Tuyển tập 500 bài toán hình không gian chọn lọc phần 4 nguyễn đức đồng

a/ Chijfng minh rkng nhi dien canh SB tao bdi cac mat phang SBA va SBC la mot n h i dien vuong.. b/ Goi I la trung diem cua SC, H la hinh chieu vuong goc cua I len CM.. Chiifng minh rang

V

A,

B

& c un

g ben so v&

i (D )

Dimg A' do

i xiJng

vd

i A

qua (D)

Lu

c do : A' va B d k ha

c be

n so

v6 i

(D ), ne

n tr

d ve t rU cf ng h op t re

n :

MA + MB >

AB <=>MA' + MA ^ AB

mi n(

MA + MB ) = m in (M A' + M B) = AB

ttf on

g Ofng :

M a M

Q = ( A' B)

o (D )

O Ke

t lu an : va

y tr on

g mo

i tr if dn

g hg

p ta x dc d in

h dif ac M t ho

a yc bt

• Xa

c di nh d ie

m

m tr en d Uo Tn

g th Sn

g (d ) de | M

A

- MB |

Ti fa ng ti

T c an p ha

n bi et h

ai tr Uc fn

g ha

p :

V

A,

B

& c un

g ben so v&

i (D )

|M

A

- MB

|

<

AB

ma x|

MA ~ MB

|

= AB

M

Mo

(d)

tuan

g lin

g M = MQ = (A

B) r^

(D)

V A,B d k hd

c ben so v&

i (D )

|M

A

- MB

|

<

|M A'

- MB

|

^ AB

Vd

i A' la hin

h do

i xijf

ng cii

a die

m

A qu

a (d), t

hi A'

va

B

a cun

g phia (D)

|

= m ax JM A'

- MB

|

= AB

ta on

g tjfn

g M

s M

Q = (A 'B

) n

(D )

t luan : va

y tron

g m

oi tri/dn

g hap ta d

a xa

c din

h die

m

M thoa ycb

t

n GlA

l T OAN T

HI

Ba

i 39

3 (D AI H OC K HO

I

A MI EN B AC

- 19 72 )

Cho mo

t kho

i t

il die

n ABCD

a/ Mo

t ma

t phan

g son

g son

g vdr

i canh B

C ca

t c

ac can

h A

B, AC, DC, DB

d c^c die

g min

h rSn

g

t uT gia

c MNP

Q la m

ot hinh thang pha

i thoa man die

u ki§

ot hinh bin

h han

h ? la m

ot hin

h ch

a nha

t ?

b/ Cho bie

t c

ac goc BAC, CA

D, DA

B la vuong, co

n BC

D la m

ot ta

m giic deu canh

a kho

i tijf die

n th

eo

a

c / Cho bie

t BC

D la m

ot ta

m gia

c deu canh

a

va c6 ta

m la die

m

O

Tin

h doan O

u ngoa

i tie

p til die

n ABCD nha

n dudn

g tro

n (BCD) lam m

ot ducfn

h mat cau trong trUdng hap ay Xa

c din

h v

i t

ri cii

a din

h A tre

n mat ca

n ABCD

I6 n

nha

n (BCD)

=>

MN //

P

Q

Va

y thie

t di6

n MNPQ la m

ot hin

h than

g (ycbt)

Muon cho M N P Q la h i n h b i n h h a n h ; tifang t u

tren ta phai c6 t h e m dieu k i e n N P // M Q , (P) // A D

Vay dieu k i e n de M N P Q la h i n h chff n h a t la BC 1 A D (ycbt)

b/ Tuf dien A B C D la tuT d i e n v u o n g d A

AB = A C A D = ^ = ^

2 2 Vay the t i c h k h o i tuT dien A B C D la :

V= - A B

c) De y dUcfng t r o n (BCD) la m o t dudng t r o n I d n cua m a t cau ngoai t i e p tiir d i e n A B C D va c6

0 la tam ciia t a m giac B C D canh a, nen t a m O cua t a m giac deu B C D cung c h i n h la t a m cua mat eau ngoai t i e p tuf d i e n A B C D

suong mat day (BCD)

12

3 0 1

i 394 ( DA

I HO

C KH OI A

- MI EN B AC

- 19 74 )

Trong m

at phln

g (P), ch

o hin

h vuo

ng ABCD c6 canh bin

g

a

Tre

n dUdn

g thin

ng goc v

di mft

t phin

g

P, ngucri

ta lay m

pt die

m

S tiiy y, r

oi difn

g m

at ph

n lira

t t

ai B',

C, D'

Bie

t (Q) i SC

a / Chu

T ng min

h rSn

g S

B vuon

g g6c AB' v

a S

D vuon

g g

oc vd

i AD'

b/ Tim c^

c qu

y tic

h ci

ia B',

C, D' kh

i S chay tren A

x

c / Xac din

h v

i t

ri cua

S tre

n Ax sa

o cho hin

h ch6p C'ABC

D c

6 t

he tic

a /

Ta c6 :

3

AB'

=>

CB

1 AB'

Ma

t kha

c t

a c

6 : S

C _ L AB' (v

i AB' n^

m trong mp(Q)

ma

S C

1

Q)

Do d6 AB' 1 (SBC

ChiJn

g min

h hoa

n to^

n tUcfn

g t

if

ta c6 AD' 1 SD (dpcm)

b/

De

y :

B' e (S AB )

AB

^

= 9 0°

; A , Bc

o din

h

= > Vay qu

y tic

h ciia nhiJng diem B'

la dudn

g

tro

n (trong m

at phln

g (SAB) dtfd

ng kin

h A

B

(Doc

gia ti

f lam pha

n dao)

• Taan

g tii ta c6 qu

y tic

h ciia nhufn

g die

m D'

trong m

at phin

g (SAD) difd

ng kin

h A

D (ycbt)

• Tucfn

g t

u t

a cun

g c

6 qu

y tic

h cua nhCifn

g die

m

C

la dudn

g tro

n trong m

at phin

g (SAC) dadn

g

kin

h A

C (ycbt)

0 /

Ha CO

1 AC

v

a

thay O

C //

SA

; SA

1

P n6n :

V

= VC 'A BC I)

=

— S

AB CD - OC '

Vi hin

h c

h U

nha

t ABC

D l

a c

o din

h, nen th

e tic

h V se

I6 n

nha

t k

hi OC la Id

ng kin

h A

C

Vi va

y OC se Id

rn nh

at kh

i n

o

la ban

ai

vi tr

i Co'

va Co"

d

oi xufn

g v

di nhau qu

a A

C cung thoa man tinh ch

at d6

:

C'oC'o

± AC

ta

i O

; C'oC'o

c (SAC)

Kh

i

do OC = O

A

= O

C

= hay n

ay ta

m gia

c vuo

ng SA

C

co OC la difd

g lin

g d

o d

o : A

S = 20C = a-s

^

Vay kh

i S

(nam tren Ax) eac

h A m

Ot doan

a

^/ 2

(c

o ha

i

vi tr

hi hin

h chop C'ABC

D c

6 t

he tic

h Idn nha

t,

va th

e tic

Bai 395 ( D A I H O C Y - N H A - D L f d C - 1976)

Cho h i n h vuong A B C D canh a Goi SA la doan thSng goc vdri mSt p h ^ n g ( A B C D ) SA = a

vk M la mot d i e m d i dong t r e n doan SD DSt S M = x

a/ Mat phang ( A B M ) cat doan SC t a i N Chitog m i n h tiif gidc M A B N l a mot h i n h t h a n g vuong

i 39 6 DA

I HO

C BA CH K HO

A

- TO NG H OP

- 19 80 )

Tro

ng khon

g gia

n c

ho ba ti

a Ox, Oy, O

z tifn

g d

oi mo

t tao

w di

nhau m

ot goc

z lay Ia

n lucf

t c

ac die

m A, B, C sa

o c

ho : O

A = a, O

B = b, O

C = c

1 /

a,

b,

c pha

i tho

a man h

? thil

c g

i d

e tam gi

ac AB

C c

6 g

oc

A vuo

ng

? Hay tim

g buoc b, c, a de tim dU

c fc

a tho

a man h

e thuf

c ay

h (

0 <

a

< 90°) v^

b =

h a d

e tam gi

ac AB

ri Id

n nha

t a

y cua go

c A bang bao nhie

u

3 / Vd

i c

ac gia thie

t cua 2 Hay tinh th

e t

ich

cu

a tiif die

n OABC uf

ng vd

i g

ia tr

(1)

(2)

(3)

Gia

i

1) AABC vuon

g t

ai

A

» BC

^ = AB^

+ AC^

(1)

Din

h l

y ham cosin tron

g c

ac ta

m giac :

AOA

B, AOB

C, AOA

C c

ho :

AB'^

= a

^ + b^

- 2abcos

a

BC

^ = b^

+ c^

- 2bcco

sa

AC

^ = + c^

- 2accos

a

Thay (2), (3) v

a (4) vao (1)

(1)

< => b^ + c^

- 2bccos

a = 2a

^ + b^

+ c^

- 2a(

b + c)cos

a

< => a^

- a(b + c)cos

a + bccosa = 0

< => g(a)

= a

^ [(

-b + c)cosa]

a + bccosa = 0 (5)

De tim dt

f cfc

a th

oa ma

n (5)

< => A = (b + c)

^c os^

a 4bccosa >

0

< => A = cosaKb + c)^cosu

- 4bc]

^ 0 (0 <

a

< 90°

=>

cos

a >

0)

< => (b + c)

^cos

a 4b

-c ^ 0 (6

) (ycbt)

(6) dieu kien ca

n v

a d

u rang bu

oc

b,

c v

a a de tim difac

a th

oa ma

n (5)

21

X

et gia thie

t : b =

c

< =>

AOAB = AOAC

Go

i H

B 1 O

A

« CH 1 O

A

^ BH = C

H

Xet h

ai ta

m giac ca

n ABC va HBC

; chu

ng

c o can

h chu

ng BC

JA

B

^ HB

^ J A O H

C

jO

H = b cos

a

BC'

^ = OB'' + OC' - 20B.OCcosa

<^

BC' = 2b'

- 2b'co

sa

AHBC BC'

= HB' + HC'

- 2HB.HCcosH

BC' = 2b'sin'a

- 2b'sin'acos

H

So sanh (7) v

a (8) th

eo

ve :

= > 2b

^ 2b^

cos

a

= 2b^

sin^

a

- 2b

^ sin^

acos

H

=>

1 cosa = sin'a

- sin'acosH

gA

C

< B Ht;

=>

m axB AC = B Ht;

ti rcJ ng iln gA

sin^acosH = sin^u + cosa - 1 = cosa - cos^o

cosH = c o s a d - cosa)

cosa.2sin^ —

s i n ^ a

cosa 4sin —.cos — 2

Bai 397 (DAI HOC BACH KHOA - TONG HCfP - Y - N H A - D L f d C - 1982)

Tren canh AD ciia hinh vuong ABCD canh a, ngudi ta lay diem M v<Ji

AM = X (0 < X < a), va tren nijfa dacrng t h i n g Ax vuong goc t a i A vcri mat p h i n g cua hinh

vuong, ngUdi ta lay diem S \6i SA = y (y > 0)

a/ Chijfng minh rkng nhi dien canh SB tao bdi cac mat phang (SBA) va (SBC) la mot n h i dien vuong

b/ Goi I la trung diem cua SC, H la hinh chieu vuong goc cua I len CM T i m quy tich cua H khi M chay tren canh AD va S chay tren Ax

c/ Tinh the tich hinh chop S.ABCM

d/Vdi gia thiet x^ + y^ = ai^, t i m gia t r i Idn nhat cua the tich hinh chop S.ABCM

Mat khac I H 1 CM => OH 1 CM (dinh ly 3 difcrng vuong goc)

Vi M e (AD) va S G Ax nen H a trong iCCt) Vay H 0 tren cung tron O H Q cua dUcfng tron dUdng kinh OC HQ la trung diem cua CD, khi M e AD va S e Ax

• Dao lai, lay mot diem H bat ky tren cung OHQ, ta c6 : O H 1 HC; CH n AD = M , tren nOfa dudng t h i n g Ox' // Ax lay mot diem I sac cho CI c i t Ax tai S Ro rang :

* I la trung diem cua SC

* I H _L CM (dinh ly 3 dUcfng vuong goc)

Ket luan : Quy tich H la cung tron OHg cua dudng tron dudng kinh OC trong mat p h i n g (ABCD) (xem hinh) (ycbt)

305

c / Th

e tfc

h hin

h chop

V(SABCM)

=

- dt(ABCM)

=

- - (A

M + BOAB.S

= (x + a).a.y (y

d/ Xet :

x ^ + y

^ = a^

- x^

Ta

C O

; max

xa

y

ra

maSV

* = — (x + a)(

x + a)(x + a)(3a

-

Ap dun

g BDT Cauchy ch

o

4 so' khon

g a

m,

ta c6 : (1)

(1)

<=>3 V

^ <

36 (x +

a)

+ (x +

a)

+ (x +

81a'

36.3.1

6

64 3a^

„

V 3

g thiif

c trong (2) xay ra <=>

a + x = 3a

- 3x

< =>

Do d6 kh

i M la trun

g die

m A

D t

hi th

e tic

h VSABC

M ci

T c da

i va

73

2

maxV =

a (ycbt)

8

Ba

i 398 ( DA

I H OC B AC

H KH OA

- TO NG H CJP -Y -NH A-D l/

dC

- 198 3)

Trong khong gia

n, cho hin

h chop S.ABC

D, day ABC

D

la hin

h ch

a nha

t v

di AB

h chop vuong go

c vd

i day, A

S = 2a

a /

M l

a m

ot difi

m tre

n can

h A

S, vd

i A

M =

x (0

^ x <

2a)

Ma

t ph^n

g MBC cM

t dien g

i ? Tin

h die

n tic

h thie

t die

n a

y the

o a, b, x

hi

X

ac din

h x sa

o cho thie

t die

n tre

n c

6 dien tich Id

n nha

t

c / Xac din

h x sa

o cho ma

t phSng (MBC) chia hin

h ch6p ra h

ai phan c6 th

e tic

m cu

a ma

t phin

g (MBC) vd

a B

C //

A

D

Ma A

D = (SAD

)

n

(ABCD)

^ (MB

C)

n (S

AD)

= M

N // AD //

B

C

Ha

n nfla

vi BC

1

AB v

B C

1

MB

Than

h thi

jf thie

t die

n MBCN

\k

m

gt hin

h thang vuong

(6

= 1

^ = 9 0°

)

B

=> fix) = 2(x - 4a)(x'^ + a^) + 2x(4a - x)^

^ f (x) = 2(x - 4a)(2x^ - 4ax +a')

DiTa vac b a n g b i e n t h i e n n^y t a t h a y :

Vay amaxS^ o 3maxS = — ^ 7 1 + 8 ^ 2 k h i va chi

8 khi: X = A M = ^^^iJ^ (yebt)

d Hien n h i e n h i n h chop S.ABC D c6 th e t i c h :

V = i A S A B A D = —

3 3

De t i n h the t i c h V cua h i n h iSng t r u t a m giac cut M A B N C D , t a difng m a t phSng ((3) qua N

1 vuong goc vdi BC; t h i m a t p h i n g (P) cat A D va BC I a n lifat t a i K va L , (P) chia l a n g t r u

I cut thanh h a i p h a n : l a n g t r u t a m gidc dtifng M A B N K L c6 the t i c h :

h

th

il ISn

g tru cut MAB

NC

D c

6 th

e tich :

V =

V

i + Va = - a

bx 6 3-2a

J

Yeu cau b

ai toan can xac dinh

x (hien nhien

2a )

- 6a

x + 4a^

= 0

Phiicfng

trinh

nay c6 nghie

m :

X =

a

(3 +

V S)

c6

t he c ho

Ba

i 399

(DA

I HOC K

T T

H SP

- NN

- 1983)

y AB

i A, AB = 2a, AC = 3a, can

h S

B vuon

g

g6c vd

i da

y S

B =

a/ Ch

i r

o tar

n va ban kin

h ma

t ca

u ngoa

i tiep

tijf

dien SABC

b/ M l

a mo

t diem d

i dong tre

i H va

K Ia

a M len cac ma

t phSn

g (ABC) v

a (SAB)

Mat

phln

g KMH

g : KMH

L la m

ot hinh chi?

nhat V

di gi

a tr

i na

o cu

a x t

hi KMHL la mot

hi

nh

vuong

c/ Tinh theo

a v

a x do d

ai dudn

g che

o ML ciia hin

h

cha

nha

t KMHL Vd

i nh

o nhat

i d

a ti

m difd

c cu

a x, ha

y ne

u le

L

d/

Ha

y tinh theo

a v

a x the tic

h V cua hin

h cho

p din

h

A, da

y PCMHL K

hi cii

a ham

V kh

i M d

i don

g tren canh

SC

e/ Xa

c dinh

I Ia

n lifa

t l

a trun

g diem ciia SC va

BC Ca

c ta

m gid

c SAC, SB

ai

A, B nen t

n (ABC)

|m

a Ol

a trun

g diem

S

C

OA = O

B = OS = O

C

Vay

O la tam cua mat cau ngoai tiep

h ma

t ca

u la :

R =

O

A = — = 2a 2

Th

at va

= SB

^ + AB

^ + AC

^

^ SC' = 33^

+ 4a^

+ Qa

^ = IGa

^

=> SC = 4

a =

> R = 2

a (ycbt)

b/ KMHL la hinh chuf nhat

M K ± (SA B)

A C 1 (SA B) '

M H 1 (A BC)'

SB 1 (A BC) J

Vay tuT giac KM H L la mot hinh binh hanh

De y den SB 1 AC => H L 1 LK : nhvt vay tit giac KM H L la mot hinh chff nhat (dpcm)

Dinh X de KM H L la mot hinh vuong Ta c6 :

o BS, M L, AC ciing vuong goc \6i A B

<=> BS, ML, AC cung nkm trong 3 mat p hing song song

309

'2 ' 1 3

x V3

-—

=

—x^(

x) (ycb

Sau day t

a kha

o sat sif bie

n thie

n va v

e d

o t

hi cu

a V trong h

e tru

e (OxV

)

Mie

n xa

c din

h : D

y = [0

; 4]

"x = 0 =i

> V = 0

Da

o ham:

V

=

^ (-3x

^ + Sax) = 0 <=

3 27

V"

—

6x + 8a)

(-= 0 «

X =

—

=>

V

= 4a^V

3

Va

y (C) l

a d

o t

hi cii

a

V trong ycb

t

4^

3 , a^ ~ ~ = V e ; d X nh Di e / 27

Xe

t : V = 4^

3 27

V S

2 ,.

,

4V 3

3

< =>

— X

(4

a

- x)

= ^

a

32 2

7

Rx) = x

^ 4ax^

+ a^

= 0 (3)

f

4a

= 0

nen t

a c

6 :

(3 )

O

X

=

9 8 32

o _

x'^

- -a

x a" ^

8V3a

^

27 '

4V3a

^ , (C

)

27

0 4a

3 8a

0

Vi X e [0; 4a] nen cdc n g h i ^m cua (3) la : x = 4a

OM^ + O'N^ = k^ (vdi k la m o t dp dai cho trifdc)

1/ Chijfng m i n h rSng do dai doan M N k h o n g doi

2 1 Vdi v i t r i nao cua M t r e n Od va N t r e n O'd' t h i tuf d i e n O O ' M N c6 the t i c h I d n n h a t T i n h gia t r i do theo h va k

3/ Muon M N tie'p xuc v d i mSt cau difdng k i n h OO' t h i h va k p h a i thoa m a n di4u k i e n gi ?

Neu each diTng M N t r o n g t r i f d n g hop do

Goi P va Q t u a n g iJng la h a i d i e m nSm t r e n Od doan O'd' G o i H l a h i n h chieu cua diem giaa K cua doan OO' l e n PQ H a y chufng m i n h r S n g k h i PQ t h a y d o i sao cho OP + Q'Q = PQ t h i H n a m t r e n m o t d i f d n g c6' d i n h H a y c h i r a dirdng c6' d i n h do

G i a i 1/ Chijfng m i n h doan M N k h o n g doi :

Vay k h i chon M e d va N e d' sao cho ; O M = O ' N =

Ifa nhat : maxV = — O M ' = (ycbt)

6 12

kV2

t h i t i i d i $ n O O ' N M c6 the t i c h

311

MN = ON +

OM

3/ Die

u kien

de MN tie

h OO' Go

i I la tiep diem cua

dadng

thang

MN vdfi ma

t ca

u diTcrn

g kin

h O'O The

o tinh chat tie

p tuye

n ti

T 1 die

m d ngoai ma

M

« MN

^ = O'N^

+ OM

^ + 20'N

0M

=> h' + k^

= k

^ + 20'N

0M ^ 20M

0'

N = h'

De

y : (OM

- O'N)

^ = OM^

+ O'N

^ 20M

- ON

> 0

o 20

M

ON $ OM

^ + O'N ^ (A

p dun

g (4))

< => h^ < k

^ «

0 <

h <

k (5) (ycbt)

(4)

• Cac

h dung doan M

N : Cho

M e

d m

a OM = a Ti

T (4) ta c6 : O'N =

2a

Ta tim di/oc Ned' difgc xa

c dinh bdi O'

N =

v6i

dieu kie

n (5)

Chiitng min

h H nkm tre

n m

ot dUdn

g co din

= y

PQ = O

P + O'

Q

=

X +

y

Ta

C

O :

Q

(d')

(6)

• [P

Q

= PH + H

Q

PH + H Q

= x + y

Dinh l

^ + 0P

2

= HK

^ + HP

2

KQ

2

= O' K^

+ 0 Q2 = H K^

+ H Q2

Do OK = O'

= HP

^ HQ^

= y^

-^ (H

P HQKHP + HQ) = (x

- y)(

x + y)

Tif (6) =

>

HP

- HQ = x

- y (7)

Ke

t ho

p (6) v

a (7) =

> H

P =

x v

a HQ = y (8)

Qua

O difn

g

di // d' =

> d' // (d'

; di) v

a (d'

; di)

1

(d;

di), ne

n go

i Q' l

a hinh chieu

cua

Q tre

n

(d; di) t

hi Q' e

d, v

a d

o (QQ'P)

_L

(d;

dj) nen goi H' la hin

h chie

u cu

a H tre

a hinh chif nh

at =

> OQ' = O'Q = y

H

P HP

X

"

' H

P

HQ '

Ma

HQ

OP

OQ '

(8)

HQ ' OQ '

OH' l

a phan giac

P O^

'

=

(3^73^

Dieu nay chijfng to

H na

m tron

g ma

t phan

g co' dinh (a) ta

o bdr

i OO' v

a ph

i // d'

Do : AOP

K = AHP

K ^ H

K = O

K = -

2

Vay H nftm t r e n diicrng t r o n co d i n h t a m K, ban k i n h — chufa t r o n g mSt p h ^ n g co d i n h

va C; (Q) la m a t p h ^ n g qua M va vuong goc AC

1/ Tuy theo M thuoc OC hay thugc OA hay chi ro each d i m g t h i e t dien m a (Q) c&t h i n h ch6p

2/ Bat X = M C T i n h dien t i c h t h i e t dien noi t r e n theo x, a, h K h i nao d i e n t i c h ay lorn n h a t

G i a i

1/ Doc gia tU p h a n t i c h va chiifng m i n h va bien l u a n , d day t a x e t h a i k h a n a n g xay r a t u y

theo v i t r i cua M t r e n A C = A O ^ OC

D C a c h dtfng t h i e t d i § n k h i M 6 A O

• Trong ( A B C D ) , qua M dUng di/cfng t h S n g song song

vdi cheo B D I a n lUOt gap A B , A D t a i N va G

• Qua N , M , G d i i n g cac difdng t h S n g vuong goc v d i

(ABCD) Ian luot gap SB, SC, SD t a i L , E , F t a dUcfc

thiet dien muon t i m la ngu giac N L E F G gom h a i h i n h

thang vuong bSng nhau co chung day lorn M E (ycbt)

• C a c h d U n g t h i e t d i $ n k h i M e O C

• Trong ( A B C D ) qua M dUng diidng t h i n g

song song v d i B D I a n luat cSt C B , C D t a i N , G

• Trong (SAC), qua M d u n g diTdng t h a n g song song

vdi SA cat SC t a i E T a m giac E N G can t a i E la t h i e t

dien muon diTng (ycbt)

21 De xac d i n h x = C M de dien t i c h S ciia t h i e t d i e n Idn n h a t t a x6t h a i k h a n a n g sau

S, = 2 x V 3 h x h V 3 Y 3a - x V 3

3a - h + 3a A

313

Si

=

—

(xV3

-

a) (3 a

- xV

S,

= -^

Sx

(-^

+4aV3

3a

(3)

•

TH 2

:

M e C

O : A

O

^ AM <

A

C

<

aV3

x <

NG //

D :

NM C M

2

x

ME //

S :

aV

s

ME C

M

AS "

CA "

aV

3

xV

s xVs, x

^h

NM

2xV3 a xV3

3a

2

XA

/3 ,

ME = h

3a

S2 =

S'2 =

3 3a

2h

x

ih =

:

^

(0

(4)

(5)

De

y den (3) v

a (5) va h

ai bie

u thiif

c S ],

S2

d (

2) va (4), ta se la

p difg

c bang bie

S

i

^ , ah

, 2a

V2 , , ^ cb (y = i x vo ng g iT rn tifc , — a = x f ma : CO ta o, d Tut)

0<x

<aV3

3 3

Ba

i 40

2 (T

T D AO T AO

v

a BO

I D UO NG C AN

B

O Y T

E TP H CM

1 99 3)

Cho ha

i diem

A , B

d

oi xilng nha

u qua m

at ph

ang (P ), I

la giao diem cu

ai (P ),

co hin

h chieu vuon

g g

o c xuon

g ( P)

la

H ,

co

n M

la m

ng tro

n dUcm

g kin

h IH

ve tron

g ( P)

1 / Chufng min

h rang

I M

la diid

ng vuon

g g

o c chun

h rkn

g h

ai die

m A

, B

luon eac

h deu dud

ng OM

3 / Cho

B =

2

a, MH

= x

, M

I =

y Tin

h the tich tu

f dien

O MA B.

Xac din

n nhat

Gi

ai

l/Tac6:ABl(P)

= ^ AB I

IM

(1)

M

thuo

c dud

ng tro

n difd

ng kin

h I

H

^ IM _L MH

(2 )

3/ De' y tha'y : O H / / ( A B M ) ^ VQ.ABM = ^UMIM

Cho tuf dien SABC c6 goc p h i n g d d i n h S vuong

V Chiing m i n h r a n g : Vs S A D Q ^ Sggc + SogA + S ' S A C

-2/ Biet rang SA = a; SB + SC = k k h o n g doi D a t SB = x Ti'nh the t i c h tuf dien SABC theo a, k ,

jva xac d i n h SB, SC de t h e t i c h tuf dien SABC \6n nhat

3/ Cho A CO d i n h B v a C t h a y do i sao cho SB + SC = k ( k h o n g ddi) T i m quy t i c h giao d i e m O

cua cac dUcmg cheo h i n h hop c6 ba canh l a SA, S B , SC

3 1 5

ai

1/

Difng : S

I 1

BC va n

oi AI, t

= -(

BC

^SI^

+SA

^SB^

+SA2

SC

= -[

BC

^SI^

+ S

A^

ISB^

Sl

2 +SA^

BC2)

4^

'

I

S = -[B

C^

.C SI

^+

SA2)] = -(BC2.Al2

)

= Si

Ap

dun

g BD

T Schwartz :

(S _ +S,„

'SBC

+ ^ IB

A + S|AC

)

2/ Go

i th

e tic

h tii

r d

ie

n SABC l

a V :

V =

- AS.S„„,, =

-SA.SB.SC = -ax(

k x) ^

24

X

+ k -

X

Dau

dan

g thijf

c tron

g (2) xa

y ra •» x = k

ak

^

24

; tuang ufng : •

SB = S

C = — (ycbt) 2

3/ Go

i O' la hin

h chie

u cu

a O xuong

(SBC)

Tron

g ma

t phang toa do (Sxy) = (SBC),

ta c6 :

y

=

SB + S

C

=

—=

> y = -x + — 2 2

A

Vi S

B + SC = k ^ 0 < S

Quy tich O' la khoang (BC) :

0 < X < k

De y thay : OO' = - SA = SG ; VB; C

2 Vay quy tich diem O la khoang (B|Ci) : •

Cho tuf dien ABCD c6 A B = x va CD = b, cac canh con lai bang nhau va bSng a Goi E, F

n luat la trung diem ciia A B va CD

1/ Chiifng minh rSng : A B 1 CD va EF la ducfng vuong goc chung cua A B va CD Tinh EF theo X, a, b

2/ Tim X de hai mat phang (ACD) va (BCD) vuong goc vdi nhau Chiifng minh rang khi do tiif

dien ABCD c6 the tich \6n nha't

G iai A

1/ Ti/ gia thiet : AC ± A D => A F _L CD

ma BC = BD => BF 1 CD

[ C D 1 A B Vay CD 1 (ABF) \

EF 4a' - b-^ - (0 < X < V4a^ - b ^ ) (ycbt)

2/ Theo tren CD 1 (ABF) A FS = Goc nhi dien do vuong khi va chi khi

) Id

n nh at k hi

SABF

Id

n nh at

Do

F

A

= F B=

i Vi l

^

ne n

VAHCD

I dn n ha

t

< => SAU

K =

iFA

.F Bs in Xp

^

= -(

4a

^ -b 2) si nA FB

(1)

(1)

3maxi(

4a

b^

^-)s in AF

n A Ff

Ba

i 40

5

(D AI H OC B AC

H KH OA T P.

HC

M -

1994)

Tr on

g ma

t ph Sn

g (P ) ch

o ta rn g ia

c AB

C vu on

g go

c ta

i

A, AB = c

; AC = b

g vu on

g go

c vd

i (P ) ta

i

A, la

y mo

t di em S s ao c ho S

A

=

h (h >

0

)

M la m

h SB Gp

i

I,

J Ia

n lu ot l

a tr un

g di em c ii

a B

C va A B.

1 /

Ti nh d

o da

i do an v uo ng g oc c hu ng c ii

a ha

i du dn

g SI v

a AB

2/ T in

h ty s6 ' g iO

Ta th

e ti ch c

^c h in

h ch op B MI

J va B SC

A kh

i do d

ai do an v

i dU cfn

g AC v

a MJ d at g ia t ri I dn n ha

t

Kh

i do

Du ng :

AN

1 (S IK )

Giai

1 /

De y t ha

y AB

1 (S AC ) va S

I c6 h in

h ch ie

u xu on

g

(S AC )

la S

K

Lu

c do : Jl

K

1 AC =>

I

K 1 ( SA C)

' AN ± S K;

N e S

K

JAN

1 IK

'AN

± S

K

N E/

/ AB ,E e SI

EF / AN F

e AB

EF

1 (S IK )

=>

EF

1 SI

Do : AB

1 (S AC )

=>

A

B

1 AN =>

A

B 1

E

F

=> EF

l

a do an v uo ng g oc c hu ng c ua A

B va S I.

Nh if ng :

E

F

= AN , ne

n ta t in

h do d

ai AN

I l

^

« AN =

2

Do d

^i do an v uo ng g oc c hu ng c ua A

B v^

S

I la :

EF = , (y

cb t)

S

Vb

^ + 4h

2

2/ D

e

y AC ± ( SA B)

3

MJ , tr on

g ma

t ph ln

g (S AB ), ito

g AH

1 MJ

=> A

H la d oa

n vu on

g go

c ch un

g ci ia A

C va M J.

Ta

CO

: AH <

A

J

= -

M.

B

maxAH = - k h i va chi k h i H = J

2

<=> M la trung diem cua SB

Luc do ti/ang ufng : —m i =

(ycbt)

Bai 406 (DAI HOC K I E N TRUC TP.HCM - 1995)

Trong mat phfing (P) cho tarn giac OAB vdi OA = OB, AB = 2a v^ difcfng cao OH = h Tren

dudng thang (d) vuong goc vdi (?) tai O, lay diem M vdi OM = x Goi E va F Ian luat la hinh

chieu vuong goc cua A len MB va OB; N la giao diem cua EF va (d)

1/ Chijfng minh MB 1 NA v^ MA 1 NB

21 Tinh BE, BF, EF, AF va the tich tuf dien ABEF theo a,h v^ x

3/ Tim vi t r i cua M tren (d) sao cho t i l dien MNAB c6 the tich nho nhat vk t i n h gia t r i nho

^

Vay : B

F = , (ycbt)

Tuan

g ti

f, tron

g tu

r gia

c no

i tie

p

MOFE t

a c

6 : ESJ

M

=>

=

BE B

Ta CO : A

a2

+ x2 )

2a

h

Tuan

g ta, t

a c

6 : AF

OB = OH

AB A

F = , (ycbt)

VA BE

F =

^AF.SBEF

= -A F.

EF B

E = „

, ,

„ ^ (ycbt)

+

n )(a + n + "

n

MN AB

l a:

V MN AB

= ^

-AF.S^^j, =

| AF M N.

OB

The tic

h tu

t die

n ABEF:

^ AF

S ^

-Ne

n :

3 min(V

^J

^A

B ' 3niin(MN)

NhUn

g : ANO

F c

o ABOM

=

> — =

=>

M

ON = OF

OB = h

^ a^

-OB

O

M

Ap dung BDT Cauchy

: MN = M

O + O

N >

2 Vh

^

- a^

^ 3m in

(MN)

= 2

V h^

- a^

O

M = O

3 mi nV MN AB

<=

> O

M = Vh^

- a

^ v

a minVMNAi

s = (ycbt)

I HO

C TON

G HO

P TP

HC

M 1995)

-Cho tarn giac deu OAB c6 can

h bSn

g a > 0 Tre

n difdn

g thin

g (d) d

i qu

a O vuong

g (OAB) la

y die

m M v

di OM = x Go

i E,

F Ian lug

t l

a ca

c hinh chieu vuong goc cu

g thing E

F cS

t d t

ai

N

1/ Chtifn

g min

h rSng A

N 1 B

M

2/ Xa

c din

h x de the tic

a nh

o nha

t

Gi

ai

A F I B

O

1/ Ta

CO

: =>

A

F 1 (MOB) =

> A

F 1 M

Ma : AE

1 MB ^ MB ± (ANE) ^ M

B ± N

A (ycbt)

2/ D

e y : AF

X (MBO) s (MNB) ^ A

F l

a chie

u ca

o hin

h cho

p A

BM

N

Ap dung BDT Cauchy, ta c6 : x + ON ^ 2 J — = aV2 (*)

3rain(x +0N) = aV2 x = ON = ^

Vay the tich tut dien ABMN nho nhat khi va chi khi x = (ycbt)

' 2

1 Bai 408 (DAI HOC XAY Dl/NG HA NOI - 1995)

Trong mat phang (P) cho hinh vuong ABCD vdi AB = 2a Tren mat phang chuTa BC va

|B1 = X K la hinh chieu vuong g6c cua diem E tren dUdng thang AI; O la trung diem cua AE

1/ Tim quy tich ciia diem K khi I chay tren doan BC

2/ Tinh do dai 0 1 theo a va X

13/ Tim X de do dai OI Idn nhat, be nhat

ma EK 1 AK => FK 1 AI (dinh ly 3 difdng vuong goc)

luon vuong, AF co dinh nen K di chuyen tren tron (C) dUdng kinh AF

321

GicJi ha

n khoan

g chay:

I

s

C =>

K ^ H

B H

c (C) ^ EK

± A

I

Vay quy tich

K l

a cun

g BH

c (C) (ycbt)

+ 2 EI

AE^

= AF

^ + EF

^ = 8 a^

AI

^ =

4

a ^ +

= > 0

1 =

V x^

ax

= fix) = x^ - ax + 2a^

; V

a e [0; 2a]

= > f (x) = (2x

- a) =

0

< =>

X = -

2

Difa v^o bang bien thien, t

a c

6 :

max f(x )

^a

^ f'(x)

max 0 1 =

S(2a ) =

2

a <2a 0«x

min O

I CUON

G 1996)

-Cho

id

dien ABCD c6 AB = C

D = 2x va

a 4 mSt) cu

a t

ri Id

n nhat

Gi

ai

1/ Nha

n thay, ca

Suy ra, die

n tic

h toa

n pha

n ciia

ti l

dien la : =

4.8^^

^

= 2.AI.C D

(1)

Vdri AI la dudng cao cua ACAD can tai A

Dau d^ng thuTc trong (2) xay ra <=> = 1 - x^ x =

Vay vdi x = — t h i dien tich toan phan cua tif dien dat gi£i t r i Idn n h a t la

maxStp = 2 (ycbt)

Bai 410 (DAI H O C Q U O C GIA T P H C M - 1996)

Cho tuT dien SABC c6 goc p h i n g d dinh S vuong

1/ Chufng minh rSng VS.SABC ^ SSAB + SSBC + ^SAC

2/ Biet rftng SA = a; SB + BC = k Dat SB = x Tinh the tich tuf dien SABC theo a; k; x va xac dinh SB; SC de the tich tiif dien SABC Idn nhat

1/ Tinh the tich V cua tuf dien theo a

2/ ChOfng minh rang AB 1 CD Tinh khoang each gifla hai difdng th^ng AB, CD theo a 3/ ChOfng minh rang cac dudng th^ng OB, OC, OD ti^ng doi mot vuong goc nhau

4/ Xac dinh diem M trong khong gian sao cho MA^ + MB^ + MC^ + MD^ dat gia t r i nho nhat

G i a i 1/ Do ABCD la tuf dien deu canh a va H la hinh chieu vuong goc ciia A xuong (BCD)

H la trong tarn ABCD

1 aVe a

^V s

a^V

2

bt)

2/ Ta C

O :

BH l

a hinh chieu cua B

A le

n (BCD)

BHiCD

= > AB 1 C

D (dpcm)

Do ABC

D la ti

J die

n de

u, ne

n BH s

i

trun

g die

m

I va B

I

= AI

Go

i

J l

a trung die

m cua A

B, th

B

Tuang tif: J

D = J

C =

>

JI 1 C

D

= > IJ l

a doa

n vuon

g go

c chun

g cua A

B va C

D

V 4

3/ Ta C

O :

A

H

= VA B^

B H^

= ja^

^ ^

(ycbt)

e dudn

g tron)

= >

OB = O

C

= OD

2

2

Ma: OB^

= OH

^ + HB^

=

- AH

^ + HB^

= — + —

4

6

3

OB = O

C

= O

D

= iV

2

Nhan tha

y : OB

^ + OC

^ = a

^ = BC^

=>

ABO

C vuon

g t

ai

O

Tifang tU, ABO

D va ACO

D vuon

; OC;

OD tifn

g d

oi mo

t vuon

g go

c nhau (dpcm)

4/ G

oi

G la trong tam t

il die

n ABC

D

= >

GA + G

B + G

C + G

I

= MA^

+ MB^

+ MC^

+ MD^

= MA+ MB+

A + MG+GB+

MG+G

C + MG

+GD

>

V

= 4

MG

2 + GA^+GB^+GC^+GD

^ + 2M

G GA+G

MG

2 +

I

= 4

MG

2 +

GA

^+GB

^+GC^+GD^

GA

^+GB

^+GC^+GD^

S > G A ^+ GB^+ GC^+ GD^ (const)

Vay : m i n i = GA^+ GB^+ GC^+ GD^ <=> M = G (ycbt)

Bai 412 (DAI HOC QUOC GIA TP.HCM - KHOI A - K I N H TE - 1997)

Tren ba canh Ox, Oy, Oz cua tam dien ba goc vuong Oxyz ta Ian li/gt lay cac diem A,B,C

vdi OA = a, OB = b, OC = c; (a, b, c > 0) Goi H la hinh chieu vuong goc ciia O xuong mfit

phing (ABC)

II Chilng minh rkng AABC c6 cac goc deu nhon va H la trifc tfim AABC

21 Chijfng minh rkng : S^ABC = S^OAB + S^OBC + S^QAC

3/ Goi M , N , P Ian lUcft la trung diem cua AB, BC, AC Chufng minh r^ng 4 mSt cua tuf dien

OMNP la cdc tam giac bang nhau Tinh the tich tuf dien OMNP theo a, b, c

1/ Doc gia tU giai c6 t h i xem De D A I HOC HUNG

VllCiNG - KHOI A - 1998 de c6 duac cdch chutng

minh ba goc AABC deu nhon

=> H la trirc tam AABC (dpcm)

: AOM

P = AON

P = AOM

N = AMN

P =

> (dpcm)

Ta CO : VQMN

P =

—

OH S MN

=> VQMNP

=

—

OH.(

— SABC )

= -(a^b

^ + b^c

^ + a^c^

Ap dun

g BD

T Cauchy, t

a c

6 :

a" + b'' ^

2

a V (1

(3)

a^b^

+ b^c

^ + a^c

^

^ a

^ + b

"

+ c' '

« a^b^

+ b^c

^ + a^c

^

^ (a

^ + b

^ + c^)

^

- 2ia%^

+ b^c

^ + a^c^

)

« i(aV+bV+aV)<—(a^+b^+c^f

4^ ''12^

'

'A

BC

12 (4)

Dfi'u

d in

= b

= c

= > max(SABc ) =

g thuT

c trong (5) xa

y r

a <

= > a = b = c

=> max(OH) =-khi

il di?

n ABCD c6 AC = A

D = BC = B

D = a; A

1/ Xa

c din

h v

i tri va tin

CD

2/ Mo

t ma

t phing (a) vuon

e AB

; J

e CD) V

hing (a) ci

t tu

f die

n

Tinh dien tic

h S(x) cu

a thi

et dien Xa

et die

n c

6 die

n tic

h Id

n nha

t v

a tin

h gi

a t

ri Id

n nha

t a

y ?

Khi do, I J l a difcrng vuong g6c chung cua

AB; CD, va I la t r u n g d i e m cua A B (vi A J = BJ)

Bai 414 ( D A I H O C N G O A I N G O T I N H O C - 1997)

4 (7)

Cho tuT dien A B C D sao cho A B = 2x, C D = 2y va 4 canh con l a i deu c6 dp d a i bSng 1

y T i n h dien t i c h t o a n p h a n cua tiir dien theo x va y

21 Xac d i n h x va y de d i e n t i c h toan p h a n dat gia t r i I d n n h a t

3 2 7

n lifat la trun

g die

m cu

a A B;

CD

Ta

CO

: AAB

^

Ttfcr

ng tir: A

N = -^

=

S AB

D

= "

A B.

DM

= x V

l -

B

Hoan loan tuon

g tif

: Sg cu =

2 1

Va

y dien tich toan pha

n S,

p cu

a t

iJ die

-Vl

- '

Ap dun

g BDT Schwart, t

l -

x2

+

Dau dan

g thijt

c trong (1) xay ra <=>

V2

Vay : maxS,p = 2

<=>

x

= y = (ycb

t)

(l -x2 )]

+ [ y^

+{

y^

l-y

<2 (1)

<=>

X = y =

Ba

i 415

(DA

I HOC VAN LAN

G

~ KHO

I

B,

D DdT 1

- 1997)

Trong m

at phan

g (P) cho dUdn

g tro

n (S) diTd

ng kin

h :AB = 2R Tre

n dudn

di (P) ta

i A, lay die

m

C s

a o cho A

C = A

B

M l

a m

ot die

m thu

pc (S ), H la h

M

1/

Chufng min

h rSn

g k

hi

M d

i don

g tre

n (S) t

hi

H d

i dgn

g tre

n m

ot dUcfng tro

n

2 1

X

ac din

h v

i t

ri

M tre

n (S ) (Tin

h do d

ai AM theo R) s

a o cho hin

h chop HAB

t Tin

h gia t

ri Id

n nha

t d

o

1/

Theo din

h l

y b

a difcJ

ng vuon

Da

CO

: BM ± A

M (2);

v

i M

e (S )

(1)

&(2 )

= i>

BM

1

(ACM)

^ (CAM)

_ L

(CBM) theo giao tuye

n C

M

O

Ca ch

d im

g :

AH

_ L

C

M

^ AH ± (CBM)

= ^ AH

1

B

C (3)

Go

i K la trun

g die

m canh B

C cu

a ABA

C vuong ca

n d A

=> AK

1

B

C (4)

(3 )& (4 )

=> BC

1

(AHK) : co din

h, bd

i v

i A

K c

o din

h

Luc do M liAi dong t r e n (S) t h i H luu dpng tren dUcfng t r o n co d i n h (Z) difcmg k i n h A K = R V2 ,

21 Cho SA = a, SB + SC = k D a t SB = x, t i n h the t i c h tuf d i $ n SABC theo a, k , x Xac d i n h

SB, SC de the t i c h tuT d i e n SABC IcJn n h a t

Vdi ch

u y :

sina =

cosa =

AB SB SA SB

AB =

aA/2sina

SA =

aV2cosa

Khi d

o : V

q»B

p

= — a-«^ a

-y 2s in a

.a-y

2c os

a

= si n2

a

Vay : max(VsABc) <

= > s in 2a = 1

<=

> a = — (ycbt)

- 1998)

n Ox, Oy, O

B = b; O

g luo

n c

6 O

A + OB + O

C + AB + B

C + CA = k khSng

I HO

C NGOA

I NGU

f H

A NOI

- PB

- 1997)

2/ Ta

CO

: O

A + OB + O

C + AB + B

C + AC = k

< = > a + b + c + Va

; b > 0

; c

> 0;

ta ap dung BDT Cauchy

(2) (3)

Va^Tc

^ >

V 2a

c

(4)

Lay : (2) + (3) + (4) v

a a

p dun

g BD

T Cauchy, t

3V2.^ /ib c

C

=

^ ab

c eV

s ABC

= ab

c

TucJng tir (1) + (5) v

_^

^

= > k > (

3 + 3V2j^ /ab

c

k >

(3 + 3 72 )3 /6

?^

,

k 1

^S AB

C

^

g

SA BC

^ 3

(6)

3 + 3 V2

(7)

Dau d^n

g thiJ

c tron

g (7) xa

va (6) xa

y r

a <

= > a = b = c

Vay: max(VsABc) =

-I 3 + 3V2J Bai 419 ( D A I H O C Stf P H A M QUY N H O N - 1998)

Cho dircrng tron (90 tarn O , dirdng kinh AB = 2R Diem M di dong tren {.f) va A M = x Tren

dilcfng t h i n g vuong goc vcJi mSt p h i n g chiJTa Cf) tai diem A, l a y mot diem co dinh S va AS = h 1/ Chilng to rSng hai mSt p h i n g (SAM) va (SBM) vuong g6c v(Ji nhau

2/ Tinh the tich tur dien SABM theo R, h , x Tim nhOng vi t r i cua M tren ('f) de the tich tiif

di?n nay dat gia t r i Idn nhat

Giai

1/ Doc gia c6 the xem C a u 1 De DAI WOQ

QUOC GIA - 1996, de" tCr: MB 1 (SAM)

Cho hinh chop SABCD co d^y ABCD la hinh vuong canh a, canh ben

SA = h va SA 1 (ABCD) M la diem thay doi tren canh CD Dat CM = x

« AH =

Vay : SH =

VsA

^

+ AH^

SH =

a^

+a V+

xV

2 2

X +

^SA

AH.B

H

Vdi : SA = h

AH =

Vx

^T a^

AB-^

AH

-^ =

ax

Suy r

a : V(x) = VSABH

= ^

• (1

>

6 x'' +

a

Khao sat ha

m s

o (1) su

y r

a :

a^h

max(Vg

^H) = k

hi x = a hay

M =

D (ycbt)

- 1999

g

kinh AB = 2R La

y die

m S sao ch

m qu

y tic

h die

m K k

hi C thay ddi

i xe

m D

e DA

;

N ban

g K

; M bang

y tic

h die

m K k

hi C thay doi l

a

dadng tron

dUcfn g

kinh A

i SB

AH

AH = SA.A

B

SB

AB

AH =

SB

R

2R 2R

RA/5 VSSH

=

V5

Goi K' la hinh chieu cua K xuong AH, khi do :

1/ ChiJng minh diem H di dong tren mot dUcrng tron Tinh do dai IH

21 Goi J la trung diem cua doan CE Tinh do dai JM va tim gi a tr i nho nhat cua JM

Vay H di dong tren cung B F ciia difcfng

tron CO difdng kinh CI tren mat phlng (ABCD)

21 Goi r la trung diem ciia CI Trong A I MC, ap dung dinh

MI^ + MC^ = 21M^ CI'

=> 2rM''^ = ( X - a)''^ + 4a^ + x^ - 5a'

=> I 'M = X - j i x +

5a'

C

O :

J l

a trungdiemC

E

ri

a trungdiemC

I JI7/

EI

jr

= iEI

Trong AMI'J vuong tai I', ta c6 : M

J =

V l'M

^ + I'J

^ =

Khi d

o : M

J >

Vay (MJ),ni„ =

— ha

y M la trung diem

- Dd

T 2

- 2000)

n ducfn

g thdn

g (d) vuon

g go

c vd

i ma

t phi

ng

(ABC) ta

i A

lay diem M Go

i H la trUc ta

2/ Kh

i M thay doi tre

n (d), tim gid

Gi

ai

1/ Go

i I la trung diem

1

BC

Ma MA

l M

I

Suy r

a : BC

1

(AMI) (*

)

fBH_

LA

C (H l

a tru

e tam

HBIM

A (do:

MAl(

ABC) 3 HE)

HB

1

(MAC); m

a : MC

C ±

BK

Tif(l)va(2)^

MCKBHK)

(3)

Do d

o : HK

± M

C

(*)

(1) (2) (3) (4) (5)

(K la true tam)

(dpcm)

(vi : H

K c

(HBK))

(vi : HK

c

Tir (4) v

a (5) =

>

HK

1

(MBC) (dpcm)

D = dlK; (ABC)] l

Luc d6, thi tich

.K

D = ^a^

• 3max(KD)

iV3

De y AHKI vuong tai K va canh hu yin

aV3

HI =

max(KD) = KoO =

12

ivdi O la trung diem H I; Ko G H I va KOO 1 HI)

Vay khi M thay đi tren (d) thi g i i tr i Idn nhát cua the tich V trong yeu cau bai toan 1^ :

max(V) = -l-.ậ—L- = _ (ycbt)

12 12 48 Dfi T L f dNG T i ; Bai 424 (DAI HOC KHOI A, B - 1982)

Cho hinh vuong ABCD canh ạ Lay M e AD va dSt AM = x (0 < x < a) Cho dudng th in g

d 1 (ABCD) tai Ạ Lay S e d va dat SA = y > 0

a/ ChUng minh nhi dien canh SB la nhi dien vuong

hi Tinh khoang cich tU M den mp(SAC)

d Tim the tich hinh chop SABCM

Al Cho x^ + y^ = ậ Tim gia tr i Idn nhát cua the tich hinh chop SABCM

Cho tU dien ABCD c6 AB = x, cAc canh con lai bSng 1 Tim x de the tich tuT di§n Idn nhát

Hxidng din

maxV = —

Bai 426 ( D A I H O C Q U O C G I A H A N O I - K H O I B - 1997)

Cho AABC can dinh A va dUcrng thiing (d) _L mp(ABC) tai Ạ Lay M e d (M ^ A),

a/ Tim quy tich trong tam G va trUc tam H cua AMBC

hi 0 la true tam AABC Tim vi tr i cua M de tuf dien OHBC dat gia tr i lorn nhát

Ch uye

n de 2

3 :

P HlT ONG P HA

P V EC TO T RON

G KHONG G IA

N

I PH UON

G PHA

P

Ca s0 cu

a phirang pha

p l

a suf dung ca

c din

h nghia v

a phep toan ciia vectc

f tron

n chu

fa

he to

a

do (Oxyz))

No tUang tif ca

c din

h nghia v

a ph

g m

at phSng (Oxy)

O

day ta lifu

y dU

c fc ca

c tin

h ch

at

ca ba

n :

• Con

g h

ai vecta th

eo qu

y t

& c

dUcrn

g che

o hin

h binh han

m : V

A,

B,

M

e (Oxy)

•

I l

a trung die

m doan A

B

AM

^ hM

B = A

; V

M

s (Oxyz)

•

G la trong tam tam gid

c ABC <=>

GA + GB + GC

VM

e

(Oxyz)

GA+G

B + G

C + GD=

• Kh

ai nie

m b

a vecta don

g phSng :

•i

Ba vec

ta

a ,

b ,

c 5t

0 dong phSn

g k

hi gia cu

a chu

ng cii

ng son

g son

g hoS

c cCing

^n

g

•2

Pha

n tic

h

m

= aa +p b

?^

0 (V

a,

p

6

R) la duy nha

t v

a luon thifc hie

n dtf

g vd

i h

ai vecta khong ciin

g phi/an

t v

a luon thiTc hie

n difc

g don

g phSng

O Gh

i ch

ii

: 1) Neu m

ot tro

ng

ba vect

a

a,

b, c

b&

ng

0 tlii ch ung d ong ph dn

g

2) Neu m

ot tro

ng

ba vect

a

a,

b, c

co

ng tuye

n thi ch iin

g cun

g don

d Sng ph dn

g

=>0 ,

A,

B,

C

e mp(

a)

n CA

C BA

I TOA

N GO B

AN

Ba

i 42

7

Ha

i hin

h binh han

h ABCD v

a AB'C'D' tron

g khon

g gia

n c6 chun

g mo

t din

min

h cac vecta DD', BB', C

C dong phSn

g

Theo h

e thu

fc Chales, ta c6 :

Gi

ai

BB'=

BA+AB' (1)

DD' = D

A + AD' (2)

Con

g the

o

ve ha

i dUcf

ng thin

g (1)

va (2)

ta dif

gc :

=> BB' + DD'

= (BA ^ DA) + (AB' + AD)

= (CD

-r

DA) + (D'C' + AD)

= C

A + AC'

= CC'

=>

(dpcm)

Ta phai chuTng minh duac rkng

Chon cdc vect0 co sd : AB = a; AD = b; AA' = c

V Cho tuf dien ABCD c6 AB = BC, AD = DC Chufng minh AC 1 BD

21 Khi t i l dien ABCD c6 AB J- CD Chilng minh rSng : AC^ - AD^ = BC^ - B D l

A' D'

C, N

e A'B sao ch

o MN la dadng vuong goc chun

0 v

a MN.BA'=

0

—>

->

• >

D =

b, AA' =

- a

Da

t : AM = xAC, B

N = yBA'

Luc d

6 : M

N = M

A + AB + B

N = -xA

C +

a + yBA'

= -x (a + b )+

a +y (c

- a)

= ( l- x- y)

a

- xb

<=>

(1

- X

- X

- y)a

^ =

0

MN.AC=

0 [(

x-y)

l-a -xb-yc](

a + b) =

- Xb

- y c](c - a) =

0

fl 2x -

0 3

Suy r

a : M

3

» MN' = -(a

- b

- c

= — » MN = ^

O Gh

i c hi

i

: Bdi

todn ndy

cd nhieu cdch gidi,

d tren

ta da

sd dung phuang phdp

n ABCD C

o

E,

F, I

theo thOf tif la trung die

m cu

a AB, CD

, E

F

a/ Chijfn

g minh : lA

* + IB' + IC + I

m M bat ky trong khong gia

n, ha

y chiin

g minh : M

A + MB + M

m O sao ch

o O

A + OB + O

C + OD = 0 ChiJn

g minh die

m O la duy n

m ch

o ta : lA + IB' =

2 IE*

; IC + I

* + IC + I

D = 2( I

E + IF*)

=>

lA + IB

* + IC + ID

* =

0 (dpcm)

hi

Tuangtif:

MA + MB-2ME;

MC + M

D = 2MF

=> M

A + MB + M

C + MD = 2(M

E + MF) = 4MI (dpcm)

c/ Cun

g tiran

g nhu the t

if ca

u a/ ta c6 :

0 = 2(0E + OF)

Ding thutc na

y chuTn

g t

o O la trung die

f

O'E + O'

O + O'F

<K.

0 = 2

0'6

+ O'

E + O'F <=

m du

y nh

at th6a : O

A + OB + O

C + OD = 0 (ycbt)

Bai 4 3 2

Cho tijf di e n A B C D v a m at phSng (P) T i m di e m M e (P) sao cho

ria tri nho nhat

Cho tijf dien A BC D Go i P, Q theo thijf tu l a t rung di e m cua c ac c an h A B v a C D H ai di e m

R, S Ian lUcft lay t re n cAc c anh A C v a B D sao cho — = — = k (k > 0) Chufng m i n h r Sn g bon

AC BD diem P, Q, R , S n am t re n cung mot m at phAng