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

10 trQng diem bai duong
VAMDK = VK AMD -

hoc sinh qioi

man Joan i>'

ih

Phi

TNHHMTVDWH

V = Vg.MA + V^.MB + V

2 2

Do 3 6 SAMD . X = — . Ha D I 1 A ' M =^ A l l

Al . A'M = AA'.d(M, AA') = a^ ^ Al =

12

nen

43^

9a^

Dl =

S.,r.nMB
+ S , , ^ „ M C + S, , M D
^ A ' C D " ' " - ' ^ A ' B D ' " " ' ' ^A-BC'
= (SACD - S ^ 3 D

>> :

V.M A ' C D

Vgy d(eK, A p ) = X =

2Vi^~2~^^

+ S^.Bc)-MA' =

_

^A'BD

%^^MK'

_

VA A ' D C

3a
^A'DC

N6n SA.MD= g ^ ' - ^ " ^ "

i.C'

SA.CD-A'B + S , . 3 ^ . A ' C + S , , 3 , . A ' D = 0

A'M

2a

.MC + V ^ . M D

Gpi A ' la giao diem ciia A M va ( B C D ) . Ta c6:

^

Mat khac

AP = a^ +

Vi^t

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

SAMD-X

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

Dl^ = DA^ +

Hhong

AA'
_

VA A ' B D

V,M A ' C D

V' A A ' C D

^MA'BD

VAA'BD

_

^A'DC

^ABD

^A'BC

nensuyra V^MB +V^MC + V^MD = kMA'
=> V cung phu'cng MA'

-.

Bai toan 13. 26: Cho hinh ch6p tu- giac c6 tat ca cac canh b i n g 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 d i n 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

Tuong tu- V cung phuang MB. Do do V = 0
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:

AF^ = BE^ = x^ + 1 - 2x.cos60° = x^ - x + 1.
Theo gia thiet: Vs ABCD = 2.Vs ABEF = 2(VSABF + VSBEF)
Ma

SABF

_

SABD

SF
SD

V,SBEF _ ^
= ''V33,,

SC

SF

^ ^2

SD

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 p h i n cua mpt

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

dien. Chung minh ring — > 288
Chu vi thiet dien: C = AB + EF + 2BE=

^{1 + Vs + 2V1O - 2>/2).

Bai toan 13. 27: Cho d i l m 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

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.

/

D


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
1
1
1
^
Ta c6: V = - SB-BB' = - S c C C =i>
= - SBSC-BB'.CC
3
3
9
Ha BH 1 AB, Ta c6 BB' < BH va C C < AC.
TLI-do suy ra

'-

-\

• Xet f(x) =
f'(x) = 0 « x =
X

f(x)
4 V, 1
Vay - < — ^ - •
•^9
V
2

rtn,,-^:ini?

r
2/3

-

f(x)

, I
2
- .
3

1/2

<

(SASBSCSD)^ . Vi d i n g thCcc khong d6ng tho

0

1

+

1/2

1/2
^^^4/9 "

Bai toan 13. 31: Cho tip dien tryc tam ABCD c6 t h I tich V (tuf di$n c6 cac canh
d6i doi mot vuong goc voi nhau). Chii-ng minh ring vai mpi di§m M n i m
trong t i j dien ta cp b i t d i n g thipc sau:

xayranenV'<
Ap dung b i t d i n g thuc BCS ta di^gc
S,+S, +Sc+S„

(3x - ^f

BBT:

Dau d i n g thu-c chi xay ra khi BAG = C A D = DAB = 90°.
2
2
Lgp luan tu'ang ty ta du'gc
< -SCSBSA,
< - S Q S A S B va
y
y

;

Ta c6: ^ = XV =
f'(x) =

< -SB.SC.-AC.BH = -SBSC.SD
9
2

9

-SASBSC. Do do V® < ( - ) V

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

_S

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

Vay

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

• 288.

<

dien SABC, V. la t h I tich ti> dien SAMN. Chung m i n h9l < V
Hiro'ng d i n giai
Gpi A' la trpng tam ASBC, I la trung d i l m BC
Ta CP A,G, A' thing hang, S, A', I thing hang
SM

SN

OSt — = X , — = y , vai 0 < X, y < 1.
• SB
SC
'
Ta CO :

TT = 7

Mat khac

Tu-ang ty:
•'SCB

Hay

SM SA'

2Ss^^A'

2x

SB' SI

SscB

3

2V

Sc;^^^. +SgN^

x +y

3

Sgj-B

3

^SCB

x+y^w
- xy — - — => y
SC
3

Dau b i n g 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

SN
— = xy
SC

SN

< 2—,

x
3x-1

> 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 -

1

2

u-ang ti^ Vs ANK =

xyV
Vl

^2

k

V

—

3xyV

xyV
+

=

, x> ^

Vi V = — - ^ 1 => — - — < 1 => X > - nen - < x < 1.
SD
3x-1
2
2

2k

a!)
. V, 3
3x2

Ta CO — = — xy =
V
4 ^ 4(3x-1)
3x2
. 1
X6t ham s6 f(x) =
vai - < X < 1
4(3x -1)
2

Do do

=5 4S =
™

= VsAMN + VsMNK

DO do X + y = 3xy ^ y =
N2

1+

^

+ ^/s^)'

tich hinh chbp cgt: V = i (S, +

^
. .„ , 3 x ( 3 x - 2 ) ^„ , „

2
Ta CO f 'X = —^
^ , f 'X = 0 c=> X = 4(3x - ^f
3
X
BBT:
2/3
1/2

+ S2)h

-

f

h.
1

f

2S.^(J^.>Sfjha.3S.h = Sh

V$y

'3
V?y S h < V < - S h .
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
1

V.

3

ninh: _ < - i < - .
va V = VsABCD. ChCrng minh:
3
V
8

SI

Hipo-ng d i n giai
SM

SN

fir

s

I < ^<.
3

V

1

+

0

3/8

3/8
^ 1 / 3

'

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 d i l m A
' djnh. Hay xac djnh vj tri cua M va N sao cho dien tich tam giac AMN dat
'91^ trj Ian nhlt.
ij J
HiFO-ng d i n giai

I

f

a)oi =

MN _ a
^ = — va I thuQC goc vuong xOy nen:
2 ~2

"r$p hgp cac diem I la phan cua du'ang tron
Ta CO Vi = VsAMK + VsANK

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.


III'

I'P' _ SO' ^ SO'
IC ~ SO

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

Vay dien tich Ian nhat khi: OM = ON =

= NM.NQ.sinMNQ

b i n g goc giOa AB va CD nen sin M NO = sina.
CO

MN

AC-x

AB

AC

NO = MR ,

MR

MN:

AM

CD ~ AC

AC

AB
AC

^

^

'

-

-

^

^

-

)

-H,

Khi lang tru c6 m^t ben la hinh vuong ta c6:
= X

» ^(a ^

X)

= x o ah - hx = ax « x =
a+h

2,
b) The tich lang tru: V = B.h = x2^/3 .h- ( a - x ) = hx/3
—
4
a
' 4 a x2(a-x)
Ap dung b i t d i n g thu-c BCS:

X

x

x^(a - X) = 4 . - . - ( a - X) < 4.-^—2
^
2

2
'
3

= 4a=
27

khix = - a.

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

AB.CD

nen SMNQR = ' '"'^^ (AC - x)x.sina <—AB.CD. sin a
AC^ .
4
01

Vay khi M la trung d i l m 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.

h
-

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

MR = ^ x
AC

AC

^

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.

(AC-x)

Tie do SMNQR max ce>AC-x = x o x =

a-x
-

V9yVmax=

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

Ta

- S O

X

SO ~ a

SO-SO'

00'

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 d i l m b i t ky thupc canh AC, dat AM = x (0 < x < AC).
Xet mat phlng (P) di qua d i l m M va song song vai AB, CD. Xac dinh vj tri
d i l m 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

\rr^

, 3 c 6 C I = ^ , P T = i ^

0H<2

khi do OMN la tarn giac vuong can.

I nnuiiy

Hu-o-ng d i n 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'

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:

uvvi

^'GpiDla trung d i l m 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- ^
f V
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.


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 nhit 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.

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
p/( + DB + DC = 3DH
DA .DA' + J ^ . D B ' + - ^ . D C ' = S.^DM = 4Di^
^ DA'
DB'
DC
3
DO A', B', C, M d6ng phIng nen 4 = ^
+
+^
> 3 3/ DA.DB.DC

DA' DB' D C
DA'.DB'.DC'
V p A B C _ D A ' . D B ' . D C ' ^ 27
W

c6:

VQABC = VMGAB + VMOBC +

VMOCA

nen - O A . O B . O C = - O A . O B . C + - O B . O C . a + - OC.OA.b
6
6
6
6
Dodo: 1 = ^ + — +
OA OB OC
Ta c6: V = -OA.OB.OC. D i l m M c6 dinh
6
tuc la cac s6 a, b, c khong doi. Do do V nho
n h i t « OA.OB.OC nho nhlt.
Ap dgng b i t d i n g thii-c BCS:
1 =

abe
OA

OB

OC

OA.OB.OC

<=> OA.OB.OC > 27abe.
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 '
VAA'BC'

V B A ' B ' C "*• VcA'B'C'

DA.DB.DC

T = VA-ABC

+ V B A B C + Vc

^ _ DA
DB
Ma 4 = DA' f DB'
AA'
DA'

Hu'O'ng din giai:
Ta

^

Dod6T

DC
DC

BB'
DB'

= V D A B C >

A B C =

VQ A'B'C

AA'

BB'

DA'

DB'"^DC

DA'+ AA' ^ DB'+ BB'
DA'
DB'

CC
+3

DC

—
64

"f
I

"64

AA'

BB'

CC

DC'+ C C
DC

CC'_

DA' ^ D B ' ^ D C

VD A B C

,

BB'

V$yminT= | ^ V D ABC khi-^"^'

64 " " " " ^ D A '

CC
—

1
w

Bai toan 13. 40: Cho kh6i chop tu 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 khli chop nho nhlt.
Hu'O'ng d i n 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.
CO

^

BC 1 SH, SB 1 OH

^

^ SHO 1^ goc giua mat ben (SBC) v^ m^t phIng dSy.
^^t-SHo = x. Khi do:
sinx

sinx

*

•
a
-tanx sinx
cosx


10 trpng diSm bSi dudng hVay

.1

1
g

icH.^o^ihPho

Cty TNHHMTVDWH Hhong Vi$t

4a^
—^ •
ScosxsitT^x

SsABCD = — SABCD-SO = -

0t f(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

(2
—
3

. 1f 1^ • vl
sinx , - + sinx

\iz

J

y' = 0 o cosx = ^ - = cosa, 0 < a < ^ <=>x = a
3
2
BBT:

X
y'

0

0

-3 .
..,2 =_a^(a + x)^(2a-4x>

f(x)= - — ('a + x)^+—3(a-x)(a
+ x)^
36
f'(x) = 0 « x =

I

BBT;

-

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 t h i 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
phIng (SAC).
Trong tam giac vuong AMH c6:
MH = x.sin45° = ^
;

™
Hinh chop S.ABCM c6 duang cao SA = y va c6 day la hinh thang
1
••'
nen di^n tich day la S = - a ( a + x)
^
The tich khoi chop S.ABCM la:V= - y . -a(a + x)= ;Jya(a + x).
3
2
6
-2 + y2' = a^' ^ y^-2 = a'
x' nen
b) Theo gia thilt x^
.2 - ..2

X

a

0

+

f

2

a

+

v6i 0 < x < a, ta c6:

0

_

f

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
= X. Oinh X dl thI tich khoi chop la Ian nhlt.
s
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
AO,SO va CO bIng nhau.
,
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= , ^

OB2 = A B 2 - O A 2 = 1-

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

=3_^^0BaV^
4

2


lOtrpng

c/u^m bSi duang

hQc sinh gini

mon Taan

Ta CO SABCD = A C . O B =

;

ly

= ^V(x^

1

V|y VsABCD = — SABCD . SH =

Cty TNHHMTVDWH Hhong Vl0t

U- H~ , ;';
+1)(3

-x^)

-XN/S-X^

2T

1 x^+3-x^

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

1

— .

Bai toan 13. 43: Cho diem IVI trong tu- dien ABCD. C^c duo-ng t h i n g MA
MB, IVIC, MD c i t mat doi dien tai A', B'.C D' tuang Ceng. Tim GTNN cua
.^_,MA
MB ^ MC ^ MP
~ MA'^MB'^MC'^MD''
Hu-o-ng d i n giai
Goi H, I l i n luo't la hinh chi4u cua A, M
len mSt phing (BCD). Ta c6 H, I, A'
^
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:
\H
AA'

_ AH

MA'

S^^^'BCD

Tuang ty"

MB'

V.

V

.| MC
MC

.T = V

V,

-1

MD

V

MD'

V,

-4
'4y

;

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
g6c 60°.
HiPO'ng d i n
Goi O la tam hinh vuong ABCD, ha OH vuong goc vai SC.
K§t qua X = a.

;

Bai t?p 13. 3: Cho tu dien d i u ABCD c6 canh b i n g a. Tinh khoang each giua
cdc cap canh d6i dien va th6 tich cua hinh tu dien d§u do.
Hifo-ng d i n
Khoang cdch giua cdc cSp canh d6i dien cua t u dien 6hu Id do dai doan noi
va VABCD = - ^ - ^ .
'^
' ^
2
12

Bai t|p 13. 4: Cho khdi t u 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

gMI-SecD
V,
V - 1V ,
MB _ V

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

2 trung dilm.K§t qua —

Ml

MA'
MA

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 d i l m 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 d i l m cua SD.
,^
, ,,

6

Ta CO the dung dgo ham hay blit d i n g thu-c Cosi:
1 fTTr.

2 BAI LUYEN TAP

1 1 1 1
= (V, + V2 + V 3 + V 4 ) — + — + — + — - 4
V V.1
V„ V,
V4 )
> 1 6 - 4 = 12.
V$y minT = 12 cx> M la trong tam tu dien ABCD.

-1

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 d i l m 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.

•

Hipangdln

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

,
Vb'+42


W trQng diem

bSl dUOng HQC s/nh gioi

Cfy TNHHMTVDWH Hhong Vi$t

mon To6n 12 - LS Ho> i - ••
, • \hd_

0uyen

b) K^t qua

ae

14:

KHOI TAON XOnV

'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.
Hiring din

Tinh cgnh day a bang each lap phuang trinh.
,

4N/2d^tan(p

Ket qua —p

^

•

,

>'i... y..

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.

ifit C

Qj^n tich mgt clu: S = 47rR^
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 , Thi tich khii clu ( hinh cau): V = -TTR^.
tarn giac ABC la tarn gi^c d§u cgnh bSng a, AA' = b
AA' tgo vai mat
3
phing (ABC) mot goc bing 60°.
Vj tri tu-cng d6i giOa mat clu va mat phing.
Hu-ang d i n
j . jj;,
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
Xac dinh hinh chieu cua A' len mp(ABC). Kk qua VACAB' = ^ ^ ^ ^
kinh r = VR^ - d^ .Dat biet, khi d = 0 thi mp(P) di qua tarn O cua mat clu,
o
m|t
phing do goi la mat phing kinh; giao tuyIn cua m^t phing kinh v6'i
Bai tap 13. 8: Cho hinh chop tarn giac SABC c6 SA = x, BC = y, cac canh con
m|t clu la du'ang tron c6 ban kinh R, goi la du-ang trbn Ian cua mat clu.
lai"d§u b^ng LTinh thi tich hinh chop theo x, y. Vai x, y nao thi thi tich
- Nlu d = R, mp(P) va mat clu S(0; R) c6 dilm chung duy nhat la H. Khi d6
hinh chop Ian nhat?
m^t phing (P) tilp xue vai mat clu tai dilm H hoac mp(P) la tilp dien cua
HiFang d i n
m$t clu tai tilp dilm H.
Gpi M trung di6m BC thi th§ tich hinh chop chia doi b^ng nhau bai mp(SAM).
- Neu d > R: mp(P) khong eo dilm chung vai mat clu.
I
2

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

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
Dung ti s6 thi tich. Kit qua VSAHK = ^ •
d = OH la khpang each ti> O tai A.
Bai tap 13. 10: Cho tCf dien ABCD c6 BAD = 90°, CAD = ACB = 60°, va AB = jjlu d < R: duang thing A cit mat clu tai hai dilm
'
AC = AD = a. Tinh thi tich tip dien ABCD va tinh khoang each giOa hai duong
d = R, (ju^ang thing A va mat clu S(0;R) c6 dilm chung duy nhit la H.
thing AC va BD.
h' do, du-ang thing A tilp xuc vai m^t clu tai dilm H hoac A la tilp tuyIn
Hu'O'ng d i n
[.'^^mat clu tai tilp dilm H.
^ U d > R: du-ang thing khong c6 dilm chung vai mat clu.
Xac dinh dgng tarn giac BCD suy ra hinh chieu len (BCD).
(J^h ly: Nlu dilm A nim ngoai mat clu S(0; R) thi qua A c6 v6 so tilp
42 va d(AC; BD) = |
Kit qua VABCD =
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.

8a^


10 tr
K/||t non, hinh non, khdi non

Cho mat cau S ( 0 ; R ) va didm M. Qua
diem M, ve 2 cat tuy^n cSt mat c i u tgi
A, B
C, D thi

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
^
Id6i, quanh A . Mat non c6 true A va goc a dinh la 2 a .
k
Hjnh non, khdi non:

^

Tryc SO. Du-dng sinh S M = /

^

M?t c l u ngoai ti4p khdi da d i ? n : Mat cau 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 ^

Goc a dinh la 2 a .

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

^2 = h ' + R '

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 .

,

Dipn tich xung quanh: Sxq = TiRC

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 t a m 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

/

^

^

rf;:)f/

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

3

Xac djnh tarn O cua mat c l u ngoai ti6p
-

''Oft.'

Ban kinh d a y R va chidu cap h t h i :

' chop do CO du-ang tron ngoai ti§p.
-

Hhang Vi$t

P h u ' c n g tich:

i M A . M B = IviC . MD = MO^ - R^.

-

cr
/ , . „ /; Phd

_ 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.

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 t a m mat cau ngoai ti§p

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

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

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

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.

a) Di 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 .

Xac djnh t a m I cua 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

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

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

khi lA = IB = 10. Vay tap h a p cac diem I la true cua du'dng trdn ngpai tidp

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

tam giac A B C .

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 v a i giao tuyen)
,

Hu-d-ng d i n giai

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

Mgt tru, hinh tru, khdi tru

BC, CA cua tam giac A B C Ian lu'p't tai

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

c^c d i l m I. J, K khi va chi khi 01 1 A B ,

khpang R khcng doi.

OJ 1 BC, O K 1 CA, 01 = OJ = OK. Gpi

Hinh trg, kh6i tru:

0 ' la hinh chidu vudng goc cua didm O

-

Tryc 0 0 ' . Du-ang sinh MM' = /

tren mp(ABC) thi cac didu kien la: O'l 1

-

Ban kinh day R va ehi4u cao h thi

AB, O'J 1

1 »\
M \

'\

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

h = / = 0 0 ' , R = OM

O'K, hay O' la t a m du'dng trdn ndi t i l p

-

Dipn tich xung quanh: S^q = 2nRI

tam giac A B C .

-

Jhk tich khdi tru: V = TtR^h

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

-

Thiet di^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

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

'


TNHHMTVDWH

Gpi O la tarn cua hinh hop, ta c6:
+ MC'2

MO^ =

M0^ =
M0^ =

MO'

4

+ MD'^
2

BD'2

+ MA'^

CA'2

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.

4
4

2
+ MB'^
2

Hu-ang din giai

DB'2

Suy ra 4MO'.2 -= -1( /r/iA2
MA'

Theo gia thiet ta c6: PQ

4
+. M

M BR' 2 +
+ MC'
+ MC'

Do do

^.)^4.2.0G.IH = k « I H = - i =

AC'^

2

M0^ =

MA' + M B ' + MC' + MD'

+
+ MD' +
+ iviA'^
MA'' +
+ MB'^
MB'

+ MD'^)

- (a' + b' + c^)

+ MD''

PA + PB + PC

f

Qpi G la trong tam cua tam giac A B C thi: P A + P B + P C = 3 P G

'Iz^^l^-k'.

2
8
Vay: Neu k' > 0 thi tap hgp cac dilm M la mat cku tam O ban kinh
k'-2(a'+b'+c')

8
Neu k' = 0 tlii dilm M trung vai O. N^u k' <0 tlii tap hgp la rong.
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:

= 2PA.PB + 2PB.PC +

nen 9 P G ' = P A ' + P B ' + P C ' . Mat khac:
PA' + P B ' + P C '

=

PG + GA

+ PG + G B

2PCPA
'

= 3PG' + GA' + GB' + GC' + 2PG(GA + GB + GC)

"

,

Tipang ty: OA' + O B ' + OC' = 30G' + G A ' + G B ' + GC'
Do do: 6PG' = G A ' + G B ' + G C ' « 3R' = 30G' + 6PG'
Tii do R' = OG' + 2PG'
OG < R

, ., ,

' nnil

Ciipn di§m I CO dinh: Pi = -PO thi 2IP +10 = 6. Khi do:
OG' + 2PG' =
= Ol' +
+ IG^
9PI^ +
0\n?- + 2IG(OI
oxciirw +
, ooi\
IG' ++ 2Pi'
+ 2IG'+
2PI)

_ 01' - 2PI'

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

Gpi I la trung di^m OG, H la hinh chi4u M len OG thi:

i • '

= 3PG' + GA' + GB' + GC'.

^. MA
MB
MC
MD
.
Do do: =
+=
+=

+=
=4
MA' MB' MC
MD'
4

'

+ GC

= 0|2 + 2P|2 + 3|G2. VayiG' =

.(MA' + MB' + MC' + MD') =

-

' ^'' " '

+PG

MA.MA' = i\^.l\^'= MC.MC = m i \

.
MO'-R'
<x> 4MG' + GA' + GB' + GC' + G D ' = 4(M0' - R')
<^ 4(M0' - MG') = GA' + GB' + GC' + G D ' + 4R' = k (*)

-

Do do P A = 3 P G nen quy tich cua Q la anh cua quy tich cua G qua phep vj

ti^ tam P ti s6 bSng 3. Ta c6:
9PG' = P A ' + P B ' + P C '

= !<'.

% 4 M 0 ' = ^ - (a' . b' . c') c M O ' =

R =

.,.,5 ^g^.. .,

+ MA'' + MB''

+ MC'

<=>

Hhang Vi?t

R2

>0

R'-0I'-2PI'

^^uyra quy tich cua Q.
^' |oan 14. 5 : Cho 2 du-ang tron (O; r), (C, r') c i t 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.

O'D' ^ AO'
Ta CO

OD ~ AO

AO-R-R'
R

AO

Tu- do mp(OMO') la mp trung tru-c cua A B .
Gpi A va A' l l n \ua\a trgc cua du-ang tron C ( 0 ;

aVe

M^AO

" r) va C ' ( 0 ' ; r') thi A va A' cung vuong goc vai A B
n6n A, A' cung n i m trong mp(OMO') va cSt

Dod6R'=

^ ( 2 - x / 3 )

nhau tai I.

M^t cku (S) CO tarn I va ban kinh R = IB la m$t
V$y: V = I TTR '^ = ^

cauphaitim.
b) Ta c6; O M = 4, O'M = 1. Xet tarn giac OMO':

(2 - 73)^ (dvtt). :^ , ,A ,.

Bai toan 14. 7: C h i j n g minh c6 mat c l u t i l p xuc vai sau cgnh cua hinh tu- di^n
ABCD khi va chi khi: A B + C D = AC + BD = A D + BC.

0 0 ' ^ = OM^ + 0 ' M ^ - 2 0 M . O ' M c o s O M O ' =^ cos O M O '

Hu-ang d i n g i a i
Gia su- c6 mat c l u tiep xuc vai sau canh

' Nen: OMO'=i2o'"vd 010'
Ta c6;

=60°

^^"81 = 2R = iM ^

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,

IM = 2%/?

BM = BS = BQ = y, CN = CP = CS = z, DN

sinOlO'
Nen

= IB^ = IM^ + MB^ = 37. Vay R =

V37 .

Bai toan 14. 6: Cho mot tu- dien deu A B C D c6 canh bSng a. Mot mat c§u (S)
tiep xuc vai ba du'ang t h i n g A B , AC, 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 , AC lam cac t i l p tuySn.

= DQ = DR = t. Do do, AB + CD = AC + BD
= AD + BC = x + y + z + t.
Oao lai, ta xac dinh cac t i l p diem tu' he
X + y = A B ; y + z = BC; 2 + t = C D

t + X = DA; X + z = A C ; y + t = BD.
Huang dan giai ra, ta du-ac:

a) Tinh ban kinh R cua mat c l u (S).
b) Tinh t h i tich kh6i c l u (S').

X=

1 [AB + A C + A D - ^ (BC + C D + DB)]

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, ODA la nhu-ng tam giac

y = ^ [ B A + BC + A D - | ( A C + C D + DA)]

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

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

deu B C D .
Ta CO A H =

1N/6

Do do R = O D =

DH =
aV2

b) Gpi O' la tam mat cau (S') vS D' IS d i l m tiSp
xuc cua (S') vai 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
tam O, tam O' t i l p xuc v a i nhau va cung t i l p
xuc v a i A D tai D va D'.

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
Gpi Mi, Ma, Ms

HC>0/)n hVO

^

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

Hipo-ng din giai
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:

OG=\jA

4 ^ 6

;OG' = OG +

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 = V 7 4 - 9 = N/65
Vd MF = 765-16 = 7
Gpi R la ban kinh duang tron ngoai tilp AABC
R=

suy ra

CM _ CM
MK " MF

r=rL^
r

V

/

N6n k = — = 1 + N/6 . Vay the tich cua tiK dien A,A2A3A4 1^:
OG

abc
4S ^

Ta c6

^ „

- R-

MK.CM

=> CM =

C M - R C M

= -^^
MF

V = Vi.k'= -r3V2(1+^/6)^
3
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:
3a
OQ
3ah
OR = PR'
= h.PF"'
27a2 + 4h2
+ h2

37

• o d d OF = 3. Suy ra

Cac tam gidc dcng dgng OKM va CFM

28

MF

7

R = OD = V O F ' + F D '

Vgy dien tich mat cdu S =

,

=—=4
=V9 + 16=5

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 c l u npi ti§p
hinh chop do.


10 tTQng diSm bSl dUCing HQC

smh gioi mdn Toon 1£

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

HiFang din giai:
Do ABC 1^ tam gi^c d4u n6n:
AM = MN = NA =

f !> tam gidc cSn A S B c6 g6c dinh Id 120° n§n A B = a V3 .
Vi A C ' + C B ' = 2 a ' + a ' = 3a' = A B ' n6n A C B = 90°.
^»
Vi S A = S B = S C = a n6n H cdch d4u ba dinh A. B, C. Do d6 H Id trung
j i l m cua cgnh A B .

2

S M M N = ^ A M . A N . s sin
in60°=

1 3V3 ,
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:
= V2 = ON

SM^ = OM^ + SO^ = 2 + 1 = 3

1

3V?

SsAM = - AM.SM =

SM = V3 , nen:

1

» '•'

3^/2

; SsAN = - AN.SN = - y -

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

vd V = ^SH.S,3,

r=

a ' , / - r-

Do d6:

S,p

2'V3+V2+r

,,f,«

Bai toan 14. 14: Cho \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,
n, p thi ta c6 h$:
Q

^/ /

m'=l(a'+c'-b')

^ / I

^ / 1

NJi.-

p'=^(a'+b'-c')

J

//"

^

V i m ' + n ' + p' = (2R)'=> ^ I l l ± i l l l P l . £ l l ^ l l ^

3V

s,„

12

V2

a

n'+p2=b'

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

2

m' + p2 = a ' =:> n2=:I(b2+c'-a')

nen:

; SAMN = ^ MN.AK =

3'2'

3V

m'+n'=c'

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

SsMN = ^ MN.SK = I

a'Vs

V3
^

OM = ATtan30° = V6 . ^

a'^/2^a'^/3^a'

4

1+2V2 + V3

8

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 d i l m 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 d i n giai

\ 4SABC =

-c')(l:^

- 3 ^ ) 0 ' +0^

4Vp(p-a)(p-b)(p-c)
= V(a + b + c)(a + b-c)(b + c-a)(a + c-b)

a) Do BSC = 60°, nen SBC 1^ tam giac d4u,
ti> do BC = a.

Tac6V=:lmnp= -^^/(a'

I\ ^

J

\^$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 t u 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
b i n g 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 d i l m B', C,
D' thoa m § n :
AB' = ACAD, A C = AB.AD,
AD' = A B A C Ta CP — = — = AD
AC
AB
nen 2 tam giac ABC va ACB' d6ng dang.
Vi d Id t i l p tuy§n cua du-ang tron (ABC).
Suy ra dAC = A B C (chdn cung AC)
Do do dAC = AC' B'

d // B'C. Tuang t y b // CD', c//B'D'

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

.-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
' J
\
11 ^ "
\\
AAB1C1 dong dang AABC
• -b

B'C

= AD suy ra B'C = BCAD
BC,
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.
Ta lai c6

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 t u a = y = 30° nen

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

Hu-ang din giai

^^'""'"-^KQ
\/$y B1C1 = cc'.

.BC = ^

'

ab

= cc'

^./^a'

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

Vi I la tam cua duang tran npi tiep AABC nen

^ O A l C D , = Al ± C i D i .

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

Tuang tu Al 1 B1C1 =:> A l l (B1C1D1). Al cdt ( A i B i d ) t?! H

Theo tinh chdt tiep tuy§n ta cc:
MAC - AGAC; AIBC = AHBC
suy ra (x = GAC, p = H B C , y = GCA = H C B .
Trong tam giac ABC ta c6:
a + p + y = 90°, suy ra trong ABCD ta c6:
a = HBD = H C b , 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^'^

TacpAH.AI =AE.AO = ADi.AM= - A D i . A D = - a b c .
2
2
.Gpi Vi Id the tich tu dien AB1C1D1.
2R
Tacr.yi^AB,.AC,.AD,
(abc)^
•
- abc
V
AB.AC.AC
(abc)'
AH =

5>

Vi = V.abc= ls.AH=

Is — . V a y S = 6VR.

-5
3
2R
.
J
^iv
ngot"
*^
A,A2A3A4 vai (0,R), (|,r) Idn luat Id mdt cdu
'^^ai tiep, npi tiep vd h| Id cac duang cao ke tu Aj. Chung minh;

Salt

Z ^
1si
h.h.


W

dii^^Tboi duanq

trQng

HQC sinh

^

qlOl mon

lOUii

- ce. , lucrnn
Cty

Hipo-ng din giai:

-^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'^,

[cpi Si 1^ di^n tich m|it d6i di^n dinh Ai, ta c6:
4

.

|;iA:.s.=o=>oi = (XSiOA,)/s^
i.1

i=1

>0|2 =

1a
OA'.OAJ = {0A.2 + 0A.2 - A^Aj^) / 2 =

- A.A.^ /2

1Si
vaS, = 3 V / h . , S,p = 3 V / r

B

.

. .J.

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 t h i tich cua khii try,
,
Hirang din giai

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

a) Gpi bdn kfnh cua hinh try Id R, hai bdn kinh

A'

ch6o nhau Id OA' vd O'D. Ve du-dyng sinh DA

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

/
/

1
1
1
1
1

thi: 9(00', A'D) = g(AD, A'D) = ADA' = a

Vi IJ Id du'ang trung binh cua tam gid
SMA n§n:
^
=-y

= R.

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

>dpcm

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

IJ

Trong tam goc AOA' cdn tgi O:

A' -

AA'^ = 2R2 - 2R^cos30° = (2 - VB )R^

vd IJ 1 (P).

^ AA' = V 2 - V 3 R

(IS.;.;'

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

Tam gidc ADA' vuong tgi A nSn: DA'^ = AD^ + AA'^

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

=^

= (2R)2 + (2-V3 )R=^ =>

Ta c6: h^ = AD^ = DA'^ - A'A^
= a'~(2-V3).-?
6-V3

4a'
6-V3

.h = 2. a

6-V3

^°AD
2
tich cua khli try Id:

Hirang din giai

3) Sxq = 27iR.RV3 =2V3 7:R'
S,p = Sxq + 2Sa,y = 2 V3 TiR' + 27:R' = 2( VS + 1)7tR'
V = K R ' . R V 3 = VBTIR'

a'= (6-VJ ) R ' R

= - ^ = ^

V6-N/3

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
hinh try.

A-

V3

:Sfp=R^-(IS,S^A^A.^)/Sj,
iS^Sj = 9 V 2 / h h ,

Vl$t

H \---T0'

= AA'tan30° = R V3 .
Nen 0|2 =

Hhang

_

^ 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:

'

R ' I S i ' + 2 X S,S,OA,OA,
i=1

TNHHMTVDWH

i

V6-^/3 _ 27iV6 - Vsa^
e-Vs'

• 6-S

(6-V3)


giSi

m6n

To6n

12 - LS Hodnh

Phd

Bai toan 14. 22: Cho hinh try c6 b^n ldu-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 ring ABCD Id ttp di^n d4u.
b) Chu-ng minh ring cac duang thing AC, AD, BC, BD luon tilp 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

Cty TNHHMTVDWHHhang

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.
Hipang d i n giai
Ha BH 1 d =^ AABH vuong tai H.
V6i

duang ch6o CD = 2R, do d6 A'C = R V 2 ,
ngoai ra AA' = R V2 nen ta suy ra AC = 2R.
C
Tu-ong ty 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' =

Vi^t

BH
^ sinBAH =

d
=

'
'

^

"

V

khong d l i .

AB
AB
Do do BAH = a khong d l i
Vdy tap hgp cac duo-ng thing d Id mdt
^
B
non nhan du-ong thing AB Idm tryc, c6
dinh la A vd goc a dinh Id 2a.
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.
Hu-o-ng d i n giai:
Gpi I id trung d i l m cua AB thi
O I I A B , S I I A B , 01 = a.

.
Ta c6: AO = SAcosSAO = — S A

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

voi mat try c6 tryc Id 0 0 ' vd c6 bdn kinh

RV2

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.

Al = SAcosSAl = - SA
2
Tu-d6:

AI
AO

• sinlAO =

V3
V6

md cos lAO =
a
OA

R = OA =

Xet tam giac SAO, ta c6: h = SO = OA.tan30° =

'^=SA =

OA
cos30°

^4l

= aN/2

2

D o d 6 V = l K 0 A ^ S 0 = ^^4^(dvtt), Sxq = 7i.0A.SA = n a ' ( d v d t ) .
^' 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 din giai:

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

' '


Cty TNHHMTVDWHHhang

10 tr
(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 AB ± OI, SI => SIO = 60°
ASOA, ASOl vu6ng tgi O n6n:
SO = d.sina, OA = d.cosa
_ - J , _ 2dslna ^, dsina *
Ol

F=

75

1

. UI =

/ i\

4i

AP = OA^ - 01^ = — (3cos^a - sin^a)
=> Al = -^.\/4cos^ a - 1 n6n:

Ta c6 SA = V O S ' + O A ' = V h ' + r '

I01A0
2d^sina F.
i
T
-SI.AB =
V4cos^a-1
2
3
Ve OH 1 SI => OH 1 (SAB), do d6 OH Id khoang cdch tu- O den m$t phing

Theo tinh chdt du'ang phan gidc, ta c6:

(SAB)

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

Vdy ban kinh mdt cau npi tiep Id: R = 10 =

^

SsAB=

=

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.

\'

"1

Ta c6:

_

A.
0,
X

Md

V3

V3

= V2

0

Suy ra :
. 2(Y+V2)_

Do do:

R'l
X

Vi$t

R3 4R'^

o x =;

R^+R=

A

IO_OA
IS ~ SA

10

^

OA

1 0 + IS ~ O A + SA

i O _ _

\

r+
rh

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 t h i 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.


Cty TNHHMTVDWH

10 trpng diem bSl dU
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

=^
i^i >

. suy ra s = ^ •. Vay
. VV=.$v3 V= r "
7
' n n i u q-•

j

kinh mat cau ngogi, npi ti§p tCp dipn
R
3
C h u n g minh: — >
. Khi nao d i n g thCrc xay ra.

2

A

•

T u o n g t y cho cac Sa, Sb, Sc roi cpng lai ta du-p-c:

. (9^^
343'''"

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

r

Trong tam giac A B C ta c6: a^ + b^ + c^ > 4 S

2(a' + b ' + c ' + a ' ' + b'" + c'') > 4 Vs .S,p. Do do SR^ >

^3

'

ban!
Dau bang xay ra khi tu- di^n A B C D d i u

*

^
X6t phep vj t y tam G ti k = j

_

thi tCf dipn A B C D b i l n thanh tu- di$n c6 4 dinh

1^ 4 trpng tam A'B'C'D' cua 4 m^t va R = 3R'.

Dgt OA = a, O B = b, O C = c

Vi R' > r => R > 3r => dpcm.
Bai t o a n 14. 3 2 : Cho \\Je di^n A B C D c6 cdc du-dng cao AA', BB', C C , DD' dong

R=iVa^+b2+c^
2

quy tai rnpt d i l m H thupc mi§n trong cua tie di$n. Cac 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,

1
abc
^

3V ^

^
VJ S^,

3V
rIVia Stp - — nen 8R^r > 3 V 3 V.

Hu'O'ng d i n giai

Ta CO r =

Hhang Vi^t

^

- ( a b + bc + ca) + - \ / a V 7 b V T c V
2

r.

^ u -

AA'

u

Ci, Di. Chu-ng minh:

^'

2

^

BB'
+

AA,

CC
+

DD'

+

BB,

CC,

8
>-.
DD, 3

A

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

a b + be + c a + V a V + b V + c

V

TCP di0n A B C D dS la tu- di^n t r y c tam nen

b^+c^

2abc

A' la tru-c t a m t a m giac BCD. Gpi J la

4.4

abc

giao d i l m cua BI v a i mat cau ngogi ti§p

4

3N/3 + 3

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

2abc

r

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

D i n g thLPC xay ra khi a = b = c.

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

2

Bai 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
=>A'H= -A'.Ai
2

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

Tu-ong t y : B'H = - B'B,; C H = - C C i , D'H
2
2
<•

Gpi O, G l l n lu'p't IS tarn mSt cSu ngoai ti4p vS trpng tSm tup di$n ABCD GPi
BC = a'. A D = a', CA = b', B D = b', A B = c, C D = c'. Gpi Sa, Sb, Sc, Sd, S,p

VHBCD ••• VHCDA + VHDAB + VHABC = V
'ABCD

lu'p't Id dien tich cdc m$t d6i d i ^ n v&\c dinh A, B, C, D vS d i ^ n tich ^o3<^
phSn cua tCr d i ^ n .
A B ^ = (OB - 6Af

i
= 2R2 - 2 0 A . 0 B ^

2 0 A . 0 B = 2R^ - A B ^

Mat khac 4 0 G = O A + O B + O C + O D

V,
=:> _JjBCD ^ " H C D A
V
V
ABCD

I

HA'

HB' H C

^

~r-~-i

+

AA'

^

a ' + b ' + c ' + a-' + b'' + c'' < 1 6 R '

^

.
^ v^HDAB
, V
" H. A B C ^ _ .j
Y/VBCD

''ABCD

^
=> 1 6 0 G ^ = 4R^ + S(2R^ - A B ^ ) , v6'i S la t6ng theo 6 canh
= 1 6 R ' - ( a ' + b ' + c ' + a-' + b'' + C ' ) > 0

2

BB'

HD' ,
+

CC

=1

DD'

AA,

BB,

CC, DD,

AA'

BB'

CC

DD'

YftBCD

A'A, B'B, C C
1+—Hj

AA

BB'

1+

CC

D'D

1-2

DD'~ M

^ A..

'


W trgng diem bSi dUOng hqcsinh

gl6i mon

Toan

1£

Lc Hoanh

L i t / iivnn mi v uvvn

'^hd

Theo bit ding thCpc B C S :
AA'

BB'
•

_AA,

+

—

BB,

CC'
+

DD'
+

CC,

DD;J

AA' BB' C C
AA, ^ BB, ^ CC,

AA'

BB'

CC'

DO-

NAhj^r)^

>16

DP' ^ 8
DD, " 3

A"i,A'2,A'3,A'4.Chu'ngminh:

i=1

; XGA,6.

i=i

n2

Dod6: i G A , < - l ZGA,
i=i

4

-14

4

4

(ab + bc + ca)

-fr.

•- ,-

r,i,r

Sxq(A'.AB'D')

—

1

..

Gpi G Id trpng tdm tip dien GA + GB + GC + GD = 0
3
3
3
1
vd GA = - ma, GB = - mt, GC = - mc, GD = - mj
4
4
4
4
Ta CO : AR^ = OA^ + OB^ + OC^ + OD^

Theo bat ding thu-c BCS :

y—^-XGAfi—
ttGA,

4tt

'ttGA,

IGA, <(R2 - 0 G 2 ) i - l - (dpcm).
i=i
i=i ^ ' ^ i
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
di$n A'AB'D'.Chung minh;

f

=>4R^>A(,^^+m^m^m,^)

«

tLK

1
—

3R'

= 40G2 + GA2 + GB2 + GC2 + G D ' + 20G(GA + GB + GC + GD)
= 40G^ + GA^ + GB^ + GC^ + GD^
GA^ + GB^ + GC^ + GD^ < 4R^

i=i

Vd 4XGAf > IGA,

3V

^(AB'D') _ Sxq(A.A'B'D')

2 ab + bc + ca 2
V(h-r) 2
= —.
< — => —^
<—
3R2
3 a^+b^+c^ 3
R2j.h ~ 3 '
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
Suy ' ra

BDT« t G ^ < ( R ^ - 0 G f t - l i=1
i=i '^'^i
GAf\ = /->A2
OAf + OG' + 20AiOG= R^ + OG' + 20G(GA. - GO)

n2

,2 3.V
R2.

_ ^tp

Ti> dien A'AB'D' vuong tai A' nen R = V a ^ T b ^ T c ^
^

^ _ 1 _ < ^ ^ .

Hifang din giai
Gpi O va R Id tdm vd bdn kinh cua m$t c^u (S).
Ta c6: GAiGA', = R^ - OG^

i=i

^tp

=

vi^

3.V

^(AB'D')

vo'i Sxq(A A'B'D')

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
'

3.V

V

nnang

mf + m^ + m^ + m^ > -1 (ma + mt + mc + md) ^ => (Jpcm.
^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 - .
r

Himng din giai
OA = a, OB = b, OC = c.

X^^zll < i .

R2.r.h 3
Hipang din giai
D$t AA' = a; AB' = b; A'D' = c. Ta c6

Tac6:l. = -L.-1, vd r = 3V
h^

l\/Id

a'

tp

-^'P - ^AOAB + S^oBc + S^oc^

3V

3V

r

3V

+ S^gg

1 1 1 1

a

b

c

h
181

.


diem

TDtri,ina

^.

hfli

1

JUcmq

1

man JOOm^ -

hoc '^inh qiol

••- "nnnil

riiu

1

t

Do do

=- +- + r h 2a b c
r 1 ^1
(^ 1 1^
<3
Ma
1

h—
[a
b 0^
y ^b^

0

n6n

+ —

l f

^1

1

ta

b cj

3

y'

1 1 1 N/3

< — = > - + - + - <

a

b c

0
+

1

V2-1
0 _

—

h

y

Do ( I 6 - - - < — = ^ - < - ( 1 + V3) .V#y - < 1 + N/3..
r h
h
r h
r
ii.».t.

h

Vay gia th Ian n h i t cua -

1^ 1 + Vs khi OA = OB = 0 0 .

r

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

tfs6-.

^ •

R

•

12'

a

„

2 + tan

2

4 tan a .
R

A„-2(t-t^)

Xet ham so y =
^

.

.

1 + t'

^j^^^,,,,,

^

Trong mpi tam gidc a, b, c, dipn tich S thi:a^ + b^ + c^ > 4 V3 S
2(AB^ + AC^ + A D ^ + 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:

+ ( O D - O B ) ' + ( O D - O C ) '
= 1 6 R ' - ( O A + O B + O C + O D ) ' = 1 6 R ' - 1 6 0 G ' < 1 6 R ' = 16,.

_8_
V3-

a

Diu ding thu-c xay ra khi va chi khi AB = BC = CD = AC =AD = BD vd O = G.
Do do ABCD la tLP di^n d^u.
n e n r = IH = g t a n 2

tan^^-tan^^

2 _,

4 + 2tan^a

Hu-ang d i n giai

= (OB - O A ) ' + (OC - O A ) ' + ( O D - O A ) ' + (OC - OB)^

4 tan a

,^

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.

Do do S,p <

I la C h a n d u - a n g p h a n g i a c c u a g 6 c S M H

a
t a nI—
—

= V 2 - 1 khi a = 2arctan V V l ^ .

A B ' + AC^ + A D ^ + BC^ + BD^ + C D ^

4h

Do h = - t a n a => R = a.

Do do; - =

#
/ / 11

a^+2h^

a

R2 = (h - R ) ' +

R

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

HiPO'ng d i n giai

Xet hinh ch6p ILF gi^c deu S.ABCD c6
canh day a, duang cao h. Gpi a Id g6c
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^
^

V^y max

1 + tan^

2

v6it = tan^2 -^^(0:'')

2 ( - t 2 - 2 t + 1)

^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 d i l m thoa
(Ji4u ki^n PD^ = PA^+ PB^ + PC^
HiPO'ng din giai

Qpi O la diem sao cho O A + OB + OC - OD = 6

«

"•"a CO PD^ = PA^+ PB^ + PC^
^ ( 0 A - 0 P ) 2 +(OB-OP)^ + ( 0 C - 0 P ) 2 -(OD-OP)^ - 0

-

^20P2 - 2 0 P ( 0 A + 0 B + 0 C - 0 D ) = 0D2 - ( O A ^ + 0 6 2 + 0 0 ^ )
^ 20p2 = OD^ - (OA^ + OB^ + OC^)

y" = 0<r>t = - 1 ± V 2 , c h p n t = V 2 - I .

(1)

(2)

^'fih phuang 2 ve cua (1), ta suy ra 20D^ - 2(0A^ + OB^ + OC^)
* DA^ + D B ^ + DC^ - (AB^ + BC^ + CA^)

(3)

4. '-t.,'"


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
a" + b" > 2

a +b

I

n

, c" + d">2

^c + d T Va, b, c, d > 0

2 ,

a" + b" + c" + d" >

a+b

c+d

>4

fa+b+c+dT

Lly a = MA', b = IVIB^ c = MC', d = MD', n = 1002, ta c6:
f(M) > 4'-" (MA' + MB' + MC' + MD')"
MSt khac : MA' + MB' + MC' + MD'

Ctj/ TNHHMTVDWH

' ^ f i l vdo khai triln tr6n ta du'P'c: A B ' + AC^ + AD^ - B C ' - C D ' - D B ' > - 4 R '
ping thuKC xay ra <=> O B + O C + O D = OA
^ A B + A C + A D = 2AO - AA' (vdi AA' la du-dyng kinh)

^ hinh hpp A B D ' 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
d^n m hinh tru : 01 = x.
0
0^y hinh try la duang tr6n c6 bdn kinh:
; ^

Sxq = 27:r.2x = 4nx

Diu '•=" xay ra khi va chi khi x' = R ' - x'

x=

b) Thi tich cua kh6i try la:

V = -6KX' + 2TtR', V = 0 o

Khai triln (OB + 0C+

OD-OAy

>Oia c6:

O B ' + O C ' + OD' + OA'

+ 2(0B.0C+0B.0D-0B.0A+0C.0D-0C.0A-0D.0A);
md 20B.OC = OB' + OC' - BC' = 2R' - BC'
tu-ang ty v6i O B . OD , O B . OA

OD. OA

V

, .x-+(R--x-) ^
,
< 47t.
^
^ = 27lR^

= 4MG'+ GA'+ GB'+ G C ' + G D '

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:

X

= 47rVx'(R'-x')

V = 7tr'.2x = 27r.x(R' - x') = -27rx^ + 27tR'x. 0 < x < R

MA^+MB'+MC'+MD'min O M ^ G

MA=MB=MC=MD

^' Tfi

r = 7R' - X' , 0 < X < R
Dipn'tich xung quanh hinh try la:

= (MG + G A)' + (MG + GBf + (MC + GC)^ + (MG + MD)'

f(M)min o

Hhong Vl^t

X =

R

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 l l n 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 ' -

Suy ra Al = V X ^ ? ^ =
IO =

=
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 t i l p (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

aV6

T h I tich cua IV = Oi.Sa = TiiK^IO
=

,

r-

naV6^2

-7i>/8x+——-X

a>/3
Ta xem V la hdm s6 theo biln s6 x, v6i x e (0; — )

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

Khido

SI'
- - ^

V =-3n>/8x^+4:^aV6x;V=0«x =
BBT

X

0

V

+

Do do

6

9

-

0

R - r _ h - S r _ I'l
R

nen 1'! =

h

~

~h

h(R-r)
R

V

1
h(R-r)
Taco: V ( H ) = -TiR^h. V(H') = Tir^
v3
R

1^
V j y V ast giS tri I4n nhjt khi va chi khi x =
"I"

• •

. ,

, ,

V

^ ^ ^ J K _ ^ 3 hay(P)iOAtailsaocho — =
SO OH 2
SO 2
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
nenR.h = { R + Vh^'+R^K
=>R(h-r)= V h ^ R '

r

Do do R ' ( h ' + ^ - 2hr) = ( h ' + R ' ) r ' =^ R '
ThI tich hinh n6n: V(h) = ^ TtR^h =

3

=

h-2r

• ^TTij:' ^ ^

Dodo

V, II')

3r-(R-r)

V,(II)

b) ThI tich cua (H') Ian nhat khi r^(R - r) Id Ian nhdt. Xet hdm s6 f(r) = r^(R - r)
vaiOTa CO f '(r) = 2Rr - 3r^ , f '(r) = 0 <^ r =

2R

9"

2R

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 d i l m 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.


10 trqng diS'm bSi dUOng h
Himng din giai
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:

'

^

x2

0 0

BBT;

X

y'
y

= X «

4V2

(vi X > 0).

a

0
+

0

r-..,f

A' = (1-3)k^-(1+6k)>0 » k > J .

,
, , Va^-x^-x
Xet y = x+ Va^-x^ vpi 0 ,
^y' =

Z

-

,

Phu-o-ng trinh ndy c6 nghi$m, ta phai c6:

2a

4Ttax

^, ^ , '
;' ,
-f •

£)|t— = k ta c6 phu-ang trinh: (1 +6k)s^ + 2(1-3k)s + 1=0

52 = 27tOA.AB + 27:OA^ = 27:xVa' - x * + 27iX
_S, _ 27tx' + 2 u x V a ^ x ' _

7xr''(l + sina)'
7i.r'(l + sin a)'
5— = —
;
3sma.cos a
3sma(l-sma)
f hi tich hinh try ngoai tilp hinh clu Id V2 = 27t.r^
(1 + sina)2
(1 + 3)2 ^.
. „
,
no d6
= — r — - — — r - r - T T ^ v6i s = sina.O < s < 1
^
V2 6sina(1-sina) 6s(1-s)

"TJO d6 Vi - —

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

b) Ta c6:

_

lii-n.

V
4
1
Vay gid trj nho nhat cua — = - i>ng vb-i s = sina = - vd OB = 3r.
V,
3
3
3. BAI LUYfiN TAP
Bai tap 14.1: Tim t|p hp-p cac diem M:
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.
Vgy ti c6 di?n tich — dgt gia tri Ian nh§t khi OM = 2x = a V 2 .
Bai t|ip 14. 2: Cho dilm A a ngoai m|t cau S(0; R).Mpt mgt phing bit ki di
81

qua AO, cit mat clu theo mOt dud-ng trbn (C). Gpi AH la mOt tilp tuyin cua
Bai toan 14. 47: Cho mpt hinh cSu npi tilp trong mpt hinh non trbn xoay M^'
(Ju-ang tron do tai H.
hinh try ngogi tiep hinh clu do c6 day du-ai n§m trong mSt phing day ciia
a) Chu-ng minh ring AH cung tilp xuc v&[ mSt clu tai dilm H.
hinh non. Gpi Vi, V2 lln lu-p't la thi tich cua hinh n6n va cua hinh tru Tim
b) Tim quy tfch cac tilp dilm H.
X
V
Hirvng din
gia trj nho nhSt cua tf so —
B
^
3) DCing AH Id mpt tilp tuyIn cua du-dyng tron (C) tgi H.
Hirang din giai
^) Kit qua du-b-ng tron giao tuyIn cua m^t clu va mp(P).
Gia su- hinh non c6 du-ang cao BH = h, ban
^ ' tap 14. 3: Cho tam gidc can ABC c6 goc BAC = 120° va du-ang cao
kinh d^y la DC = a, g6c giua duang sinh v^ \
' 0
'*^H = a V2 . Tren du-ang thing A vuong g6c vb-i m^t phIng (ABC) tai A lly
tryc IS a; ban kinh hinh cau npi tiep hinh non
^ai dilm I vd J a v l hai phia cua dilm A sao cho IBC Id tam gidc diu vd
•, 1^ r. .
.
"^^C la tam giac vuong can.
Tiha'
^) ChCeng minh ring BIJ, CIJ la tam gidc vuong
Ta c6: Vi =
1^) Xac djnh tam va tinh theo a ban kinh cua m|t clu ngogi tilp tu- di$n

j ^ ^ ^ _ r ( 1 + sina)^
r(1 + sina)^^^^
iJBc.
,:
M(3\ = OB + OD =
sin a
sin a
sin a


nOtrQng

diem boi dUcrng

hQC Sim

giOl

man

roon

TX - L B nuunii

ng Vi^t

rinj

Hipang 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
^ jj^n
hinh vuong canh 2R.

a) Dung d u a n g t h i n g vuong goc v a i m$t p h l n g .
b) K e t q u a R = 3a.

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

.

Bai tap 14. 4: Cho tu- dien SABC c6 SA 1 mp(ABC). (SBC) 1 (SAB). Cho bi^,

SB = a V 2 , B S C = 4 5 ° .
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<

^

day R- K i t qua Sxq = 47tR2; S,p = STIR^; V = 2%R^
t,) K i t qua VuT = 4R^

,f^,. ,

gai t?P l^- 9: Cho hinh non c6 goc dinh 2 a . 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.
Hirangdln^-^^^'*-'^*^^^^'®*
TSm mgt c l u ngoai t i l p va tam mat c l u npi t i l p hinh non la tam d u a n g tron
ngoai t i l p va tam d u a n g tron npi t i l p t h i l t di^n qua tryc.

K •"

Bai tap 14. 5: Cho tCp dien A B C D c6 mat c l u npi tiep (I, r). Cac mat p h i n g ti^p
xuc v a i m$t cau do v^ song song v o i 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.

cot 4 5 ° Kit qua

..x>< -

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

Dung ti s6 dien tich, ti s6 t h i tich cua cac hinh dong dang.
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
khacA.
a) C h u n g minh t u dien SABC chf c6 mot cap d6i dien vuong goc v a i nhau
b) Xac dinh tam mat c l u ngoai ti§p t u dien SABC. Tinh the tich mat cau
(SBC) tai v a i (ABC) mot goc b i n g 60°.
Hu'O'ng d i n

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

a) T u dien SABC chi c6 mot cap d6i dien SA va BC vuong goc vd'i nhau.
b) K i t qua V =

36
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

Dung b i t d i n g t h u c A M - G M .
[^^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 l u g t la ban kinh mat
cau ngoai tiep va mat c l u npi t i l p cua hinh chop do. T i m gia trj Ian n h l t cua

tls6 ^

'

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

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 M O N O ' c6 thI tich
t nhlt.

^' a) Xac dinh khoang each

Hu'O'ng d i n

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 v a i m$t trung t r y c va mgt
f^fian giac t u a n g Crng.
. ,
'

tqua

R.-\f3

vai ON goc 90°.

npi

tilp trong t u dien A B C D dat gia tri nho n h l t .

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

b) K i t qua O M hp'p

y

-

dien A B C D biet r i n g DA =BC, DB = CA, DC = A B va ban kinh m$t c l u

Hu'O'ng d i n

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

,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

a) Xac dinh tam va ban kinh hinh c l u ngoai tiep tif dien S A B C .

b) Tinh tan cua goc a = A S B

Tinh t h I tich khoi ISng tru t u giac d§u npi t i l p hinh trg.

.

1+

cos^
«.A. [OA

..:


W trqng diSn* b_oi_ duang

cnur^n

HQC sinh giSTmSn

loQn

I a - le noann

rno

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

f

T h i tich hinh ISng trg A B C . 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 v e c t c p h ^ p t u y l n

1 . KlfiN T H I J C T R Q N G T A M

p h i n g (Q) c6 v e c t c phSp tuyen n ' thi

O i 4 m v a s^cXo
^

is

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

Ba v e c t a do-n vj i , ] , k tren 3 tryc Ox, Oy, Oz :
i=(i;0;0),

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

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

kXo

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

1-k

-(Oxy)la

Izol;

(Oyz) 1^

|xol;

z = z^.1--kkz .

d(Mo, P) =

'AXo^yo^C^o^DI
VA^+B^+C^

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

Hai vecta: u = (x,y,z) va v = (x'.y'.z') t h i :

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

u ± V = (x ± X ' ; y ± y ; z ± z ' ) ; k u = (kx; ky, kz)

[u;v] =

[AMo

I = Vx^ + y^ + z^

y

z

z

X

X

y

V y'

z'

z'

z'

x'

y'

cos(u,v) =

(Ozx) 1^ |yo

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

1-k

u

- 4

sin(d, (P)) = |cos( u , n )|

D i l m M chia doan t h i n g A B theo ti so k ^ 1:

u . V = XX' + yy' + z z ' ;

d
n>

d CO V T C P u v ^ (P) CO V T P T n

AB = ^ { x 2 - x , ) ' + ( y 2 - y i ) ' + ( z 2 - Z i ) '

MA = k M B o

d'

cos(d, d') = I cos( u , v ) I

AB = ( x 2 - x i ; y 2 - y i , Z 2 - z i )

X-

• 'in 1 •

d c6 V T C P u va d ' c6 V T C P v thi

j=(0;i;0), k=(0;0;l).

Hai diem A ( x i , y i , Zi) va B(X2, y2, Z j ) t h i :

nu..;(

n vS m^t

c6 V T C P u = A B thi d(IVIo, d) =
u
Khoang e a c h giira hai dipo'ng t h i n g c h e o nhau:
di qua M l v ^ c6 V T C P u i ;d2 qua M2 va c6 V T C P U2 t h i

x.x'+ y.y'+ z.z'

.M^Mg I
d(d,,d2) =

-

3 v e c t c a , b, c dong p h i n g :

-

3 v e c t c a , b, c khong dong p h i n g :

ill hJ

[ a , b ].c = 0
[ a, b ] . c

0

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

, , ::/V cr. iV'

•^It p h i n g qua Mo(xo,yo) v ^ v e c t c phap tuyen n = (A,B,C)
Di^n tich v a t h i tich
Di$n tich tarn giac ABC:

The tich ti> di^n ABCD:

Ax + By + Cz + D = 0, A^ + B^ + C^;^0
S = ^ I [ AB, AC ]
V =

I [ A B , A C ]. A D I
6

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

j^ay

i:U,0;tK,

'

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 v e c t c chi

Phucng

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