Tài liệu tong hop hinh hoc khong gian 12

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 THỂ TÍCH KHỐI ĐA DI Ệ N
I/ Các công thức thể tích của khối đa diện:
1. THỂ TÍCH KHỐI LĂNG TRỤ:


B: dieän tích ñaùy
h : chieàu cao





 Thể tích khối hộp chữ nhật


 Thể tích khối lập phương






2. THỂ TÍCH KHỐI CHÓP

1
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
B : dieän tích ñaùy
h : chieàu cao



3. TỈ SỐ THỂ TÍCH TỨ DIỆN
 !"###
$%&'()*+,""
"-
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SA 'B'C '
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SA SB SC
V SA ' SB' SC'
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Chú ý:

./012345,62
7

012345899:2


0123459;8-
7 7 7
a b c
+ +

7/0124&2$<,

7
a
/=5-9<,5-9-$(2$<,$><,?2
,@A-$(2$<,5B,4CD'2E&4$(
F/GH2DI<,H2DI 2-$(2$<,
BÀI TẬP
.59"-$(&2$,6212J2",622-&9
@B
a
"
 K%-9"L
 MNOD,2%&4"OL
75-9&2$<,"-$(?2>?27MNOD,2%&4

  2&",622-
 K%-9"OL
5-9"-$(&2$,62>",622-$(B

"K%-9"
.
F5-9"P-$(P5,62>",622-$(
>"?2
a

 K%4-9"P
  2&D,2%&4"%&$<,$C45-9"P
Q5-9"-",622-,R26&B"

a
K%4-9"
S-9"-&A"&2$<,?&D2&9,622-
,B.%4-9"
T-9"-$(&2,62E5B,,622-4">@
D'2DN2E&M4&2$B"+9$(2-
U
SU
α
=
K%4
-9"
V-9"P-$(P5"&2$<,
""PK%4-9"P
W-9"P-$(P5;8",622-&A$(@P
B"7
K%4-9"
.U-9"P-$(P52,62X",622-&A
$(@P"P7
K%4-9"P

..59"P-$(5,62",622-&A$(@P2-
2;"$(@PFQ
U
K%4-9"P
.7-9"-$(&2$,62X70C"$<,&A
>@"+9&A$(@2-SU
U
K%-9"
=P.7

.H2DI###-$(&2$<,?2>?2
a
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B. MẶT NÓN, MẶT TRỤ, MẶT CẦU.
I) MẶT NÓN, HÌNH NÓN, KHỐI NÓN:
1) Mặt nón:
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2-α@UmαmWU
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f,12J2∆2N&A-
* d: đường sinh
*
∆
: trục
* O đỉnh
* 2
α
: góc ở đỉnh
2) Hình nón:
=5-Dnd(5YDX&
&2$,62f,(f,&
2-,62
oDiện tích xung quanh:"
df

π
Dl
l: độ dài đường sinh
r: bán kính đường tròn đáy.
3) Khối nón:
=5-'29*D24-

F
+2N-
oThể tích khối nón: 
π

.
D
7

h: độ dài đường cao
r: bán kính đường tròn đáy
II) MẶT TRỤ, HÌNH TRỤ, KHỐI TRỤ:
1) Mặt trụ:
12J2∆Y2Y2
,$,&p2?2D
[ADnd(YX12J2
f,(f,∆2N&ADI
* d: đường sinh
*
∆
: trục
2) Hình trụ:
=5DIDnd(5YDX&
5;8f,(f,&
oDiện tích xung quanh:"
df
7
π
Dl
l: độ dài đường sinh

r: bán kính đường tròn đáy.
3) Khối trụ:
=5DI'29*D24-
+2NDI
oThể tích khối nón: D
7

h: độ dài đường cao
r: bán kính đường tròn đáy
 Chú ý: DIl.
III) MẶT CẦU, HÌNH CẦU, KHỐI CẦU:
1) Mặt cầu:
%&ikYqD
K89+9$%&[D2622
$%&i&p2?2D+
2N&A*,E&i$D
r!,"@iD
{ }
Di[[
=
Chú ý: oisD
⇔
?&2@"
oimD
⇔
?&D2@"
oiD
⇔
?&D>@"
2) Vị trí tương đối của mặt phẳng và mặt cầu:

&A*,"@iD&A9J2@tMN=5B,4iD>&9@ti=p2$
RiB&9@t
osD
⇔
@t62l@"(@t
∩
@"
φ
oD
⇔
@tB9du@"=
r-(S): tiếp diện, (H): tiếp điểm
omD
⇔
@tl@"L12Dn@-E&=$
77
D
−
 Chú ý:B,U(i≡=5@tl@"L12Dn@iD
3) Vị trí tương đối của đường thẳng và mặt cầu:
&A*,"@iD12J2∆MN=5B,4iD>∆i=p2$R
iB∆
osD
⇔
∆62l@"(∆
∩
@"
φ
oD
⇔

∆B9du@"=
r-∆: tiếp tuyến(H): tiếp điểm
omD
⇔
@tl@"%&9E!
Q
FP!d,2f,5*,%*,
oP!d,2f,5*,"
df
F
π
D
7

oK%*,

F
π
D


BÀI TẬP
.5-Dnd(-127U&$$(D7Q&
K!d,2f,45-]
K%4-
[B!f,C45--p2$RE&4$(B&A9J2 B!.7
&K!B!-
7[5DI-$$(DQ&-p2$2;$(?2T&
K!d,2f,45DI%4DI
lDIX&A9J2Y2Y2-DI$DI&K!4B!+

>
l5-?2&&A9J2f,DI4-+&B!&&2$<,7K
!d,2f,%45--
F[5DI-$$(D<,D


K!d,2f,!9*45DI
K%4DI
%&*+?&D>12Dn$(Y2-2;DI45DI
?2U
U
Kp2$2;DI45DI
Ql5-C"X&A9J2f,DI+&&2$,62E-,(<?2
7

K!d,2f,!$(%-
E(,2412Dn$(5-Y&A9J2@"&A9J2 
$(5-&2-SU
U
K!&2$"
S[A9J2f,DI45DIl5DILB!5,627b
K!d,2f,!9*45DI
K%4DI
K%H2DI 2$<,B95DI
T[--2-XC?2.7U
U
-$$(?2DK!4B!f,
12Y,622-,
V[H2DI 2-<,-$(&&2$<,K%4DI
2B9H2DI(

W[ !<,-?2B9D2&-K%4--
.U[DI2NB9D2&*,B,12Dn$(4DI?&D>&A4
*,
K!d,2f,%4DIB9D2&*,$bB,B
124DI
K2$Dkv4%DIB9D2*,$bD
..5899:2P###P#
K!d,2f,%44DI-12Dn4$(2B9$5
,62P###P#
K!d,2f,%4--CE&i45,62P$(
12DnB95,62###P#
.75899:2P###P#j$kE&$&A*,f,V
C45899:2]
. !P-P⊥@PQ&2$,62F
j$kE&$&A*,f,C4 !
.F5-9&2$<,"-$(?2>?2
j$kE&$&A*,f,$C5-9
.Q  !P-P⊥@PF&2$,62S
Vj$kE&$&A*,f,CP4 !
.S5-9"P-$(P5,62"⊥@P"7
j$kE&$&A*,f,QC"P
.T5-9 2$<,"P-$(?2>?2j$kE&
S
$&A*,f,QC"P
.V)5H2DI###-vp$<,?2
 j$kE&$&A*,f,$C4H2DI
 K!&A*,%4*,:2 2
.W 5-9"P-$(5,62"⊥@PPq2&9@tf,
,622-"[A9J2@tl"""P##P#
 [bT%&P###P#,6?&D>&&A*,

 K!&A*,%4*,+
7U 5-9&2$<,"-$(?2&A>+9$(&2-?2SU
U

 j$kE&$&A*,f,$C4H2DI
 K!&A*,%4*,:2 2
7. 5-9"-"""-<,
 j$kE&$&A*,f,$C4H2DI
 K!&A*,-
T
PHƯƠNG PHÁP TỌA ĐỘ TRONG KHÔNG GIAN
OHỆ TỌA ĐỘ TRONG KHÔNG GIAN
A. KIẾN THỨC CẦN NHỚ
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=!2w&DIidi(ix&,622-+2N!DIN,622-D2
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kji 
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ikkjjikji
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yy@yy@
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zyxvzyxu
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Rnmnzmznymynxmxvnum
∈+++=+
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7
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7
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z
z
y
y
x
x
kzz
kyy
kxx
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



=
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→→→→→→
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
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
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vu
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→→
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xz
xz
zy
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→→

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→→→
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→→→
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U}|
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→→→
wvu
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7
.
=
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[ ]
Z
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=
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ABCD
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.
=
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7
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7
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b
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−++
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W
II.PHƯƠNG TRÌNH MẶT PHẲNG
OKIẾN THỨC CẦN NHỚ.
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oL:
→→
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Un
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
α
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α
B
l
@
α
⊥
→
n
o\B,
yy@yy@
777...
zyxvzyxu

==
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@
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
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77
..
77
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77
..
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yx
xz
xz
zy
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OOBI TP.
1.i qua im v vuụng gúc, song song mt phng
1)Vieỏt phửụng trỡnh maởt phaỳng (P) ,qua :
M(2;-4;1) ,N(3;-2;-4) vaứ

(P) : 3x +4y-2z 5 = 0.
ẹS :15x -13y-2z-82 = 0
E(-4;1-2) vaứ

(P) : 2x -3y+5z 4 = 0 vaứ (Q) : x +4y-2z +3 = 0

ẹS : 14x -9y-11z + 43 = 0
F(3;-2;1) vaứ chửựa giao tuyeỏn hai mp(P) : x +2y-4z 1 = 0; (Q) : 2x -y+3z +5 = 0.
ẹS : 14x +13y-23z +7 = 0
Qua B(1;2) va giao tuyeỏn hai mp(P) : 2x -3y-15z +3 = 0;(Q) : 4x -2y+3z -6 = 0.
7KD2622!DIid(x%&@&A9J2@d7(xU
G899:2D5&A9J2@tf,,622-@
S:x - 2y + z - 2 = 0
KD2622id(x%&@~.yy~7@~T~.V&A9J2@t7d~(x.UB9:2
D5&A9J2 ,622-&9@t
S: 2x + 5y + z

11 = 0
FTrong không gian với hệ tọa độOxyz cho hai đờng thẳng d và d lần lợt có phơng trình :
d :
z
y
x
=


=
.
7
và d :
.
Q

7
7

+
==

z
y
x
.
Viết phơng trình mặt phẳng
@

đi qua d và vuông góc với d
S: 2x+y-z-2=0
QB9:2D5&A9J2D2&D12+9Y,B9:2D5&A9J2@t
,%&[
U
@d
U
y(
U
yx
U
*+Y2Y2$&A9J2N@id(@i(x@idx
,$5B,4%&[
U
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U
y(
U
yx
U

@d
U
(
U
x
U

U

>$idi(ix
,%&[
U
@d
U
y(
U
yx
U
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U
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U
x
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U

*+ $DIidyi(yix
QB9:2D5&A9J2D2&D12+9Y,B9:2D5&A9J2@t
0f,%&@~.y7y@7y~Fy@FyQyS

0f,%&[
U
@.yy~7,622-DIi(
0f,[
U
@7y~.y,622-@Uy7y.@.yFy.
0f,[@.yy7Y2Y2&A9J27d(xFU
L0f,%&@y.y~.@7y~.yF,622-&A9J2
d(7xU
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U
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U
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K5&%%&@.y7y.@.yyU@.y~7y.P@.y.y.,&&A9J2
2. Quan hệ với đường thẳng: Chứa đường thẳng, vng góc với đường thẳng và song song với dường
thằng
1)Cho(d
1
) :
2 4 0
2 2 4 0
x y z
x y z
− + − =



+ − + =

; (d
2
) :
1
2
1 2
x t
y t
z t
= +


= +


= +

a)Viết phương trình mặt phẳng (P) chứa (d
1
) và song song (d
2
).
ĐS : 2x – z = 0
b)Cho M(2;1;4) .Tìm toạ độ điểm H thuộc (d
2
) sao cho MH nhỏ nhất ?
ĐS : H(2;3;3)

2)Viết phương trình mặt phẳng (P) :
a.Qua A(3;-2;4) và (d) :
3 2 1 0
2 3 0
x y z
x y z
+ − + =


− + − =

.
ĐS : 29x + 17y – 8z -21 = 0 .
b.Qua B(1;4;-3) và
2
2 1
1 3
x t
y t
z t
= −


= −


= −

.
ĐS: 7x - y – 3z -12 = 0 .

c.Qua C(2;-1;5) và
3 1
2
2 3
x z
y
+ −
= + =
.
ĐS: x +7 y – 3z +20 = 0
VB9:2D5&A9J2$<,12J2
.

7
B

.
7
 7

x t
d y t
z t
= +


= +


= −



7
. 7 .

7 . Q
x y z
d
− − −
= =
ĐS:3x – y – 4z +
T U=
2)Viết phương trình mặt phẳng (P) qua M(2;-4;1) và chắn trên ba trục toạ độ theo:
e) Ba đoạn bằng nhau .ĐS : x + y+z+1 = 0
f) Ba đoạn thành cấp số nhân công bội bằng 2. ĐS : 4x + 2y+z-1 = 0
g) Đoạn trên Ox bằng 3 lần các đoạn trên Oy và Oz. ĐS : x + 3y+3z+7 = 0
 Ba đoạn a,3a,5a .a
*
∈ ¡
. ĐS :15 x +5y+3z -48 = 0
5)(d) :
1 3 2
2 3 4
x y z− + −
= =
,(d’) :
2 1 4
2 3 4
x y z+ − −
= =

a.Chứng minh : (d) // (d’) .
b.Viết PTmp chứa (d) và (d’) .ĐS : 10x + 16y – 17z + 72 = 0 .
SKD2622!DINid(x12J2∆
. 
. . F
x y z− −
= =
%&[@Uy~7yUB
9:2D5&A9J2@tf,%&[Y2Y212J2∆w21p2$2;12J2∆
&A9J2@t?2F
ĐS:4x - 8y + z - 16 = 0. Hay 2x + 2y - z + 4 = 0.
.7
3. Quan hệ giữa hai mặt phẳng
Qj3kD:24$A9&A9J2X$9:2D5Y,
d•(€7x•FU.Ud•.U(€7Ux•FUU
7d•d•x€QWd•S(•Wx•QU
d€(€x•.U7d€7(•7x€U
S&A9J2-9:2D57d•&(€x•SU@&€d•7(€@Q&€.x•.UU
2$Dk4&5&A9J2-
"2Y2,
KD'2,
l,,622-,
4. Một số bài tập liên quan khác
.D22!Nid(x%&@.Uy7y~.12J2-9:2D5





+=

=
+=
tz
ty
tx
.
7.

G899:2D5&A9J2@tf,Y2Y2p2$R@tv
ĐS:7x + y -5z -77 = 0
712J2@P-9:2D5
7
7
7 7
x t
y t
z t
= − +


= −


= +

MN
∆
12J2f,%&@FyUy~.Y2Y2
@PO@~7yUy75B,,622-4D>@PKD2$&A9J2f,
∆

](B9:2D54&A
9J2-p2$B@Pv
ĐS:
7d~x~WU
.
%&
( )
7yQyA
12J2
. 7
 
7 . 7
x y z
d
− −
= =
B9:2D5&A9J2
( )
α
 
d
Y
p2$R
A
B
( )
α
v.
ĐS:
7 7 .Q Ux y z+ + − =

FKD2622!DINid(x$%&@~.y.yU@UyUy~7@.y.y.=](B
9:2D5&A9J2@tf,%&w21p2$R&A9J2@t?2


Đs: x-y+z+2=0 7x+5y+z+2=0
QKD2622!Nid(x[@.y7yG899:2D5&A9J2f,[lidi(
ixY% !i†v
ĐS:6x+3y+2z-18=0
SKD2622!id(x%&O@.yQyU12J2

.
 F
. 7
x t
y t
z t
=


∆ = −


= − +

y
7
7

.  
x y z−

∆ = =
− −
B9:2D5&Y412J2f,%&Olp12J2
.
∆

7
∆
B9:2D5&A9J2@
α
f,%&OY2Y2
.
∆

7
∆
ĐS:9x + 5y -2z – 34 = 0
T%&
( )
7yQyA
12J2
. 7
 
7 . 7
x y z
d
− −
= =
B9:2D5&A9J2
( )

α
 
d
Y
p2$R
A
B
( )
α
v
ĐS:
F  Ux y z− + − =
VKD2622!NOxyz%&B@7y~7y.C@~7yUy.B9:2D5
&9J2@ABC5&%&M,&A9J27x€7y€ z •UYMAMBMCĐS: M( 2 ; 3 ; - 7 )
.FTrong kh«ng gian víi hÖ täa ®éOxyz cho hai ®êng th¼ng d vµ d’ lÇn lît cã ph¬ng tr×nh :
.