ip

3R^ 3.R^

vai S,xq(A.A'B'D) = - (ab + bc + ca)

Tu dien A'AB'D' vuong tai A' nen R = Va' +b' +c-

Suy ra -"xqlA'AB'!)) 2 ab + be + ca 2

^ — V(h-r) 2 ^:^^3- 3R' 3"a'+b'+c' 3

Bai toan 4: Cho r, R Ian lugt la ban kinh mat cau noi tiep, ngoai tiep cua mot tu dien c6 the tich la V.

Chung minh ring: 8Rr ^ 3 V3 V. Giai

Goi O, G Ian lugt la tam mat cau ngoai tiep va trong tarn tu dien ABCD. Goi BC = a', AD = a', CA = b', BD = b', AB = c, CD = c'. Goi Sa, Sb, Sc, Sd, Sip Ian lugt la dien tich cac mat doi dien vai cac dinh A, B,

C. D va dien tich toan phan cua tu dien.

Ta CO AB' = (OB - OA)" = 2R- - 20A.0B => 20A.0B = 2R' - AB'

Mat khac 40G = OA + OB + OC + OD => 160G^ = 4R^ + Z(2R^ - AB^), vai Z la t6ng theo 6 canh

= 16R^ - (a + b^ + c + a' + b' + c') ^ 0 :^ a + b^ + c + a' + b' + c' < 16R^ Trong tam giac ABC ta c6: a + b^ + c > 4S V3 Tuang tu cho cac Sa, Sb, Sc roi cgng lai ta dugc:

2(a + b + c + a' + b' + c') > 4 V3 .S,p. Do do 8R^ ^ V3 S,p Dau bang xay ra khi tu dien ABCD deu.

Bai toan 5: Chung minh rang trong mot mat non, goc a dinh Ion hom hay bSng bSt cu goc nao do hai duang sinh tao nen.

Giai Thiet dien qua true SO la tam giac can SAB,

goc a dinh ASB = 2a nen OSB = a. Ve duang sinh SC, goi H la trung diem cua BC

thi tam giac SBC can tai S nen

BSC =2p, p-BSH. ^'

150

OB HB Trong hai tarn giac vuong OSB, BSH: sin a = , sin p =

SB SB Vi duong kinhi A B > BC OB > HB => sina > sinp Hem niia goc a. P nhon nen a > p: dpcm.

Bai toan 6: Chung minh rang the tich V va dien tich xung quanh S ciia mot hinh 6V

V 71 y non tron xoay tuy y thoa man bat dang thuc:

Dau dang thiic xay ra khi nao? Gidi

Ggi ban kinh day va duong sinh hinh non la x va y

Ta c6: V = - TTX^ - x ' va S = Tixy voi 0 < x < y.

B D T « 4 x V - x ^ ) < ^ « - - ^ ^ ^

2S

r \^ X X

+

3V3 y y' (Vs)-'

3 1

y (V3)-^ (V3)-^ > 0 (1)

X X 71 Vi 0 < — < 1 do do dat — = coscp vai 0 < (p < — thi

y y 2

(!)<!> cos-'cp -1

coscp-1

> 0

f 1 ^ ( ^ 1 coscp-v

cos" +

1 y ( 2 ^ coscp- 41] coscp+ — 7 =

>0

Dau "=" xay ra khi coscp =

> 0: Dung

^ hay y = V3 X .

DANG TOAN •

4. GIA TR[ LdfN NHAT, NHO NHAT VE KHOI TRU,

KHOI NON, KHOI CAU Phuang phdp tim gid tri Ian nhdt, nhd nhdl: - Nhom hinh phuang vd so sdnh - Dung hat dang thuc Cosi: vai cdc so a.h.c khong dm thi

> -JaJ), dau hangxdy ra khi a = h

151

a + h + c > \lahc, ddu bang xay ra khi a b = c

- Dung lam thicc bgc hat - Ditng dgo ham, dua vao dnh chat dan dieu hay lap Bang bien thien de ddnh gid,... Chity: 1) Dien tick xung quanh cua mat non: Sxq = nRl,

The tich cua khoi non: F = - n-R'h. 3

2) Dien tich mat cdu: S = 47rR^

The tich khoi cdu: V = — nR^. 3

Bai toan 1: Trong cac hinh hop noi tiep mat cau ban kinh R, hay xac dinh hinh hop CO dien tich toan phan Ion nhat.

Giai Mat phang qua moi mat hinh hop cat mat cau

theo mot duong tron, do do mat hinh binh hanh noi tiep duong tron nen la hinh chu nhat.

Vay hinh hop ABCD.A'B'C'D' noi tiSp mat ck la hinh hop chil nhat.

Goi cac kich thuac la AA' = a, AB = b, AD = c thi a + b^ + c - 4R^

Dien tich toan phan cua hinh hop la: S = 2ab + 2bc + 2ca < a + b^ + b^ + c + c + a = 8R^

2R Dau khi: a = b = c =

V3 nen hinh hop la hinh lap phuong.

Bai toan 2: Cho tu dien OABC trong do OA, OB, OC doi mot vuong goc voi nhau. CO duong cao OH = h. Goi r la ban kinh mat cAu noi ti^p tu dien. Tim gia

tri Ion nhat cua —.

Dat OA = a, OB = b, OC = c.

^ . 1 1 1 1 -Ta co: — = — + — + — va r =

h" a" b" c"

Giai

3V 1 S tp

Ma^ 3V

^AOAD ^AOBC "*" ^A(X:A ^AABC | | |

3V

r 3V

\ 1 a b c

•1_ h

1 1 1 1 1 Do do = - + - + -

r h a b c

152

Ta CO bat dang thuc: 1 1 '

+ — + -a b c

< 3 1 1 1

\di b c

nen { \ 1V' 3 1 1 1 V3 l a b c j h ' a b c h

Do d 6 - - - < — i ^ - < ~ ( l + V3) r h h r h

Vay ~ < 1 + V3 .DAU "=" xay ra khi a = b = c.

Vay gia tri Ion nhdt cua - la 1 + Vs khi OA = OB = OC. r

Bai toan 3: Cho hinh chop tu giac deu, goi R, r Ian luot la ban kinh mat cau ngoai

tiep va mat cau noi tiep ciia hinh chop do. T im gia t r i Ion nhat cua t i so — . R

Giai Xet hinh chop tu giac deu S.ABCD c6 canh day a, duong cao h. Goi a la goc

bai mat ben voi day. Goi O, I Ian lugt tam mat cau ngoai tiep va noi tiep ciia inh chop thi 0 . 1 e SH.

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

R^ = (h - R) ' + ^ 1 V 2 ,

R = a^ + 2h^

4h

„ , a 2 + tan^a Do h = — tan a ^ R = a.

2 4 tan a 1 la chan duong phan giac cua goc SMH nen

a a r = I H = - tan -

A D

4 tan a. tan 2 Do do: - =

R 4 + 2 t an -a

a

2 _ tan - t a n

2 2

1 + tan

2 ( t - t " ) — Xet ham s6 y - ^ voi t = tan^ 9 ' t e (0; 1)

1 + t ' „ , , 2 ( - t - - 2 t + l ) , Ta co: y = • " =

(1 + tO y' = 0 « t = - l ± V 2 , c h o n t = V 2 - 1 .

153