Kiến thức cơ bản và nâng cao hình học 12 (tái bản lần thứ nhất) phần 2

62. Cho hinh tru c6 cac day la hinh tron tarn O vh tarn O', ban kinh ddy bang
chi^u cao va bang a.
Tren ducmg tron day tarn O la'y diem A, trdn ducmg tron day tarn O' la'y
diem B sao cho AB = 2a.
Tinh the tfch kh6'i tir dien OO'AB.
(Trich de thi Dai hoc - Khoi A - 2006).

ChUffng in
PHl/ONG P H A P T O A D O T R O N G K H O N G G I A N
I. Ht TOA DO TRONG KHONG GIAN
A. L t THUYfeT C A N N H 6
1. Toa do cua diem va cua vecta
/. He tog. do: Trong khong
gian CO ba true toa d6 vuong goe
vdi nhau doi mot va c6 dinh hudng
Ba true nhu vay duac goi la ht
toa d6 vuong goc trong kh6ng gian
Didm O goi la g6'c toa d6
- True hoanh, dinh hudmg
duang X Ox, c6 vecta don vi
i = (l;0;0)

o

Hinh 34

True tung, dinh hudng duong y'Oy, eo vecta dan vi
} =(0; 1;0)
f True cao, dinh hudng duang z'Oz, c6 vecta dan vi
^=(0;0;1)
- Cac mat phang xOy, yOx, xOz d6i m6t vudng goc vdi nhau duac goi

1^ cac mat phang toa dd.
Vi i , } , it la cac vecta dan vi tren true, ma cac true vu6ng goc nhau d6i
ni6t, nen ta cof
= 1


va

T.J =0; i.k

=0;

j . k

2. He qua

= 0

Trong khdng gian vori he toa dp O x y z con dupe gpi la khong gian O x y z

2 . Toa dp mot diem
Trong khong

gian O x y z

a, = b,, a2 = b2, a, = bj.

b)

0).

0

=

(0; 0;

cho

m6t diem M tuy y, m6i diem
hoan toan xac dinh boi vecta

a) a = h «

c) a va b ciing phuong

M

d) Trong khong

OM

gian

c6 mot so k: a, = kb,, aj = kbj, a, = kbj.
Oxyz

c6 A ( a , ; a2; a,) va B(b,; b2; b,) thi

.45 = ( b - a , ; b2-a2; b j - a j ) .

V i 3 vecto i , j , k la 3 vecta
khong dong phang nen c 6 b6 3 so

3 . T i c h v6 hudmg

duy nha't (x, y, z) sao cho:

1) Bieu thAc toq do
OM

=

X.

i +y.j

+ z.k
D i n h If. Trong khong gian O x y z tich v6 hudng cua hai vecto a (aj; a.^, a-,;

=

OM,

+OM3

+ OMj

Hinh

35

va A (b,; bj; h^) la m Or sty duac xac dinh bcfi cong thiic:

Nguoc lai vdfi (x, y, z) ta c6 didm M duy nhat trong khong gian thoa man
OM

=

X.

a .b = a,b| + a2b2 + ajbj

i + y.j + z.k

Bp ba so (x, y, z) gpi la toa dp cua diem M doi v6i he true Oxyz da cho
va k i hieu la M ( x , y, z) hoac M = (x, y, z)
3. Toq do cua

. .

.

2) Ifng

dung

* Do dai cua vecto

vecta

Trong khong gian Oxyz, cho vecto u , ta luon c6:

* Khoang each giua hai diem A
AB =

M

= a,, i +

3.2-j

7(XB

-

x^f

+ (ye - yj'

+ (z^ - z^f

.

+a3 k

Bp ba so (a,, a2, a,) xac dinh duy nhat va gpi la toa dp ciia vecto u, ki

* G p i 9 la goc giiia 2 vecta a va b

hieu la u (a,, aj, a,)

cos 9 =

2 . Bieu thirc toa do cua cac phep toan vecto

S ( b i ; b2; bj) k h i do
± b = (a,± b,; a2± bj; a j i b,)

b) k. a = (ka,; kaj; k-a,) = k(a,; di^; aj) (k la so thuc)

a,b, +a2b2 +a3b3
^/af+af+a^

-^/bf+bf+b

=> a J . b <=> aib] + a2b2 + a3b3 = 0

/. Dinh U. Trong kldong gian Oxyz cho cac vec ta a (a,; aa; aj)

d) a

A ( X A ; yA; Z A ) . B ( X B ; yg; Zg) la

4. T i c h CO hudng cua hai vecto (hay tich vecto)
/) Dinh

nghia:

Tich c6 hu6ng (hay tich vecto) cua hai vecta a (ai, a2, a,)

Va b (b„ b2, bj) la mot vecto duac k i hieu \di[a,b]

d6 duac xac dinh nhu sau:

hay a A 6 va c 6 toa


f
a, a, a, a,
Vb,b3 b3b, b,b, )
3

2) Tinh chat
[a,b]=0
{a,b]

o a =k.S
l a v a [ a , 6 ] I . 6 ( r a , b ] . a = 0; r a , b 1 . b = 0)

a,b = a

B. Vf DU

>

sin(a,b)

Vi du 1
Cho tii dien ABCD:

3i)Chvtngm\nh: AD+ BC = BD + AC
b) Goi M la die'm chia dudng trung tuyen AA, ciia mat phang ABC theo
so 3: 7 (
= — ). Chung minh rang:
MA, 7

3)Apdung
J. Tick dien tick cua hinh binh hanh va thetich khoi hdp
• A B C D la hinh binh hanh SABCD = A B . A D sin A
AB,AD

• A B C D A B C ' D ' la hinh

hop

• VABCD.A'B'C'D'= T A B , A D

.AA

Ba vecta a, c, bdong phang <=> a,b .c = 0
Ba vecta a, b, c khdng d6ng phing <» a,b
I

5. Phuong trinh mdt c^u
* Dinh li: Trong khong gian Oxyz,
mat cau (s) tarn I(a;b;c) va ban kinh r
CO phuong trinh:
(x-a)^ + ( y - b ) 2 + ( z - c ) 2 = r2.
Nhdn xet:
Phuong trinh

x ' + y ' + z ' + 2Ax + 2By + 2Cz + D = 0v6i:
' Hinh 36
A^ + B^ + - D > 0 la phuong trinh mat c^u tarn I(-A;-B;-C) va
r = ^A^ + B' + C' - D .

lai:
a)

DM = —DA + DB + DC
20
10
20
AD - AC = CD
BD - BC = CD

AD - AC = BD - BC

AD+ BC = BD +AC
)Cdchl. MA _ 3
MA, ~ 7 ^

D

AM =- MA.
AM = DM - DA
MA, = DA, - AM
Nensuy ra DM - DA = - (DA, - AM)
DA + — DA,
>DM = —
10

10 '
1
Ma DA, = -(DB
+ DC)
T>oA6 DM = — DA — DB + — DC.
10
20
20

X-^'-

-V-


Cdnh 2: Bien ddi ve phai:

= - - b+a
2

— DA ^ — DB ^ — DC
10
20
20

= - — b + a + -{c2
2

+ M^)+ —(Z)M+ MB)+—-{DM

= --(DM

10

20

+ MC)

20

_L DM + — DM +— Z)M +— M4 +— MB +— MC
10

20

20

10

20

= DM + — MA + — (MB + MC
10
20

10

MA + — M A i

,

1 MA^^ = MN^= - (a - b +cf=

4

= ld^^MN =
2

MN

0
•

20

2

2

—(a-b+c)

1
- {a^+ b+c+l{-a.b
4

+ a.c-b

c)

^

dV2

AB = d . Theo c6ng thiic ta c6:

M N . A B = M N . AB

VP = VT do la dieu phai Chung minh.
<»COS(MN, A B ) =

V i du 2:
Cho tur dien deu ABCD canh d,
M va N Ian luot la trung di^njL AG '
va BD.

cos ( M N . A B )
=> ( M N , A B ) = 45°.

3). M N . A C = - ( a - b + c)b = - ( a . b - b" + c.b) = 0 .

MNl

1) Tim do dai MN.
2) Tim goc giua MN va AB.

AC.

MN . B D = MN ( A D

3) Chiing minh M N l AC, MN 1 BD.

- AB)

= - ( a - b + c) (c - a)
2
\^ ^(c + a)(c - a) - b(c - a)

Giai.

21Hinh 38

Dat vecta AB = a, AC = b,
AD = c,

a)=

2) MN . AB = - ( a - b + c)a = - ( a ' - b . a + c . a] = y ,con
,c6

2MAi

= DM +

+ -(AD - AB)
2

:b

= - ( c ' - a' - b . c + b . a ) = 0

= d.
=> M N

a.b = c.a = b.c = a

cos 60"= -d\
2

1) MN = MA + AB+ BN
1 r - 1
= - - b +a+- BD
2
2

1BD.

Vi du 3.
U m toa d6 hinh chie'u cua die'm A ( l ; - 3 ; -5) trdn:
1) mp Oxy;

2) mp Oxz

3) mp Oyz;

4) True hoanh;

5) True tung

6) True cao.
A

1

Giai:

Vf du 6:

1) Tren mp Oxy thi do cao z = 0 nen toa d6 hinh chie'u cua diim A la

Cho a = (3; - 1 ; 5) va 6 = (1; 2; - 3 ) . Tim c thoa man cac di6u kidn

A,(l;-3;0).

sau: c ±Oz, ca = 9, cb = - 4 .

2) Tr6n mp Oxz thi tung d6 y = 0 nen toa do hinh chieu cua diem A la
A j d ; 0; - 5 ) .
3) Tren mp Oyz thi hoanh do x = 0 nen toa do hinh chieu ciia di^m A la

Giai: Goi c= (x. y, z). V i c 1 Oz o ck = 0 o

x .0 + y .0 + z .1= 0

=> z = 0 vi c a = 9 « 3x - y = 9 V i cc = 4 <=> X + 2y = - 4

A3(0; -3; -5).

4) Tren true hoanh Ox thi tung do y = 0, do cao z = 0 nen tea d6 hinh
chieucuaAlaA4(l;0;0).
5) Tren true tung Oy thi hoanh do x = 0, do cao z = 0 nen toa do hinli
chieu ciia A la A,(0; -3; 0).
6) Tren true cao Oz thi hoanh do x = 0, tung do y = 0 nen toa do hinh

chieu ciia A la A^iO; 0; -5).
Vi du 4: Cho A(-3; 2; -1). Tim toa do diem doi xiing cua A qua gdc toa

Giai he: P'^-y^^ ^
• [x + 2 y = - 4

1^ = 2
[y=-3

c , (2; - 3 ; 0)

Vi du 7.
Cho tii dien ABCD c6 A ( l ; -2; -1), B(-5; 10; -1), C(4; 1; 11), D(-8; -2; 2).
ViS't phuang trinh mat ciu ngoai tie'p tii dien ABCD.
Giai:
Goi I(x; y; z) la tarn mat c&u ngoai tie'p tii dien ABCD, thi ta phai c6:

d6, qua eac true toa do, qua cac mat phang toa d6.
IA = IB = IC = ID
Giai: Qua gdc toa do: toa do diem doi xiJng ciia A la: (3; - 2 ; 1).
IA=IB
Qua true hoanh x' Ox: toa dd di^m ddi xiing ciia A la: (-3; 2; -1).
Qua true tung y'Oy: toa do di^m ddi xiing etia A la: (3; 2; 1).

IA=IC

(1)

IA=ID

Qua true cao z'Oz: Toa do di^m ddi xiing cua A la: (3; - 2 ; - 1 ) .
Qua mat phang Oxy: toa do diem ddi xiing ciia A la: (-3; 2; 1).
Qua mp Oyz: toa dd diem ddi xiing cua A la: (3; 2; -1).

lAJBJC,

lA

IB

IC, ID ,ID

lA = ( l - x ; - 2 - y ; - l - ? )

Qua mp Oxz: toa do di^m ddi xiing ciia A la: (-3; - 2 ; -1).
Vi du 5:

=> lA = ^il-xr+i2

Cho AB = (2; - 3 ; -1). Tim toa dd diem A, biet B ( l ; - 1 ; 2).

IB = (-5-x; 10-y; -1-z)

Giai:

+ yy+il + zr

=> IB

Goi A(x;y;z) ta c6: AB = (1-x; - 1 - y ; 2-z).

Suy ra

2=1-X

X

-3 = - 1 - y

y =2

-l = 2-z

z=3

IC=

=-1
A ( - l ; 2; 3)

(4-x; l - y ; l l - z )

IC = V ( 4 - ; c ) ' + ( ! - > ' ) ' + ( 1 1 - ^ ) '

A1


ID = ( - 8 - x ; - 2 - y ; 2-z)
ID

T a c o h a : • 18x - 22y - 5z = 0

( l - x ) ' + (2 + y ) ' +(1 + zf
( 1 - x ) ' + (2 + y ) ' +(1 + zf

x)^+(l-y)'+(ll-z)

= (8 + X ) ' + ( 2 + y ) ' + ( 2 -

Vidu9.
Cho cac die'm A ( 3 ; 0 ; l ) , B ( - l ; 4 ; l ) , C(5;2;3), D(0; - 5 ; 4 ) .

-2
^

Chirng minh rling b6'n diem A , B, C, D la 4 dinh ciia hinh t i i dien.

I (-2; 4; 5)

z = 5

I

Khoang each l A =

=

-^{l-xf

+ ( 2 + >;)' + ( l + z ) ' = 9. Mat e^u

ngoai tiep t i i dien A B C D c6 tarn I(-2;4;5) va c6 ban kinh r = l A = 9

Tinh ^AO (O la trong tam ciia mat BCD ciia hinh t i i dien).
G i a i : a. Neu bon d i ^ m A , B, C, D la b6n dinh cua m6t hinh tir diSn thi

ba vector A B , AC, AD khong ddng phang.

Vay phuong trinh mat cau la:

A B = (-4;4;1), AC = (2;2;2) = 2 ( 1 ; 1 ; 1 ) , A D = ( - 3 ; - 5 ; 3 )

(x+2)' + ( y - 4 ) ' + ( y - 5 ) '

=81

Cdch I: Xet bieu thirc

Vidu8
Cho a = (3,;2;2)va 6 = (18;-22;-5). T i m cbie't

= 14,c 1 a,c

c tao vdfi true tung goc t u .

±bva

-4

1

1

-5

3

+ 4

1

1

3

-3

[AC,AD].AB

+

1

1

-3

-5

= -4.8 - 4.6-2 = -58 9^ 0 (dpcm)

Giai

Cdch 2. Ta khong tim dugc cap so x, y thoa man A B = xAC

Goi c = (x; y; z)
V i : fl 1 c

< 0 < = > y < 0 nen chon y = - 6

Dap so: c = (-4; -6;12)

- 2 y + 10 = 0

3x-z + ll = 0
y = 4

z = -2y
y = ±6

V i c tao vdi true tung goc til nen c.j

zf

<»<{x + y + 4 z - 2 2 = 0

X =

»

x ' + y ' + z ' = 14'

= (5 + X ) ' +(10 - y ) ' +(1 + z)

(1) c> ( 1 - x ) ^ + ( 2 + y ) ' + ( l + z)^ = ( 4 +

X

2

3x + 2y + 2z = 0

nghia la he sau v6 nghiem:

= 0 <=> 3x + 2y + 2z
^ 1 c o

^ = 0 »

=0

18x - 22y - 5z = 0

= 1 4 c : > x ' + y W = 14'

- 4 = X - 3>'
4 = X - 5y ^
1=

X

+ 3y

he nay v6 nghiem

+ yAD,


67. Toa d6 trung diem cac canh cua tam giac ABC la (1;3;2), (0;2;0).
(2; -2; 4). Tim toa do cua cac dinh tam giac ABC.

b. Goi 0(x, y, z) ta c6:
AO = AB + BO

68. Tim tren true hoanh mot diem each deu hai diem A(l; -3;7) va B(5;7;- -5).
AO = AC + CO
69. AABC CO A(l;2; -1), B(2; -1;3), C(-4;7;5). Tim d6 dai ducmg phan
AO=AD+DO

giac trong BD.

3AO = AB + AC + AD-(OB + OC + OD).
Vi O la irong tarn tarn giac BCD ntn OB + OD + OC = 0.
1
1
Suyra: AO =-{AB + AC + AD) =-(-5;l;5)

70. AABC CO A(-4; -1;2), B(3;5; -10). Tim toa d6 dinh C bie't trung di^m
canh AC thu6c true tung,^ trung diem canh BC thu6c mpOxz.
71. AABC CO A(6;2;3), goc toa d6 la trung diem canh AC. Trong tam G ciia

AABC thu6e true tung. Tim toa do B, C.

VsT

A0\ ^^i-5y+l'+5'

72. AABC CO A(-l;2;3), trong tam G trung vdi g6c toa do, Be Ox,

= ^

C e mpOyz. Tim toa d6 B, C.
73. Tim the tich tii dien ABCD biet toa d6 cac dinh A(2; -1;1), B(5;5;4),

C. BAI TAP

C(3;2;-1),D(4;1;3).
74. Cho hinh hop ABCD.A'B'C'D', biet A(-1,0,1), B(2;l;2), D(l;l;2),

63. Cho tii dien ABCD. Tim diem O sao cho:

C'(4,-5;l).

OA + OB + OC + OD = 0.

a) Tim toa d6 cac di^m eon lai ciia hinh h6p.

Chiing minh di^m O la dilm duy nha't.

b) Tim the tich hinh hop tren.

64. Cho tii dien ABCD. Goi A', B', C , D' la cac di^m theo thir tu chia cac
doan thang AB, BC, CD, DA theo ty s6 k:

75. Cho lang tru diing ABCA,B,C, c6 day ABC la tam giac vudng
AB = A C = a, AA, =. aV2 . Goi M, N 1^ lugt la trung diem ciia doan

A'A

BB

CC

DD

AB

BC

CD

DA

AA, va BC,. Chiing minh MN la dirdng vu6ng goc chung cua cac ducmg

=k

1. CMR vdfi moi di^m O bat ky trong kh6ng gian, ta lu6n c6:
OA + OB + OC + OD = OA' +OB+OC'

+OD'

2. Vol gia tri nao ciia k thi b6'n diem A',B',C,D' d6ng phing?
65. Cho a = S,b

=l,(a,6) = 30". Tinh goc tao bdi tdng va hifu hai

thing AA,va BQ.Tinh

VMA,BC, •

76. Trong khong gian toa d6 Oxyz cho 0(0; 0; 0), B(a; 0; 0), D(0; 1; 0),
O' (0,0,a) la bon dinh ciia hinh hop chu nhat OBCD.O'B'CD'.
Tim add 'BD

L^T:.

(Trich d6 thi DHXD, 1999).
77. Trong kh6ng gian toa d6 Oxyz cho hinh tii dien ABCD, bie't toa d6 cac

vecta a,b.
66. Tarn giac ABC c6 toa d6 cac dinh A(3; -1;6); B(-l;7; -2), C(l; -3;2).
Chiing minh tam giac ABC la tam gidc vu6ng.

dinh A(2; 3; 1), B(4; 1; -2). C(6; 3; 7), D(-5; -4; 8).
Tinh do dai ducmg cao ciia tii dien xua't phat tur A.
(Trieh de thi DH Duoc, 1999).


II. M A T P H A N G

A. THUYfeT CAN NH6
1. Phuofng trinh long quat cua mat phdng
* Vector phap tuyen: Vecta n^O vu6ng goc vol mat phang («) goi la
vecta phap tuyen cua mat phang (a).
Dinh ly: Trong kh6ng gian Oxyz ne'u m|t phang (a) c6 cap vecto chi
phaang la a (a,; a^, a,) va b (b,; h^, b,) thi ( a ) c6 m6t vecta phap tuy^fn
CO

toa do n =

V

>

J

Kh6ng CO mat x va y (A = 0, B = 0) thi mat phing do song song hoac
Itrung voi mat phang Oxy. Tuong tu mat phang Ax +D = 0 song song hoac
ttrung vdi mat phang Oyz, mat phang By + D = 0 song song hoac triing v6i
lat phang Oxz
f

* Ne'u A,B,C,D khac 0, bang each dat a = - — , b = - — , c = - — ta c6
A.
B
C
\ dua (1) \i dang: - + ^ + - = 1(2)

a b e

Khi do mat phang (a) cat cac true Ox, Oy, Oz \in luat tai cac di^m:
(a;0;0); (0;b;0); (0;0;c).
Phuomg trinh (2) la phuang trinh ciia mat phang theo doan chan.
2. Vi trf tuong doi cua hai mat ph^ng
Cho hai mat phang (or,): A,x + B,y + C,z + D, = 0

= (ajbj - ajbj-, ajb, - a,b3;a,b2 - a^hi)
•
Nhuvay n =
Dinh nghia: Phuong trinh c6 dang Ax + By + Cz + D = 0 trong d6 A, B, C
(«2):A2X + B2y +C2Z + D2 = 0
khong d6ng thcri bang 0 (A^+ B^+
0) dugrc goi la phuong trinh t6ng quat
cua mat phang.
1. ( a , ) n ( a 2 ) ? i ^ o A,:B,:C, ^ A2:B2:C2
* Nhan xet
B, C, A
* Neu mat phang ( a ) c6 phirong trinh t6ng quat laAx + By + Cz + D = ti
2. (a,)songsong{a.^)<^ A.i ! L - i ^ - C
thi vecta phap tuyen cua no la n (A,B,C)
* Phuomg trinh mat phang di qua di^m M„(Xo,yo'Zo) nhan vecto i
A, B, C2 A
n (A,B,C) 0 lam vecto phap tuySn la:
ngoaira(a,)±(a2)<»A,A2+BiB2+C,C2= 0
A(x - Xo) + B(y - yo) + C(z- ZQ) = 0
3. Khoang each tiir mot diem den mot mat phang
* Cac truorng hop rieng
* Cho mat phang ( a ):Ax + By + Cz + D = 0 va Mo(Xo;yo;Zo). khoang
Trong khong gian Oxyz cho (or): Ax + By + Cz + D = 0 (1)
each tiir diem MQ de'n mat phang a, duac tinh theo c6ng thiic:

* Neu D = 0 thi (or) di qua gd'c toa do va ngugfc lai.
* Neu trong phuomg trinh (1) khong c6 mat x(A = 0) thi mat phang
AXQ + Byo +Czo+D
tuong ung se song song hoac chiia true Ox.
d(Mo,a) =
^A'+B'+C
Tuomg tu vdri y va z.
* Khoang each gifia hai mat phang song song la khoang each tiir m6t.
* N6'u plijong trinh mat phang c6 dang Cz + D = 0
diim bat ky cua mat phang nay den mat phang kia.
49


'

B. Vf DU

Mat phdng (R) vu6ng goc vdi hai mat phang (?) va (Q) nen nhan hai
veeto phap tuye'n ciia hai mat phang nay lam cap veeto chi phuomg. Vay

V i du 1: Viet phuong trinh mat phang di qua diem ( 2 ; - l ; - 1 ) va vu6ng

;vecto phap cua mp (R) la:

goc vdi true eao.

r

0

0

—

Giai: Mat phang phai tim vu6ng goc vdi true cao nen nhan veeto

0

1

1

0

0

1)

>

\

k = (0;0;1) lam vecta phap tuydn. Vay phuong trinh mat phang phai tim la:
0(x - 2) + 0(y + 1) + l(z + 1) = 0 « z + 1 = 0.

^1

1

- 1

-1

0

= (0; 1; 1)
Mat phang (R) di qua A(-l;2;3) nen phuong trinh mat phang (R) la:

V i du 2:

0(x + l ) + ( y - 2 ) + ( z - 3 ) = 0

Viet phuong trinh mat phang qua ba diem
c:>y + z - 5 = 0
M(3;-1;2);N(4;-1;-1);Q(2;0;2)
Vi du 4:
Giai:

Trong kh6ng gian vdi he true toa dd Oxyz, cho tii dien ABCD vdi

Veeto phap tuyen ciia mp (MNQ) la:
—

0

-3

(2; -1; 6); B(-3; -1; -4); C(5; -1; 0), va D ( l ; 2; 1).

-3

1

0

- 1

•

n=
V

1

0

1. Chitng minh rang tam giae ABC vudng. Tinh ban kinh va dudng tron
= > « = (3;3;1)

Phuomg trinh mp (MNQ): 3x + 3y + z + D = 0
Mp di qua Q(2; 0; 2) suy ra D = -(3.2 + 1.2) = -8
Vay pt mp (MNQ): 3x + 3y + z - 8 = 0 ,

jndi tie'p tam giae ABC.
2. Tinh the tich ciia tii dien ABCD
(Trich de thi D H Thuy san, 1999)
Giai:

V i du 3:

1.

Trong khdng gian Oxyz cho di^m A ( - l ; 2; 3) va cac mat phang

=>C5.C4 = (-3).8 + 0.0 + 6.4 =0

(P):x-2 = 0 , ( Q ) : y - z - l =0

C4 1 Cfi nen A ABC vudng tai C.

Viet phuong trinh mat phang (R) di qua di^m A va vuong goc vdi hai
mat phang (?) va (Q).
(Trich de thi vao D H LuSt Ha Noi, 1999)
Giai: Veeto phap tuye'n ciia (P):
Veeto phap tuye'n cua (Q):

= (-5; 0; -10); CA = (-3; 0; 6);Cfl = (8; 0; 4)

= (1; 0; 0)

CA = 7(-3)'+0'+6' =3V5
CB = V 8 ' + 0 ' + 4 ' = 4V5, AB = 5V5
vay

SABG = ^ .

3V5 . 4V5

= 30

(dvdt). Goi P la nita chu vi A A B C va r la

= (1; 1; -1)
ban kinh dudng trdn ndi tiep tam giae A B C .

en


^ABC - pr=>r - ———

Khoang each

5 + Zo = — ^ —
«49(5 + z„)= (6zo-9)^

6V5

3. Mat phang (ABC) c6 cap vecta chi phuong:

o Zo^ + 46zo + 88 = 0

CA =(-3;0;6) =3(-l;0;2)
C5 =(8;0;4) =4(2;0;1)
Nen vecta phap tuyen cua mat phang(ABC) la:
0 2 2 -1 -1 0
nABC
=
0 1 1 2 2 0
= (0;5;0) = 5(0;1;0)
Vay mp(ABC) c6 phuang trinh y + 1 = 0

Khoang each tir D den mp (ABC) la:
2 +1
d(D, (ABC)) = Vo'+i'+o'

Dap so:
Vi du 6:

z„ = - 2
z„ = - 4 4

M,(0;0;-2) va M2 (0;0;-44)

Tim tap hop nhung di^m M(x,y,z) each mp (P):
4x - 4y - 2z + 3 = 0 m6t khoang bang 2.
Giai:

= 3

The tfch tir didn DABC la:
j3.30 = 30(dvtt)

Vr du 5:

Tim mot diem \xtn true cao each d6u diem A(l; -2;0) va mp
(P): 3x - 2y + 6z - 9 = 0.
Giai:

Goi M € Oz CO toa dp M(0; 0; z^). Ta c6:
MA = (1; -2; -Zo)

M de'n mp (P) la:

6z„-9
V3'+(-2)'+6^
6z„-9
OZ "

V a y r = ^ = V5.

V=^h.SABc=

di^m

MA = ^1^ + (-2)'+ (-z^)

Khoang each tijr M d6n mp (P) duac tinh theo cong thiic:
4A: - 4;; - 2z + 3
2 = V4'+(-4)^+(-2)^
<::>±12 = 4 x - 4 y - 2 z + 3
Vay tap hop cac di^m M g6m hai mat phang:
4x - 4y - 2z - 9 = 0 va 4x - 4y - 2z + 15 = 0
C . BAI T A P

78. Viet phuang trinh mat phang di qua diem (3;-2.-7) va song song mp
2 x - 3 z + 5 =0.
79. Viet phuang trinh mat phang qua hai diem A(l;-l;-2) va B(3;l;l) va
vu6ng goc vdi mp (P): x - 2y + 3z -5 = 0.
80. Viet phuang trinh mat phang qua true tung va di^m A(l;4;-3).
81. Viet phuong trinh mat phang qua hai diem A(7;2;-3), B(5;6;-4) va song
song vdi true hoanh.

82. Tim dien tich tam gidc do mp (P): 5x - 6y + 3z +120 = 0 cat mp Oxy.
83. Tim the tich tii dien do mp(P): 2x - 3y + 6z - 12 = 0 cat cac mat phang
toa d6.
84. Tim tren true tung mot diem each mp: x + 2 y - 2 z - 2 = 0 m6t khoang
bang 4.
=0.

THUY^T CAN

thang. Vay trong khdng gian Oxyz ta xem ducmg thang (d) la giao cua hai
mat phang phan biet (a) va (a') wdi:

=0

87. Viet PT mp phan giac ciia goc nhi dien tao boi hai mp:

( a ) : Ax + By + Cz + D = 0

x - 3 y + 2z-5 = 0 v a 3 x - 2 y - z + 3 = 0

( a ' ) : A'x + B'y + C'z + D = 0

88. Xet xem diem A ( 2 ; - l ; l ) va goc toa do O c6 nkm cung phia, hay khac
phia dd'i vdi mp.(P), mp(Q).
(P): 3 x - 2 y + 2 z - 7

=0

(Q):5x-3y + z-18

=0

(Q): 3x + 2 y - z + 3 = 0

2. (P): 2x + 3 y - 5 z - 1 5

A'x + B'y + C'z + D ' = 0
A'+B'+C^^

ditn (P; Q) hay nam trong hai goc ke nhau cua nhi dien nay?
=0

'Ax + By + Cz + D = 0
Do do (d):

89. Xet xem dia'm A(2; - 1 ; 3) va goc toa do O ciing n^m trong goc ciia nhi
1 (P): 2 x - y + 3 z - 5

NH6

Ta da biet giao tuyen cua hai mat phang phan biet cat nhau la mot dudng

86. Viet pt mp each d^^u hai mp: (P) 3x + 2y - z + 3 = 0 va (Q)
3x + 2 y - z - l

A. L t

1. Phirong trinh tong quat cua^ircmg thang

85. Tim tren true hoanh mot diem each deu hai mp:
(P): 1 2 x - 1 6 y + 1 5 z + l = 0 va (Q): 2x + 2 y - z - 1

III. P H U O N G T R I N H O U O N G T H A N G

0, A ' +B'' +C'' ^ Ova A : B : C ^ A' : B': C

He phuong trinh (1) la phuong trinh tdng quat ciia ducmg thang.
2. Phuong trinh tham so cua ducmg thdng
• Dinh ly: Trong khong gian Oxyz cho ducmg thang A di qua diem

=0

M() (xo; Yo; Z()) va nhan vecto a (a; b; c)

(Q): 5 x - y - 3 z - 7 = 0

0 lam vecta chi phucmg. Di^u

kien can va du de diim M (x; y; z) nam tren A la c6 mot s61 sao cho:

90. Viet PT mp phan giac ciia goc nhi dien (P; Q) biet P:
2x -14y +6z - 1 = 0, Q : 3x+5y -5z +3 = 0 va g6c toa do thudc goc nhi
didn CO mp phan giac.
91. Tim m de hai mp c6 PT: 3x -5y + mz - 3 = 0 va x + 3y + 2z + 5
vuong goc voi nhau.

(1)

=0

92. Trong khong gian Oxyz cho hinh lang tru diing ABC.A,B|C, v6i
A(0;-3;0), B(4;0;0); C(0;3;0); B,(4;0;4).
a) Tim toa do cac dinh A, va C,. Viet phuofng trinh mat c^u c6 tarn la A
va tiep xiic vdi mat phang (BCC,B,).
b) Goi M la trung diem ciia A,B, .Viet phuong trinh mat phang (P) di
qua hai diem A, M va song song vdi BC,.

x =

x^+at

y = yo+bt
Z =

trong do a^ + b^ + c^

0

(2)

ZQ+Ct

• Djnh nghla: He phuong trinh (2) la phuong trinh tham so ciia ducmg
thang A, trong do t la tham s6'.
Khi do A di qua diem (XQ; yo; z^) va vecto chi phuong la a (a, b, c)
3. Phucmg trinh chinh tic cua ducmg thang
Tit (2) va neu a, b, c deu khac 0, khii t d cac phuong trinh tren ta c6:

y-yo

^-^0

a

2-^0

b

c

(3)

Dinh nghla: Phuomg trinh (3) v6i a, b, c

3) A//

0 duoc goi la phuomg trinh

chinh tic cua ducmg thang

4) A = A'

4. Vi trf tuong doi gifia ducmg thang va mat phdng
Cho duomg thing A :

x =

XQ+at

y =

yo+bt

Z =

ZQ+Ct

^

a,a

.= 0

a,MM

^0

a,a

a,MM'

= 0

6. Khoang each
1) Khoang tic mot diem din mot ducmg thang
Cach 1: Mu6'n tim khoang each tut mdt diem M de'n duomg thang (A) ta

va mat phang ( « ) : Ax + By + Cz + D = 0

lam nhu sau:

De tim so giao di^m cua A va (a) ta giai he phuong trinh g6m cac

• Viet pt mat phang (a) qua M va vu6ng goc vdi duomg thing A

phuong trinh cua A va (a) va cho ta phuomg trinh

• Tim toa do giao di^m H ciia (A) va (a)

(Aa + Bb + Cc) t + Axo + Byo +

CZo

+ D = 0 (*)

M6i nghiSm ciia phuong trinh(*) ^n t umg vdi 1 giao di^m cua A va (a):

• Tinh d6 dai M H , do chinh la khoang each tir M de'n A ky hieu la
d(M, A)

1) Ne'u Aa + Bb + Cc 7i 0 thi phuong trinh (*) c6 nghiem duy nha't =>

Cdch 2: Sit dung c6ng thirc d(M, A ) = M H =

A cat (a).
2) Ne'u Aa + Bb + Cc ^ 0 va AXo + Byo + Cz^ + D ^ 0 thi (*) v6 nghiem,

(Mo € ( A ) , a

vecto chi phuomg ciia ( A ) .

Icliido A / / ( a )

2) Khoang cdch giita dudng thing vd mat phang song song

3) Neu Aa + Bb + Cc = 0 va
nghiem Idii do A c

A' <=>

AXQ

+ Byo +

CZ^ + D

= 0 thi (*) v6 so

(a).

sau:

5. V| tri tirong doi cua hai duomg thang
I.. ^

Trong Ichong gian Oxyz cho ducmg thang A di qua M va c6 vecto chi
phuong a , duomg thang A' di qua diem M ' va c6 vecto chi phuomg a', ta c6:

1) A va A ' cheonhau

De tinh khoang each gifia duomg thing A va mat phing (a) ta lam nhu

o

• La'y m6t diim tuy y MQ e A
• Tinh khoang each d(Mo, a) tiif Mo den A
Khoang each nay chinh la khoang each giua A va (a) va duoc ky hieu la
d(A, a).

a,a

3) Khoang cdch giQa hai ducmg thang cheo nhau
2) A va A ' cat nhau o

•

MM'

a,a
a,a

^0

=0

Cdch I. Di tinh khoang each gifia hai ducmg thing cheo nhau A va A' ta
lam nhu sau:
• Lap phuong trinh mat phang a chfia A va song song \di A'.

• Lay m6t di^m M tuy y trdn A' r6i tfnh khoang each tiir M d6n a.
Khoang each nay chmh la khoang each gifla hai ducmg thang cheo nhau A va
A' ky hidu la d(A. A')
a,d MM'
Cdch 2: S\x dung cong thiic: d( A ;A') = h =
a,a
(M e A, a la vecta chi phuofng ciia A. M ' e A', a'la vecto chi phuong
ciia A'.

Giai: Ta c6:
+

x + y-z + 3 = 0

(1)

2x - y + 5z - 4 = 0

(2)

1 4
3x + 4 z - 1 = 0 <:> x = - - - z
3 3
1 4
Datz = t,thayvao(1): - - - t + y - t + 3 = 0<::> y =

10

7
+ - r
3 3

1

4
1
3 3
10 7
y =
+ -t
^
3 3
z =t
x=

B. CAC V I DU
Ta CO PT tham s6':

V i du 1.
Viet PT tong quat cua ducmg thang biet PT tham so:
x = 2-t

I Duong thang nay di qua didm ( - ; - — ; 0) va vector chi phuong
3
3
4 7
1

z = - l + 2t
x-2 = -t
Giai: Ta c6:

x-2

^ =t
2
z+1
=t

_y

2
z+1_ y

»

2x +

y-A-Q

[y-z-\ 0

I 5 x - 7 y + 2 z - 3 = 0vampOxy
^ Giai: Mat phang Oxy c6 vecto phap tuyen k = (0; 0; 1) va di qua goc
toa d6 nen c6 phuorng trinh: z = 0.

V i du 2.

Vie't phucmg trinh tdng quat ciia ducmg thang biet PT chinh lie:
x-\

3

tz = 0

0

x-\
2
3
z-3 = 0

Vay pt ducmg thing phai tim la:
f5x-7y + 2z-3 = 0

z-3

2
Giai: Ta c6:

V i du 4. Viet phuong trinh ducmg thang la giao tuyeii ciia 2 mat phang:

0

^

3x-2>;-7 = 0
z-3 = 0

.

VfduS.
Viet phuong trinh ducmg thang song song
voi hai mp:

V i du 3.
Viet phuong trinh tham s6' (tiir do suy ra PT chmh iic) ciia ducmg thang
' ;c + ; ; - z + 3 = 0
biet PT t6ng quat:
l2x-;; + 5z-4 = 0

3 x + 1 2 y - 3 z - 5 = 0 , 3 x - 4 y + 9z + 7 = 0,
va cat hai ducmg thang:
x + 5 _y-3
-4

Hinh 40

_z + \' + l
3

-2

z-2


G i a i : Ducmg tiiang (A) piiai t i m la giao tuye'n cua liai mp (P) ^'a mp(Q)
M p (P) cluia (d,):

ciii phuong

_ z+1

x + 5 _ y-3
2 - 4

= (2; -4;

3

G i a i , M p phai t i m thudc chum mp:
m(2x - 3y + z - 3) + n(x + 3y + 2z + 1) = 0, (m^ +

d i qua di^m (-5; 3; - 1 ) va c6 vecto

n^^0)

ci> (2m + n)x + (-3m+3n)y + (m + 2n)z - 3m + n = 0

3). V i (A) la giao tuye'n ciia mp (P) va mp (Q),

f V i mp phai t i m song song vdri true Ox c6 vecto ehi phuong T = ( 1 ; 0; 0)

ma (P) // (P'), (Q) // (Q') nen (P) c6 cung vecto phap tuyen vdi (P);

nen ta c6: 2m + n = 0 <=> 2m = - n . Chpn m = - 1 , n = 2, ta duoc FT mp phai

3x + 12y - 3z - 5 = 0

tim la: 9y + 3z + 5 = 0.

^

= (3; 12; - 3 )

= 3(1; 4; - 1 ) , va (Q) c6 cung

vecto phap tuye'n vdfi ( Q ) : 3x - 4y + 9z + 7 = 0.

V i d u 7.

.^=(3;-4;9)

Viet PT mp qua dudng thang:

rx-3jv + 7z + 36 = 0

Vecto chi phuong u^ ciia (A) la tich c6 hudng ciia n, va n ^ .

^

=^ Vecto chi phuong
4
"A

. «2 ] =

=[«,

-4
V

-1 -1

1 1

4

3

3 3

-4J

9

9

= (32;-12;-16) = 4(8;-3;-4)

M p (P) C O cap vecto chi phuong u, va u^ nen c6 vecto phap tuyen:
Up

= [u,

]=
v

-4

3

3

2

2

-4

-3

-4

2

8

8

-3

= (25; 32; 26)

15

0

'^^'^^ S^'^

m(x - 3y + 7z + 36) + n(2x + y - z -15) = 0 ( m H n^ ^ 0).
<=> (m + 2n)x + (-3m + n)y + (7m - n)z + 36m - 15n = 0
Theo eong thiic khoang each, ta c6:
36m-\5n
^|(m + InY

25x + 32y + 26z + 55 = 0

M p (Q) chiia (d^):

x-3

y+\

-2

3

vecto chi phuong U2 = (-2; 3; 4) va

, d i qua diem (3; - 1 ; 2) c6 cap

4

= (8; - 3 ; -4) nen c6 vecto phap tuye'n
3

"o

= ["2;

MA ] =

4

4

-4 -4

-3

-2 -2

3

8

-3

8

»

PT ducmg t h i n g ( A ) phai t i m la:

4 y - 3 z + 10 = 0

Vidu6.
Viet phuong trinh mp qua ducmg thang:

•2x-3:>; + z - 3 = 0
X + 3;; + 2z + 1 = 0

va song song v6i true Ox.

+ {Im -

nf

= 3

85m^ + 19n^ - 104mn = 0

^ =1
n
m
19

4 y - 3 z + 10 = 0

25x + 32y + 26z + 55 = 0

+ (-3m + nf

<!> 8 5 ( — )^ - 104— + 19 = 0 ( v i n ^ 0)

= (0; 24; -18) = 6(0; 4; - 3 )
PTmp(Q)la:4(y+ l ) - 3 ( z - 2 ) = 0 »

^^^S 3.

G i a i : M P phai t i m thuoc chCim mp:

PT mp (P) la: 25(x + 5) + 32(y - 3) + 26(z + 1) = 0
o

"^^^

85

m = n
%5m = \9n

Chon m = n = 1, ta C O mp phai t i m la: 3x - 2y + 6z + 21 = 0
Chpn m = 19, n =85, ta c6 mp phai t i m la: 189x + 28y + 48z - 591 = 0.
Vf d u 8. T i m toa do giao didm cua ducmg thang:
.x-1

y-A

z-5

va mp 3 x - y + 2 z - 5 = 0


Giai: PT ducmg thartg c6 thi vie't dudri dang tham s6:
h^7

=2
yQ=-3=^Q(2;-3;2)

+ 5t

y = 4+t

ZQ =

2

z = 5 + 4t
Vi du 10. Tim diem B doi xiing v6i A ( l ; 3; -4) qua mp: 3x + y - 2z = 0.
Thay chung vao PT mp: 3(7 + 5 t ) - (4 + t) + 2(5 + 4t) - 5 = 0 o t = - l .
Thay lai vao PT dudng thang, ta dugc toa do giao di^m (2; 3; 1).

Giai: Trudc het, tim PT ducmg thang
y^B. Theo tinh chat d6'i xiing, ducmg

Vi du 9.

thang AB di qua A va vuong goc vdi (P)

Tim diim Q doi xiJng vdi P(4; 1; 6) qua dudng thang:

ndn

'x-y-4z
2x + y-2z

+

2x + y - 2 z + 3 = 0

3 x - 6z+15 = 0 o

phap

tuyen

n = (3; 1; -2) ciia (P) lam vecto chi

+3=0

phucng, suy ra phuong trinh ducmg thang
x = \ 3t
AB la: • y = 3 + t . Toa do giao di^m I

z = -4-2/
(1)
(2)

ciia AB va (P) la nghiem ciia he PT:

x = -5 + 2z.

Hinh41

Dat z = t va thay vao m6t trong hai PT ciia (d), ta c6 PT tham s6:
x = -5 + 2t
(d) \ = 7 - 2t

vecta

+ l2 = 0

Giai: Vie't PT ducmg thang (d) da cho
du6i dang tham so:
x - y - 4 z + 12 = 0

nhan

di qua diem (-5; 7; 0) va vecto chi phuong

z =t

x = l + 3t

x = l + 3t

y =3+t

y =3+ t

z =-4-2t •

z = -4-2t

3x + y - 2 z = 0

3(l + 3t) + 3 + t - 2 ( - 4 - 2 t ) = 0

x = -2

u = (2; - 2 ; 1). Viet PT mp (a) qua P(4;l;6) va vu6ng goc v6i (d) nSn

y=2

nhan u ciia (d) lam vecta phap tuydn.

z = -2

V a y P T m p ( a ) l a : 2 ( x - 4 ) - 2 ( y - l ) + z - 6 = 0 o 2 x - 2 y + z - 1 2 = 0.
Tim toa d6 giao diem A cua (d) va ( « ) bang each thay PT tham s6' ciia
(d) vao PT ( a ): 2(-5 + 2t) - 2(7 - 2t) + 1 - 12 = 0
t = 4. Thay lai vao PT tham s6' ciia (d) dugc toa d6 A(3; - 1 ; 4).
Theo tinh chat doi xiing thi A la trung die'm PQ, de dang c6:

t = -\

I(-2; 2; -2).

Theo tinh chat doi xiing thi I la trung di^m AB nfin de dang tinh dugc
toa do diem I .
-2 =|(1+^.)

3 = ^(4 + ^ , )
y,

=\
+>^z,)

2 = ^ ( 3 + ;^,)
- 2 = l ( - 4 + z,)

^A

= T ( 2 / ' + Z O )

4 = ^(6 +

Ze)


Vecta chi phuong ciia (D2):

=-5

XB

B(-5;

/

1;0).
U2

z,=0

=
\

Vi du 11.

-1 -1

2
-1

M,M^

3

\ + 2z-2

3

2
[M1,W2]=

=0

1 1

2

2 2

- J

= 5(1;-1;-1)

=(l;-3;-3)

Viet phucfng trinh hinh chieu ciia ducmg thang:
J5x-4y-2z-5

5

3

-1

3

1 I

-1 -1

1

2

II

= (l;4;-3)

-1

=0
[M,,M2].M,M2

= 1.1 +4.(-3) + (-3).(-3) = -.2

trSn m p 2 x - y + z - l = 0
U„U2

Giai: Ducmg thang da cho thupc chiim mp:
m(5x - 4y - 2z - 5) + n(x + 2z - 2) = 0
«(5m

+ n)x - 4my + 2(-m + n)z - 5m - 2n = 0 (P)

= V l ' + 4 ' + ( - 3 ) ' =:,V26

f A p dung c6ng thiic tinh khoang each ciia hai dufimg thang cheo n hau vao
bJli toan ta c6:

V i P 1 Q (mp Q: 2x - y + z - 1 = 0) ndn phai c6:
Hp

±

Hp

HQ

[u,,

d(D,A) =

=0

. HQ

U2

.M,M2

2
^1/26

_U,,U2_

2(5m + n) + 4m + 2(-m + n) = 0 ce> 3m = - n .
T.

Chon m = - 1 , n = 3, thay vao (P) duoc mp:
-2x + 4y + 8z - 1 = 0

BAITAP

93. Cho ducmg thang (d) c6 PT:
'-2x + 4y + 8z-l

FT hinh chieu cua ducmg thang phai tim la:

2x-y

+ z-l

=0
=0

V i du 12. (Dai hoc su pham thanh pho Ho Chi Minh - A, B - 20(X))
Trong khSng gian vdi he true toa d6 Oxyz cho cae ducmg thing:

x - z s i n a + cosaj = 0

( a l a t h a m so)
y-zcosa -sina = 0 •

1) Xac dinh vecta chi phuong ciia (d).
2) Chiing minh (d) tao vdi true Oz m6t goe kh6pg phu thu6e a.
3) Viet PT hinh ehi^i (d') ciia (d) trdn mp Oxyj
4) Chiing minh vdd moi gia tri a ducmg thing (d') lu6n tiS'p xuc vdi mdt

(D,):

•

(D^):

1 "

2

"

x + 2y-3
2x-y

+ 3z-5

3

'

=0
=0

dudng tr5n e6' dinh thu6e mp Oxy.
94. Dudng thing (d,) CO pt:

' x + h.-k
[(1 -k)x-ky

=^
=^

ring khi k thay ddi dudng thing d^ ludn:
Tinh khoang each giiJa (D,) va (Dj).
Giai. M , e D„ M , = (1; 2; 3) vecta chi phuong ciia (D,): ui = (1; 2; 3)
M 2 = ( 2 ; - i , n ) e (D^)

1) Di qua 1 diem cd' dinh.
2) Thudc mdt mp cd dinh.

vdi k

0, bat ky. Chiing minh

95. Viet PT mat phang chiia ducmg thang

x-2z^0
3x-2y

va vuong goc

+

z-3^0

v6i mat phang: x - 2y | f z + 5 = 0.
x^7
x-1
2

96. Chimg to rang hai ditcmg thang:

_y + 2 _
~ -3 ~

+ 3t

z-5
va < y = 2 + 2t
4
z = \-2t

\5x-3y
|03. Chiing to rang ducmg thang:•< ^
4x - 3y + 7z - 7 = 0.
04. Cho hai duofng thang c6 PT:
;c + >; = 0
(d,):

Cling thuoc mot mp, vijet PT mp do.
97. Tim PT hinh chieu cua ducmg thang
X

x-2
^

_ y + 2_
^

z-1
len mp
^

+ 2z-5 = ^ ,
, „ nam trong mp:

x + 3y-\ ^
, (d,):

x->' +z +4 = 0 ' ' ' '

[>' + z - 2 = 0

1) Chung to rang hai ducmg thang do cheo nhau.
2) Tinh khoang each gifla chiing.

+ 2y + 3z + 4 = 0.

3) Vie't PT ducmg thang qua M(2; 3; 1) va cat (d,), (dz).

98. Chumg minh rang hai iducmg thang sau day cat nhau:
(d,): X = 2t - 3, y = 3l[ - 2, z = 4t + 6.

;c = 2^ +1
5. Cho hai ducmg thing : (d,)

(d^): X = t + 5, y = -4rtj - 1, z = t + 20.

=0

- X - V +i 4

=0

(da)

vaCdj)

z = 3r-3

99. Tinh khoang each giiJj^ hai ducmg thang:
2x~z-\l

y = t^2

x-s-vl

3x + y-2

=0

3y-3z-6

=0

y=

ls-3

z = 35 +1

1) Chiing to rang (d,), (d2) la hai ducmg thang cheo nhau.
2) Tinh khoang each gifla (dj) va (d2).

100. Tim tap hop cac dien'^ M trong khong gian each deu ba diem A ( l ; 1; 1),
(Dai hoc tong hop Ha Noi, khoi A, nam 1994).

B ( - l ; 2 ; 0), C(2; - 3 ; 2J).
101. 1) Viet PT ducmg thalng di qua diem A(0; 1; 1), vuong goc vol ducmg
x—\+7

thang — ^ =

z

- Y

-

•>

ducmg thang •

'x + y-z

+2=0

x+l =0

3 '~ -2

_z-2
~

2

^

•

^

(Dai hoc Bach khoa Ha Noi, nam 1997).

a) Chiing minh rangi ducmg thang (d) va dudng thang A B cung nam
trong mot mp.
|
b) Tim diem I e (d) sao cho A I + BI nho nha't
102. Cho mat phang (P): 2x + y + z - 1 = 0, va du6ng thang (d):

. x + l _y-2
_ z-2
thang (d) CO PT:
3 — 2 2

Goi N la diem doi xiing cua M qua ducmg thang (d). Tinh do dai cioan
thing M N .

2) Cho A ( l ; 2; - 1 ) , B('[7; - 2 ; 3) va duomg thang (d) c6 PT:
x^\\_y-2

/. Trong khong gian v6i hd toa do Oxyz cho diem M ( l ; 2; -1) va ducmg

=y =

Viet PT ctia ducmg, thang qua giao diem ciia (d) va (P), vu6ng goc vdi
(d) va nam trcmg (I^).

. Trong khong gian, cho hinh binh hanh ABCD c6 hai dinh C(-2; 3; -5),
7
D(0; 4; -7) va giao diem hai duomg cheo M ( l ; 2; - - ) .
1) Viet PT ducmg thing chiia canh AB.

2) Tinh khoang each tiir goc toa do den mp chiia hinh binh hanh.
(Dai hoc dan lap Dong D6 Ha Noi, khdi A, 1997).


112. Trong khong gian v d i h6 tea d6 Oxyz, cho mat cSu

108. A A B C CO A ( l ; 2; 5) va PT hai trung tuye'n 1^:
x-3

_ y-6

-2

2

_ z-1

. x-4

^ y-2

1

-4

1

^

z-2

(S):

1

(P): 2x - y + 2z - 14 = 0

1) Viet PT chinh tac cac canh ciia AABC.
2) Viet PT chinh tdc duomg phan giac trong goc A .
(Hoc vien K y thuat quan su B6 Qu6'c phong).
109. Trong khong gian vdi he toa do Decac vu6ng goc Oxyz cho hai'ducmg

- 2x + 4y + 2z - 3 = 0 va mat phSng

1) Vie't phuong trinh mat phang (Q) chiia true Ox va cat (S) theo mot
ducmg tron c6 ban kinh bang 3.
2) T i m toa do diem M thuoc mat cSu (S) sao cho khoang each tiir M den
mat phang (P) Idn nha't.
(Trich d^ thi vao dai hoc khoi B - 2007).

thang:
Ix-2y+z-4=0
[x + 2y-2z

+ 4 = 0

y = 2+ t

va A ,

z = \ 2t

a) Vie't phuong trinh mat phang (P) chiia ducmg thang A, va song song
vdi ducfng thang A2.
b) Cho diem M (2; 1; 4). T i m toa d6 diem H thu6c ducmg thang A2 sao
cho doan thang M H c6 do dai nho nha't.
(Trich de thi vao dai hoc khoi A - 2002).
110. Trong khong gian vdi he toa do Oxyz cho hinh chop S.ABCD c6 day
ABCD la hinh thoi, A C cat BD tai goc toa do O. Biet A(2; 0; 0), B(0; 1; 0),
S(0; 0; 2 \/2 ). Goi M la trung diem ciia canh SC.
a) Tinh goc va khoang each gifla hai ducmg thang SA, B M b) Gia sir mat phang ( A B M ) cat ducmg thang SD tai d i ^ m N . Tinh the'
tich khoi chop S.ABMN.
(Trich de thi vao dai hoc khoi A - 2004).
111. Trong khong gian Oxyz cho 2 du5ng t h i n g :
d,
'-2

+ y^ +

z + 2
-1

1

'x = -l + 2t
vad.,: \ = \ t
z = 3

1) Chung minh d , va 02 cheo nhau
2) Viet phucmg trinh dudng thang d vu6ng goc vdi mat phang (P):

7x + y - 4z = 0 va cat hai ducmg thang d , va d2.
(Trich de thi vao dai hoc khoi A - 2007).


C. 3x - 3y + 2z + 1 = 0

+

3y + 2z + 1 = 0
2 + 4;
y = -6/ the tlii
z = -l-8r
A: =

ON TAP CHl/ONG III
1.

D. X

120. Cho phuong trinh tham so ciia ducmg thang (A)

CAU HOI T R A C NGHlfiM
phuong trinh chinh tSc cua dudng thang (A) la:

113. Trong khong gian Oxyz, toa do cua vecto a = 3 i + 2 k la
A.a=

(3; 0; 2)

B. a = (3; 2; 0)

C. a = ( 3 ; l ; 2 )

D. a = ( l ; 3 ; 2 )

A.

x-2 _y__ z + \

B.

x-2_
y _ z+1
~4~~ f 6
I

114. Cho vecto a {2; 3; -1), b(0; 1; 4), c ( l ; 0; -3). Xac dinh toa d6 cua
vecto 2 a -b - 2 c.
A.(2;5;l),

B. (2; 5; 0),

C.(l;4;l),

x+2
C.
4

A.

115. Cho 4 diem M ( - l ; 5; -10), N(5; -7; 8), P(2; 2; -7), Q(8; -10; 11)
. A. Hinh binh hanh
C.

i

IC.

B. Hinh vuong

ffinhthoi

-2

_z + l
~~5~

B.

x-3_y-6_z+l
-2 ~ -4 ~ 5

y-6
2

z +1
5

D.

x-3
4

x-Z_y-6
x-3
4

+

+ 4x - 2y - 20 = 0

A.(1;-2;0);R = 5,

B. (-2; 0; 1); R = 5,

C.(-2; 1;0);R = 5,

D. (-2; 1; 0); R = 4
1..

A. Cat nhau;

>

117. Viet phuong trinh mat phang qua goc toa do 0(0; 0; 0) vk hai diem
P(4;-2; l),Q(2;4;-3)
A.

X

+ 5y + lOz = 0

B. 2x + 5y + lOz = 0

.

B. | ;

C. 2;

D. 3

199. Viet phuong trinh mat phang qua diem M(2; 3; -1) va song song vdi
mat phang 5x - 3y + 2z - 10 = 0
A. 5 x - 3 y + 2z+1 =0

B. 5x + 3y + 2 z - 1 = 0 '

z+7
-5

fx = 1 + 2/

[ x = 6 + 3/'

A: ] y = l + t
z = 3 + At

A': ly = -l-2t'
z = -2 + /'

B. Cheo nhau;

C. Song song;

D. Triing nhau.

II. BAI TAP.
5. Cho tii dien ABCD. E, F, I theo thu tu la trung diem ciia AB, CD, EF.

C. X + 7y + lOz = 0
D. 3x + 7y + lOz = 0
118. Tinh Idioang each tir diem M ( l ; - 1 ; 2) den mat phing (P) c6 phuong
trinh lOx + lOy + 5z + 2 = 0
A.l;

y-6
2

It. Xac dinh vi tri tuong ddi cua hai ducmg thang

D. Hinh chu nhat

116. Tim tam va ban kinh hinh ciu c6 phuong trinh la:
+

x + 2 _ [y _ z + 1
D.
4
6

8

>1. Cho 3 diem M(3; 6; -7), N(-5; 2; 3), P(4; -7; -2). The thi phuong trinh
^ • duomg thang QP, Q la trung diem cua MN la:

D. (2; 4; 0)

TirgiacMNPQlahinhgi?

y
z-1
6 - 8

I a) Chiing minh lA + IB + IC + ID = 0
b) Vdi diem M bat ky trong khong gian, hay chiing minh:
4 MI = MA + MB + MC + MD
'124. Cho tii dien ABCD ma M la diem di dpng trong khong gian, G,, G2 Ian
lugt la trong tam tii dien va trong tam tam giac BCD
1) Chung minh G,C +G,5 +G,D = 0


2) Chvoig minli GA + GB + GC+ GD = 0
Ghi chu: Trong tarn cua tiJ dien la giao didm cac ducmg ndi m6i dinh ciia
tii didn tdi trong tftm cij a mat doi dien.
3) Tim tap hop diem M thoa man he thiic:
MA+ MB + MC + MD = 4MB + MC + MD
125. Trong khong gian vdi he toa do Oxyz, cho tii dien ABCD voi
A(3; 2; 6;), B(3, - 1 , 0), C(0, -7, 3), D(-2, 1, -1).
a) Chiing minh tur dien c 6 cac cap canh d6'i vuong goc v6i nhau.
b) Tim goc giua dirong ithang (d) di qua hai diem A, B va mp (a) di qua

badiemA,B,C.
c) Thiet lap PT mat ciu ingoai tiep tii didn
(Dai hoc Bach khoa Ha ]Noi, nam 1996)
126. Cho mat ciu (S) c6 PT: (x - 1)^ + (y - 1)^ + z^ = 6 va hai duofng thing:
(di): X = 1 + 2t, y = 3 - 2t, z = 1 + 2t
(dj): X = 1 - t, y = 2 + 2t, z = 1 - 3t
Viet PT mp tidp xuc mat c^u (S) dong thdi song song vdi (dj) va (d2).
127. Trong kh6ng gian vcti l?^ toa do Oxyz cho ba diim A(l; 0; 0),
B(0; 2; 0) va C(0; 0; 3).
1) Viet 0iuong tnnh tong quat cua cac mp (OAB), (OBQ, (OCA) va (ABQ
2) Xac dinh toa d6 tam I ciia mat ciu n6i tiep tii dien OABC.
3) Tim toa d6 diem J d6'i xiing vdi I qua mat phang ABC.
(Dai hoc Hue'-2000)
128. Trong khdng gian vdi he toa d6 Oxyz, cho diem A(l; 2; 1) va ducmg
thang (d) CO PT: - = — = z + 3.
3 4
1) Vict PT mp di qua A va chiJa ducmg thang (d).
2) Tinh khoang each tilt die'm A da'n ducmg thang (d).
(Dai hoc Kien true Ha Noi, nam 1997).

.4

19. Trong khdng gian v6i ht toa d6 Oxyz cho ba diem H -;0;0
K 0;1;0 ,1

a) Viet PT giao tuya'n ciia mp (KHI) va mp x + z = 0 of dang chinh tac.
b) Tinh cosin cia goc phang tao bdi mp (KHI) va mp Oxy.
(Dai hoc Giao thdng Van tai Ha N6i, nam 1997).
T30. Cho hai ducmg thkg c6 PT: (d) j
(A): x-5.

2

y-2

[x-y-z

^^ ~ ^

z-6

+5=0

1
3
1) Xac dinh vecto chi phucmg ciia ducmg thang (d)
2) Chiing minh hai du5ng (d) va A cung thudc mdt mp, viS't PT mp do.
3) Vi6t PT chinh tac ciia hinh chie'u song song cua (d) theo phuong (A)
I6n mp 3x - 2y - 2z - 1 = 0.
1; (Dai hoc Xay dung Ha Ndi (HS chua phan ban), nam 1997).
r31.,.Viet phuong trinh dudng thang di qua dilm A(3; -2; -4), song song vdi
X — 2. v + 4 z — \
mp 3x - 2y -3z -7 = 0, ddng thdi cat dudng t h ^ g
=
=
3
2
(Dai hoc Thuy Igi Ha Ndi nam 1997).
132. Viet phuong trinh mp chiia gdc toa dd va vudng gdc vdi hai mp cd PT:
X - y + z + 7 = 0 va 3x + 2y - 12z + 5 = 0.

(ViSn Dai hoc Md Ha Ndi, khdi A, nam 1997).
133. Cho hai didm A(0; 0; -3), B(2; 0; -1) va mat phang (P) cd phuong trinh
la: 3 x - 8 y + 7 z - l =0.
1) Tim toa dd giao di^m ciia dudng thang di qua hai di^m A, B vdi mat
phang (P).
2) Tim toa dd diem C nam trfen mp(P) sao cho tam giac ABC deu.
(Dai hoc Qudc gia Ha Ndi - A - 2000).
4Z

7^


4) Chiing minh rang nS'u di^u kien tren duac nghiSm diing t h i mot trong
hai so b, c nho hem a, s6' con lai 16n ban 2a.

ON TAP CUOl N A M

j39. Trong mp (?) cho du6ng thang d co dinh va mot diem c6' dinh O g d, mot
goc vuPng Oxy quay quanh O, Ox va Oy cat d tai A va B. Cho d' ± P

134. Mot hinh h6p chu nhat c6 do dai dudng cheo d, no tao vdri day goc a
va

mat ben

\dn goc

p . Chiing minh the tich hinh hop bario

d^'sin a sin /? -yjcosia + P) cos(a -

tai O. Lay S € d' thoa man SO = ^ ,

SA = | o A .

Khoang each t i i O

P).
d€n d bang a va O A B = a .

135. Day hinh chop la tam giac vuong c6 canh huyen a va goc nhpn a . Mat

a) Tinh a.

ben qua canh huyen vuong goc v6i day, hai mat con lai tao vdi day goc
B. Chiing minh the tich hinh chop bang —Q

sin

b) Ke OE 1 SA, OF 1 SB. T i m quy tich E, F k h i xOy quay quanh O.

latgP

c) Gpi G la trpng tam A SAB, I la tam mat cau ngoai tiep tii dien SOAB.

2 4 V 2 s i n ( a + 45°)

Chiing minh O, G, I thang hang.

136. (Dai hoc Quoc gia thanh pho Ho Chi M i n h A - 2000).

tie'p xiic

Cho tam giac din ABC canh a. Tren dudng thdng d vuong goc vdi m;ii
phang (ABC) tai A lay diem M . Goi H la true tam tam giac ABC, K I t
true tam tam giac BCM.

. Cho hinh c^u (O, R)

v6i mat phang (P). Cho hinh non (nam

1) Chiing minh rang M C 1 (BHK) va H K 1 (BMC).

bang X .

2) K h i M thay doi tren d, tim gia t r i Idn nhat cua the tich t i i dien KABC.

a) Cho X < 2R va X < h. Tinh t6ng dien tich S cua hai thiet dien. Bieu

137. Cho hinh chop t i i giac deu S.ABCD vdfi day la hinh vuong A B C D co

thiic t i m dupe co con thich hop kh6ng n€\x h < x < 2R (keo dai cac

cdng phia vdi hinh cin doi vdfi (P), day thupc (P), dudng cao h, ban kinh
day bang R. Cat hai hinh bang mp (Q) // (P), each nhau mot khoang

canh bang a. Mat ben tao vdi day mot goc 60".

i duofng sinh ciia hinh non dd chiing cat (Q).

Mat phang (P) chiia canh A B va cat SC, SD l^n luot tai M va N. Cho biet

\h) Khao sat sir bieh thien va ve do thi S (x la doi so). Bien luan cac
trucmg hpp.

goc tao bai mat phang (P) va mat day hinh chop la 30".
1) T i i giac A B M N la hinh gi? Tinh dien tich t i i giac A B M N theo a.
2) Tinh the tich hinh chop S.ABMN theo a.
138. Cho goc tam dien dinh O, cac goc b dinh deu bang 60". Tren cac canh
Ox, Oy, Oz ta lay cac didm A, B, C sao cho O A = a, OB = b, OC = c.
1) Cho a = b = c, thi hinh chop OABC co gi dac biet? Tinh khoang each

141. Cho hinh chop S.ABC co day ABC la tam giac deu canh bang a, SA
vuong goc vori mat phang (ABC); SA = a; I la trung diem ciia BC.
a) Tinh khoang each tir A den mat phang (SBC).

nl

b) Viet phuong trinh mat ci\x ngoai tiep tii dien SAIC.

•2. Trong khong gian vdri he toa dp Oxyz cho hai diem: A(0; - 2 ; 0), B(2; 1;

tir O den mp (ABC) va tinh the tich ciia hinh chop nay.

4) va mat phang (a): x + y - z + 5 = 0.

2) V 6 i a 5^ b 9i c, tinh cac canh cua AABC theo a, b, c. Chiing minh dien

a) Viet phuong trinh tham so ciia dudng thang d diilqua A va B.

kien can va du de B A C = I v la be + 2a^ = a (b + c).

b) T i m tren dudng thang d diem M , sao cho khogng each tir M den mat

3) Cho biet a va b + c = d, B A C = Iv. Tinh the

phang (a) bang 2 V3 .

tich cua hinh chop the(

a va d. Lap phuong trinh de tinh b, c trong truomg hop nay. T i m diei

c) Viet phuong trinh mat cin (S) co dudng kinh AB. Xet vi t r i tuong doi

kien de tinh dupe b, c.

giiia mat c^u (S) va mat phang (a).


143. Trong khdng gian vori he tea d6 Oxyz cho 4 di^m: S(2; 2; 6), A(4; 0; Oj
x=t
x + 3>'-l = 0
B(4; 4; 0), C(0; 4; 0).
A,: y = -t va A j : • [y^z-2=Q
a) Chung minh rSng hinh chop SABCO la hinh chop Hi giac d^u.
z = -A-2t
b) Tmh th^ tich cua khoi chop SABCO.
la) Vie't phucmg trinh mat phang (P) chiia A, va song song v6i A2.
c) Vie't phuang trinh mat ciu ngoai tie'p hinh chop S.ABCO.
l b ) Tinh khoang each giiia A, va Aj.

144. Trong khong gian v6i he toa do Oxyz cho hai ducmg thang c6 phuang |c) Viet phuang trinh ducmg thang A 3 di qua M(2; 3; 1) va cat ca A,
trinh Mn luat la:
fva A 2.
\2x-y + 3z-5 = 0
2x-2y-3z-n
= 0 va diem A(3; 2; 5). l48. Trong khong gian Oxyz cho 2 mat phang:
A,: [ x + 2y-z = 0 A , :
2x-y-2z-3
=0
(«):2x-y + 2 z - l = 0
a) Tim toa do diem A' doi xiing vdi diem A qua ducmg thang A2.
(/?):x + 6y + 2z + 5 = 0
b) Lap phuang trinh mat phang di qua ducmg thang Aj va song song vui
a) Chiing minh rang ( or) va (/?) vu6ng goc voi nhau.
ducmg thang A2.
b) Lap phuang trinh t6ng quat cua mat phang (P) di qua goc toa do O va
chiia giao tuyen ciia hai mat phang ( a ) va (/?).
c) Tinh khoang each giua hai ducmg thang Aj va A2.
145. Trong khong gian vdi he toa do Oxyz cho 2 duotng thing:
f x + y + z-4 = 0
^ ^ x-l y_z-2
A,:• <
va A,:
2x-y + 5z-2 = 0
-2 3 1
a) Xet vi tri tuang doi cua A, va A 2.
b) Cho diem A(0; 1; 3). Tim diem M trtn A 2 sao cho doan AM ng. '
nha't.
146. Trong khOng gian vdfi he toa d6 De cac vu6ng goc Oxyz cho hai die
A(l; 2; 1), B(2; 1; 3) va mat phang (P): x - 3y + 2z - 6 = 0.

a) Viet phuang trinh mat phang (Q) di qua A, B va vuong goc v6i m.'
phang (P).
b) Goi ducmg thang A la giao tuye'n ciia hai mat phang (P) va (Q). Hay
vie't phuang trinh chinh tac cua dudng thang A.
c) Goi H la hinh chieu vu6ng goc cua A tren mat phang (P). Tim toa do
ciia diem H.
147. Trong khong gian vdi he toa 66 Oxyz cho hai du&ng thang


HLfdNG

1.

D A N GIAI - D A P

S O

a) Trong khoi da dien m6i
canh la canh chung ciia dung
2 mat
b) Cung sir dung tinh chat
tren

2.

Chia khoi lap phuofng thanh 6
khoi tii dien.

3.

Cho tir dien deu ABCD. Tarn
ctia cac mat ABC, BCD,
ACD, ADB la M , N, P, Q

Hinh 45

Wi

Hinh 43

Xet khoi tii dien MNPQ. Dung dinh ly Talet ta chimg minh M N =
NP = ^ ; M Q =

^ . D o d 6 M N = NP = NQ = MQ = MP = PQ=

Khoi tii dien deu canh a c6 the tich ^ ~ ^

Khoi tii dien MNPQ canh - c6 M ti'ch V =
.3
3
12

6

l A B , BC 1

SA

BC 1

(SAB)

BC

A B ' ± SC; A C X SC => SC 1

(AB'C).

(AB' 1 (SBC) nen AB' 1 B'C). Ta c6 AB' = ^
3

~

1

AB'

c) Cho S.AB'C CO chieu cao SC, day la A AB'C vuong b B' vi

^.

va A C . SC = a.AC

12
.a SC = a
ma

nen A C = ^

B'C = ^

3'2'

6.

2

" 6

va SC =

^

6

Vay F^^^,^. = T ( T - ^ - — ) • —

b) VAEDF = VAEFC vi S AADF = S A^pcva ciing chidu cao EH. Chiing minh
tuong tu ta suy ra:

'

3

= —

36

3
Vs.ABc = 6 VS.ABC

•

Tren canh CD la'y E chia trong CD theo ty so —.
n
No'i BE va AE. Mat phang ABE chia khoi tii dian ABCD thanh 2 khoi

V BEFC-

c) Neu ABCD la tii dien deu khi do ta chimg minl> (EDC) la mat phang

chop CO cung chieu cao. Ta dh dang chiing minh

doi xiJng cua hinh, (ABF) cung la mat phang dd'i xiing cua hinh. Khi do
ta chimg minh dugc 4 khd'i til dien AEDF, AEFC, BEDF va BEFC bang
nhau theo nghia c6 phep ddi hinh (doi xiing qua mat phang) bien khoi

^
3

324

a) AEDF - AEFC - BEDF - BEFC.

nay thanh Lhoi kia.

2

:> AB' 1 (SBC) ^

* ABCD - ^ ' • V MNPQ

^AEDF - ^AEFC " ^BEDF "

3

b) Tit BC

AD

= 27. V .

4.

a) V 3 . , B c = ^ - ( ^ a ^ ) a = ^ -

i

AC ED

"

-


Hinh

Hinh

46

Hinh

'

47

i f ;

7. a) Day lang tm la A ABC c6 difin tich — ^ .
4

Tarn day Ik H v i A ' A =

= A ' B = A ' C nen A ' H 1 (ABC) va A ' A H = 60° la goc gifta A A '

Dien tich ASAC =-SA.S;G:sin ASC
2
,
Dien tich AS A C ' = - SAS C' sin A 5 C
2
i
1

-S^,.BH

mat
'

Dodo

day. Ta c6 A H = ^ . - ^
3
2

= ^ n e n

A H ' = A H tg60° =

1

A H , BC

1

AH

=>

A'AH

= 60°

=> A A ' H

= 30°

SA

3

_

SA

7-

SC

SB
r.

SC

.
r(dpcm).

SB

A p dung di^u nky vao bai tapjt s6 5 vdi SC =
BC

1

(AA'H)

=>

BC

1

AA'

=> BC 1 BB' => BCC'B' la hinh chG nhat.
c)

^S.ABC

= a

Vay the tich lang tru A B C . A ' B ' C =
b) BC

48

AA'

= 2AH

Vs.^sc _ SA
S.ABC

= 2a => SBCC-B' =

2a'

Tiir E ve EK 1 A A ' => BKC la thiet diSn thing cua lang tm. Ta tinh duoc

SC

a) A A ' 1 A B , A A ' 1

, SC = a VI.

1 i =1

SB

~ SA'SC'SB'

I

" n'2'3
AC,.\ 1

6'
A C => A B 1 ( A A ' C C ) => B C la

ducmg xien c6 hinh chie'u tren, ( A A ' C C ) la A C => B C ' A = 30°.
=> A C i= A B . cotg30°' = AB> /3 ma A B = AC.cotg30° = b V3 => A C = 3b.

Vay

= a l Vl3 + 2a' = a'(2 + Vo

) don v i dien tich

Ve chop S.ABC. TiT B va B' ha B H 1 (SAC), B ' H ' ± (SAC). K h i do
BH
B'H'

b) ^ . . c . V c

=SAABC.CC=

= (lb.bV3). V%^=

^ b ^ ^ = b l V 6

^ _SB_
~

SB''
81


Dat A ' H =

AA

AM

-AM'

1

AC ( H e

AC)

A A , B D can ( do A , B =

BD X A , 0 . Mat khac BD 1 A C

BD 1 ( A , A O )

A,D)

That vay: ve A , K 1 A D => H K 1 A.K
a
AH
AK
coscp cos— =
. T2
AA, AH

i

1-

cos a
2 oc
cos —

AK

•
^ a sin« .
2

a
cos —
2

\o
Jcos^ —

11. Ke A ' H 1 (ABCD),
PiK l ' A B ( K e AB)

2

^ M p ~ ^MEHK • A A i

cos

2

a

fSi

2

cos a

'^

1

•

cos^a = 2a si

a
2

\

cos 2 a

H e (ABCD) va ke H M 1

_

AD (M e

A ' M H = 60°, A ' K H = 45°

A'K.

= xa,

8^.82

S2

—

— x y . s i n a .a

= ya.
1

S,

2

a

a

.sin a .a

sin a
2a

cos 2 a .

Theo dinh ly 3 ducmg vuong goc ta c6 /s.D 1 A ' M , A B 1

82

xy.sin«

A , H = a sincp.

a
cos —
2

Vay the tich khdi h6p V = (AB.AD) sin, a . A , H =
^

I

SMEHK=

cosa

a
cos —
2

= cosa.

AA,

V = a.b.x = abc . ^ (don vi the tich).
G o i 0 0 , la giao tuyen ciia 2 mat
cheo ( A , C i C A ) va ( B , D i D B ) . Qua
l € 0 0 , ke lin lugt 2 ducfng thang
K E va M H deu vuong goc v 6 i
0 0 ] . K h i do a la goc gifia M H va
: E va M E H K la thiet dien thang
:iia h6p. Dat K E = x, M H = y t h i

a
cos a = coscp c o s y . (*)

= a

=.x = c J - .

=

BD 1 A , H

Dan den A , H 1 (ABCD). Dat A, A C = c(? ta c6 he thiic:

Tir (*) suy ra coscp =

3c'-4x'

vay the tich lang tru la

Hinh 49
10. Ha A , H

S

= xcotg45" =

3c'-4x'
X

sin 60"
=

= HK, nhung H K

=>

2x

A'M =

X

•

Goi O la tarn hinh vu6ng ABCD,
SO 1 (ABCD)
AD),

Goi E H la ducmg trung binh
cua ABCD, v i A D // (SBC)
=> khoang each ttr A den (SBC)
chinh la khoang each tir E
de'n (SBC). Ke E K
1
SH
EK 1 (SBC) => E K = 2a.

x