Sáng kiến kinh nghiệm Rèn luyện kỹ năng tính thể tích khối chóp

I. LÝ DO CHỌN ĐỀ TÀI

 !"#$%&'()*
+,*-.-#+/*0
12!3,1!&#+124005678.
"$#&09:&.1"$;1
$<&&'-"%=>?(@A
.+,*-!,.:&-.+BB4
C.0509 $#&!&05
"$#&(D<E&FF.+GHE
10I%.4052.2-#3<
050J3"H."4(K!-<-
2I2#"$#&#
LM09NA-A#%O*PQ&.3<
1,&,J0J..+O*
H10B&.+ O<2I,1.(
 09C-I.. %
,1!&I#R?RC,S%T2(K!-
<0J%,.&,GHE<*&%
U1,JL&,0J%%-VS1S05.$
I<1U,22! 1.H10B&%!&
%-V ",1!& $2(
W..FXYZT[#\"
$<]&A/-2*^,0&A1!&!^
,$%(KR?,.#,1.
I2"$#&#L2.+091
*":&UTM#I2-.$<&
-"%?(
_,U1??S .+HO2,-`
B05%$<<O<.103""0J.+
"$#&(abB&.+ !&OZ110

.;&'I$?XcFd%S&-HA?XcFe=
XcFd2O<fRèn luyện kỹ năng tính thể tích khối chóp $
%&FX
F
II. CƠ SỞ LÝ LUẬN VÀ THỰC TIỄN
1. Cơ sở lý luận
g--05"'5&.. .%052
+O*$05!#(_>I-
1$10 Polya ",1.Pf
• fN"1.
• fN.,
• fN1-9,+O.105&
• fh"9,
iU<3-.0^&':1<
0J%12&T1*-05%OIU,
22!12O005.,(
j,.I.12U'I..3"
< 1,!:09k.LM%OlO11G
C+B%. %(D05.,1.&.(
m",1.,.O<+.10%f
• fm&T
• fKO&n5& 1(
• fo<A./#I
",1.(
N-IO2<SU.10"!3,
.!^,$%T-I.H05.
.I.#<p-<0-.(K!-I%$1.3
""0J.+I+A.103"!12.1!&
H10B&.+R?RC, ,I-2.$
C,-I.,.^%E".+S%T2(

2. Thực trạng trước khi thực hiện các giải pháp của đề tài
 
N-"%=>?XcFeHI $qq
F
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>#1."$#&(_.+#,!
T-#057(g"'.+-T
40ZC.C.0509 $#&BB
.%(
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I.0J- .<2&2 %(N-
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1."$#&

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hS&3U2A#U10!3"f
r_.21!&AZ1!.&,0522+OHs2#
B/..^1,B%B&.+I+A.
.#I(
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A0J.+!CQ"BU1<bS%T2
p-<R?,1..(
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PHẦN MỘT
Hệ thống kiến thức liên quan đến bài toán tính thể tích khối chóp
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SA ABC⊥
( ) ( )
SAB ABC⊥
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m. f
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v 1
( )

SH ABC⊥
( )
SO ABCD
⊥
HỆ THỐNG KIẾN THỨC CƠ BẢN
1. Tam giác
'156("6#7

F
X
ABC a
S ah
=
w
F
X
b
bh
w
F
X
c
ch

F F F
% % %
X X X
ABC
S bc A ac B ab C
= = =


e
ABC
abc
S
R
=
Y)1.09L2&
[ABC∆

ABC
S pr=
Y1.09L2&
ABC∆
[

Y [Y [Y [
ABC
S p p a p b p c= − − −
f
X
a b c
p
+ +
=

Các tam giác đặc biệt:
Tam giác vuông
_.q8_q <05.f
m@f

X X X
BC AB AC= +


X X
(   (BA BH BC CA CH CB= =

( (AB AC BC AH=


X
(AH BH CH=

X X X
F F F
AH AB AC
= +

e
Hình chóp
S
A
C
B


S
B
A
C

H
H
S
A
B
C
M
N
A
B
C
D
O
A
B
C
H
h
a
a
b
c

A
B
C
H

S



K<.q8_f
F
(
X
ABC
S AB AC=

_.q8_q Nx%$05..f

% %
AC
B C
BC
= =


% %
AB
C B
BC
= =

 
AC
B C
AB
= =



 
AB
C B
AC
= =

_.q8_Tq Tam giác cân
j" 8_
AH BC⇒ ⊥


( (AH BH B CH C= =
K<.q8_f
F
(
X
ABC
S AH BC
∆
=
_.q8_1b Tam giác đều
j" 8_
AH BC⇒ ⊥
m09f
u
X
a
AH =

K<.q8_f

X
F u
(
X e
ABC
a
S AH BC= =
2. Tứ giác
Hình vuông
K<q8_K1b

X X
ABCD
S AB a= =

m09Qq8_K1b

X XAC BD AB a
= = =

Hình chữ nhật
K<U!q8_K

(
ABCD
S AB BC=

m09QU!q8_K

X X

AC BD AB BC= = +

Hình thoi
K<q8_K

F
(
X
ABCD
S AC BD=

d
A
H

B
C
)
(
)
)
A

H
C
B
A

O
B

B
A
C
D
H
C
C
D
A
B
O
O
O
A
B
C
D
(
09Qq8_Kf
AC BD⊥
_q8_K#
y yAB CD
 Hình thang
K<q8_K

( )
F
X
ABCD
S AH AB DC= +


KOf
AH DC
H DC
⊥

⇒

∈

q09 
q8_K
3. Góc giữa đường thẳng d và mặt phẳng (P)
j#U09>:&>Y@[#U
09>09>z2#
 :&>Y@[
Thực hiện:
(N2z Y@[
(h##UY@[#Uz
4. Góc giữa hai mặt phẳng (P) và (Q)
j#U:&>#U09>A
05?:&>n#
-2"
Thực hiện:
({.-2 Y@[Yg[
(NY@[09>#
Yg[09>1#
(h##U:&>Y@[Yg[#
U09>1
5. Thể tích khối chóp

_"$#&f
F
u
V Bh=

N#f8<.
 #&
6. Thể tích khối tứ diện đều
_<q8_K1b N*,..1b
N*,.:.
jT .8_K
( )
AH BCD
⇒ ⊥

#q09 <(
N"$<f
|
CD
A
B
H

P)

A
d
d’
H
(

M
ϕ

c c
c }c
ϕ
≤ ≤

P
Q
c c
c }c
ϕ
≤ ≤

(
(
a
b
d
)
ϕ

S
A
C
H
B
C
B

D
A
M
H

H
(xem bài toán 20)
X u
F F X u X
(W ( (
u u e FX
u
ABCD BCD
a a a
V AH= = =
Y[
PHẦN HAI
Thực hiện bài toán tính thể tích khối chóp
A. Phương pháp thực hiện. m",1."$#&AO<+
.10%f
r(mR1&T!$#&
r. {.09 $#&(
r(KO"< ,24
r(o!&"$#&
r8(N<. 09 $#&
B. Nội dung, biện pháp thực hiện các giải pháp của đề tài.
 Yêu cầu đối với giáo viên:
+ _.21!&AZ1!.22+OHs2#A
B/.1!&^1,B%B&.+I+A
..#I(

r8!&%..09*&&!-0J,
.AG%$,2&n5&,?! 
%(
 Yêu cầu đối với học sinh:
+ _ASU&A/-2O.2
1!&(
+ gl1!&A%-V"S%T^%E".1!&
+-A .(
C. Thực hiện bài toán tính thể tích khối chóp
1. Khối chóp có cạnh bên vuông góc với mặt đáy
*#(_#&W(q8_# q8_.81Wq
#:  (82 
X uAB a AC SB a= = =
  
c a< ∈¡
(N" $#&
W(q8_+
a

Hướng dẫn Hình vẽ
{.09 $#&
Wq#:
⇒
Wq09 $#&
m .q8_8
o!&"$#&
 
N"$#&W(q8_f
(
F

(
u
S ABC ABC
V SA S=

N.q8_8f
~
S
A
B
C

N#8_
N<.q8_
N09Wq
N"$#&W(q8_
X X X X
u XBC AC AB a a a= − = − =

K<.q8_f
X
F X
(
X X
ABC
a
S AB BC= =

( )
SA ABC SA AB⊥ ⇒ ⊥

N.Wq8qf
X X X X
u XSA SB AB a a a
= − = − =
u
(
F X
(
u |
S ABC ABC
a
V SA S
= =
Y[
*#(_#&W(q8_# q8_.T81Wq
#: (82 
X uAC a SB a= =
  
c a< ∈¡
(N"$#&
W(q8_+
a

Hướng dẫn Hình vẽ
9:;f8.X0^O1.FG
.f .q8_T8
{.09 $#&
Wq#:
⇒
Wq09 $#&

m .q8_8
o!&"$#&
N#
( )
AB BC=

N<.q8_
N09Wq
N"$#&W(q8_
 
N"$#&W(q8_f
(
F
(
u
S ABC ABC
V SA S=

N.q8_T8f
X X X
X XAB AC a AB a
= = ⇒ =

K<.q8_f
X
X
F
X X
ABC
a

S AB
= =

( )
SA ABC SA AB⊥ ⇒ ⊥
N.Wq8qf
X X X X
u XSA SB AB a a a= − = − =
u
(
F X
(
u |
S ABC ABC
a
V SA S
= =
Y[
*#(_#&W(q8_# q8_.1b
Xa

c a< ∈¡

1Wq#: (82
dSB a=
(N"$#&W(q8_+
a
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Hướng dẫn Hình vẽ
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⇒ ⊥ ⇒ ⊥

⊥


( ) ( )
( )
( )


SCD ABCD CD
SD SCD SD CD
AD ABCD AD CD
∩ =


⊂ ⊥ ⇒


⊂ ⊥

#  U    :
&>YW_K[Yq8_K[
·
c
|cSDA =

N.WqKqf
·
c c
  |c ( |c u
SA
SDA SA AD a
AD
= = ⇒ = =
u
(
F u
(
u u
S ABCD ABCD
a
V SA S
= =
Y[
*#=(_#&W(q8_K# q8_K1b 

Xa
 
c a< ∈¡
1Wq#: (_1W_: #
c
|c
(
N"$#&W(q8_K+
a
Hướng dẫn Hình vẽ
9:;f8.~0^O1.|f#
 q8_KX
a
 #
U1W_Yq8_K[1b|c
c
{.09 $#&
Wq#:
⇒
Wq09 $#&
m q8_K
Xa
o!&"$#&
N<q8_K
N09QW_ 
q8_K1b
Xa

{.#UW_Yq8_K[
 

N"$#&W(q8_Kf
(
F
(
u
S ABCD ABCD
V SA S=
K<q8_Kf
X
e
ABCD
S a
=

q_09Q q8_K
1b
X X Xa AC a
⇒ =

( )
SA ABCD
⊥ ⇒
q_2# 
W_Yq8_K[
⇒
#UW_Yq8_K[
·
c
|cSCA
=


FF

60
0
(
D
C
B
A
S
N09Wq
N"$#&W(q8_K
N.Wq_qf
·
c c
  |c ( |c X |
SA
SCA SA AC a
AC
= = ⇒ = =
u
(
F • |
(
u u
S ABCD ABCD
a
V SA S= =
Y[

*#>(_#&W(q8_K# q8_K1b
a

c a
< ∈
¡

1Wq#: 
·
c
FXcBAD
=
i" 8_
·
c
edSMA
=
(
N"$#&W(q8_K+
a
(Trích đề tuyển sinh khối D năm 2013)
Hướng dẫn Hình vẽ
9:;7
o!&"$#&
N<q8_K
N09Wq
N.WqiTq
N"$#&W(q8_K
 
N"$#&W(q8_Kf

(
F
(
u
S ABCD ABCD
V SA S=

q8_K#
·
·
c c
FXc |cBAD ABC
= ⇒ =

ABC
⇒∆
.
u
X
a
a AM⇒ =

K<q8_Kf
X
u
X (
X
ABCD ABC
a
S S AM BC

∆
= = =

N.Wqiq#
·
c
edSMA
=

u
X
a
SA AM⇒ = =

u
(
F
(
u e
S ABCD ABCD
a
V SA S= =
Y[
*#?(_#&W(q8_K# q8_KqK
 uAD CD a AB a
= = =

c a
< ∈
¡

1Wq#: 1W_
 #1b
c
ed
(N"$#&W(q8_K+
a

Hướng dẫn Hình vẽ
9:;7
FX
S
A
B
C
D
M
120
0
(
B
A
C
D
M
120
0
)
60
0
A

S
B
C
D
ua

a

a

A
B
C
D
(
45
0
o!&"$#&
 
N"$#&W(q8_Kf
(
F
(
u
S ABCD ABCD
V SA S=

N<q8_K
N09Wq
N.Wq_Tq

N"$#&W(q8_K
K<q8_Kf

( )
X
F
X
X
ABCD
S AD AB CD a
= + =

ADC
∆
K
X X
XAC AD DC a⇒ = + =

( )
SA ABCD
⊥ ⇒
q_2# 
W_Yq8_K[
⇒
#UW_Yq8_K[
·
c
edSCA
=


SAC
∆
q
·
c
edSCA
=
XSA AC a
⇒ = =
u
(
F X X
(
u u
S ABCD ABCD
a
V SA S
= =
Y[
*#@(_#&W(q8_K# q8_K1b 
a
 
c a
< ∈
¡
1Wq#: #UWK:&>YWq8[1b
c
uc
(
N"$#&W(q8_K+

a
(Trích đề tốt nghiệp THPT năm 2013)
Hướng dẫn Hình vẽ
9:;f
 m"09Wq $
#&AC.05#UWK:
&>YWq8[
 _AC.2# 
WK:&>YWq8[
_Af
( )
AD SAB⊥

o!&"$#&
N<q8_K
N09Wq
N"$#&W(q8_K
 
N"$#&W(q8_Kf
(
F
(
u
S ABCD ABCD
V SA S=

K<q8_Kf
X
ABCD
S a

=

( )
( )
SA ABCD SA AD
AD SAB
AB AD
⊥ ⇒ ⊥ 

⇒ ⊥

⊥



⇒
Wq2# WK
YWq8[
⇒
#UWKYWq8[
·
c
ucASD =
SAD
∆
K#f
·
c c
 uc ( uc
u

SA a
ASD SA AD
AD
= = ⇒ = =

Fu

A
30
0
D
C
B
S
u
(
F u
(
u u
S ABCD ABCD
a
V SA S
= =
Y[

2. Khối chóp có mặt bên vuông góc với mặt đáy
*#(_#&W(q8_# q8_.Tq:1W8_
.
a


c a
< ∈
¡
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$#&W(q8_+
a

YTrích đề tuyển sinh khối D năm 2014[
Hướng dẫn Hình vẽ
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Nếu hai mặt phẳng vuông góc với nhau và
cắt nhau theo một giao tuyến, đường thẳng
nằm trong mặt phẳng này và vuông góc với
giao tuyến thì nó vuông góc với mặt phẳng
kia.
{.09 $#&
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o!&"$#&
N<.q8_
N"$#&W(q8_
 
N.W8_
a

" 8_
u
X
SH BC
a
SH

⊥


⇒

=



( ) ( )
( ) ( )
( )
( )

SBC ABC
SBC ABC BC SH ABC
SH SBC SH BC
⊥

∩ = ⇒ ⊥


⊂ ⊥


N"$#&W(q8_f
(
F
(
u

S ABC ABC
V SH S
=
ABC
∆
Tq#f
X X
AH BC
BC a
AH
⊥



= =


K<.q8_f
X
F
(
X e
ABC
a
S AH BC
∆
= =

u
(

F u
(
u Xe
S ABC ABC
a
V SH S
= =
Y[
*#(_#&W(q8_# q8_.q
·
c
ucABC =
:1
W8_.
a

c a
< ∈
¡
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N"$#&W(q8_+
a
YTrích đề tuyển sinh khối A và A
1
năm 2013[
Hướng dẫn Hình vẽ
9:;7
8.}0^O1.•
m .q
 _A.,2fm .

Fe
H
S
A
B
C

S
H
A
C
B

(
q8_q
·
c
uc ABC BC a
= =
ABC
⇒
k.
BC a
=

{.09 $#&
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N<.q8_
N"$#&W(q8_

 
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a

" 8_
u
X
SH BC
a
SH
⊥


⇒

=



( ) ( )
( ) ( )
( )
( )

SBC ABC
SBC ABC BC SH ABC
SH SBC SH BC
⊥

∩ = ⇒ ⊥



⊂ ⊥


N"$#&W(q8_f
(
F
(
u
S ABC ABC
V SH S
=
N.q8_q#f

·
c c
% %uc (%uc
X
AB a
ABC AB BC
BC
= = ⇒ = =


·
c c
u
% %uc (%uc
X

AC a
ABC AC BC
BC
= = ⇒ = =
K<.q8_f
X
F u
(
X •
ABC
a
S AB AC
= =

u
(
F
(
u F|
S ABC ABC
a
V SH S
= =
Y[
*#(_#&W(q8_# q8_.8
u  e BA a BC a
= =
X uSB a=

c a

< ∈
¡
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·
c
ucSBC =
(N
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a
YTrích đề tuyển sinh khối D

năm 2011[
Hướng dẫn Hình vẽ
9:;7
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#s05< (

( ) ( )
SBC ABC⊥
#GA
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⇒
09 $
#&
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 
j2# W8_
SH BC

⇒ ⊥

( ) ( )
( ) ( )
( )
( )

SBC ABC
SBC ABC BC SH ABC
SH SBC SH BC
⊥

∩ = ⇒ ⊥


⊂ ⊥

Fd
30
0
Tam giác ABC là nửa tam
giác đều cạnh
BC a
=

C
A

B
u

X
a

a

S

B
A
C
H
)
30
0
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N<.q8_
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N"$#&W(q8_f
(
F
(
u
S ABC ABC
V SH S
=
N.W8
·
c
ucSBC =
·

c c
% % uc (%uc u
SH
SBC SH SB a
SB
= = ⇒ = =
K<.q8_f
X
F
( |
X
ABC
S BA BC a
∆
= =

u
(
F
( X u
u
S ABC ABC
V SH S a= =
Y[
*#(_#&S(ABCD# :1Wq8.b
:&>#:&>YABCD[(82
aBDaAC eX ==

c a
< ∈

¡
(N
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Hướng dẫn Hình vẽ
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u
S ABCD ABCD
V SO S
=
K<q8_Kf
X
e
ABCD
S a
=
q_09Q
Xa

X XAC a
⇒ =
(K##f
X
X
AC
OA a
= =

N.W€q€(

X X X X
u XSO SA OA a a a
= − = − =
u
(
F e
(
u u
S ABCD ABCD
a
V SO S
= =
Y[
*#?(_#&.S(ABCD# 1b
Xa

c a
< ∈
¡
1
 #
c
ed
(N"$#&S.ABCD +a

Hướng dẫn Hình vẽ
9:;7
8.F}0^O1.F•

XAB BC CD DA a

= = = =

j#U1: 1b
c
ed
{.09 $#&
o!&"$#&
N<q8_K
{.#U1
N09W€
N"$#&W(q8_K
 
j
( )
O AC BD SO ABCD
= ∩ ⇒ ⊥

N"$#&W(q8_Kf
(
F
(
u
S ABCD ABCD
V SO S
=
K<q8_Kf
X
e
ABCD
S a

=
( )
SO ABCD BO⊥ ⇒
2# 
1W8Yq8_K[
·
SBO
⇒
#UW8Yq8_K[t
·
c
edSBO =
N.W€8€(
·
c
  ed X
SO
SBO SO OB a
OB
= = ⇒ = =
u
(
F e X
(
u u
S ABCD ABCD
a
V SO S= =
Y[
*#@(N"$<q8_K1b

a

c a
< ∈
¡

F}
B
C
S

A
B
C
D
O
45
0
)
Hướng dẫn Hình vẽ
9:;7
_.,0^O1F~
<#&
_.:.

(ABCD A BCD
V V=
{.09 $<
o!&"$<
N<.8_K

N09q
N"$<q8_K
 
N.8_K
a
i
" _K
u
X
BM CD
a
BM
⊥


⇒

=



jT 
( )
BCD AH BCD∆ ⇒ ⊥

N"$<q8_Kf
(
F
(
u

ABCD A BCD BCD
V V AH S
= =
X X u u
(
u u X u
H BM
a a
BH BM
∈



= = =


K<.8_Kf
X
F u
(
X e
BCD
a
S BM CD
∆
= =
N.q8(
X
X X X
X

u
u
a a
AH AB BH a
= − = − =
u
(
F X
(
u FX
ABCD A BCD BCD
a
V V AH S
= = =
Y[
4. Một số dạng khác
*#(_#&W(q8_K# q8_K1b
Xa

c a
< ∈
¡
(2# GW:&>Yq8_K[" 
q8(j#U:&>YW_K[:&>Yq8_K[1b
c
|c
(N"$
#&W(q8_K+
a


(Trích đề kiểm tra học kỳ II lớp 12 năm học 2014 – 2015 của Tỉnh Đồng Nai)

Hướng dẫn Hình vẽ
9:;7
 2# GW
:&>Yq8_K[" 
q8
( )
SH ABCD⇒ ⊥
Ys    C.  
Xc
B
D
A
M
H
C

A
S
D
B
H
M
(
60
0
09 $#&[
{.#U?&>YW_K[
Yq8_K[

o!&"$#&
N<q8_K
{.#U:&>
YW_K[Yq8_K[
N09W
N"$#&W(q8_K
 
( )
SH ABCD⊥
N"$#&W(q8_Kf
(
F
(
u
S ABCD ABCD
V SH S=
K<q8_Kf
X
e
ABCD
S a
=
ji" _K
HM CD
⇒ ⊥
( )
( )
SH ABCD SH CD
CD SHM SM CD
HM CD

⊥ ⇒ ⊥ 

⇒ ⊥ ⇒ ⊥

⊥



( ) ( )
( )
( )
·


SCD ABCD CD
SM SCD SM CD SMH
HM ABCD HM CD
∩ =

⊂ ⊥ ⇒


⊂ ⊥

#U
:&>YW_K[Yq8_K[t
·
c
|cSMH
=


N.Wif
·
c c
  |c ( |c X u
SH
SMH SH HM a
HM
= = ⇒ = =
u
(
F • u
(
u u
S ABCD ABCD
a
V SH S= =
Y[
*#(_#&W(q8_ # q8_.1b
a

c a
< ∈
¡
(2# GW:&>Yq8_K["
q8%
XHA HB
=
(j#UW_:&>Yq8_[1b
c

|c
(N"$
#&W(q8_+
a

(Trích đề tuyển sinh khối A và A
1
năm 2014 )
Hướng dẫn Hình vẽ
9:;7
 2# GW
:&>Yq8_["
q8%
XHA HB
=
( )
SH ABC
⇒ ⊥
YsC.09
 $#&[
{.#UW_Yq8_[
o!&"$#&
 
( )
SH ABC⊥
N"$#&W(q8_f
(
F
(
u

S ABC ABC
V SH S
=
N.q8_
a
i"
 q8
u
X
CM AB
a
CM
⊥


⇒

=


XF
C
S
(
60
0
A
B
C
H

M

N<.q8_
{.#UW_Yq8_[
X
| |
MA MB
AB a
MH
HA HB
=

⇒ = =

=

N09W
N"$#&W(q8_
K<.q8_f

X
F u
(
X e
ABC
a
S CM AB
∆
= =


( )
SH ABC AC⊥ ⇒
2# W_
Yq8_[
⇒
#UW_Yq8_[
·
c
|cSCA
=
i" q8
| |
AB a
MH⇒ = =
N.i_if
X X
~
u
a
CH CM MH
= + =

N.W_f
c
XF
( |c
u
a
SH HC
= =

u
(
F ~
(
u FX
S ABC ABC
a
V SH S
= =
Y[
*#(_#&W(q8_K# q8_K1b
a

c a
< ∈
¡
(jiaA05" q8qKt" _aKi(
82 W #  :&>Yq8_K[  
uSH a
=
( N "  $ #&
W(_Kai+
a
(Trích đề tuyển sinh khối A năm 2010 )
Hướng dẫn Hình vẽ
9:;7
N"$#&W(_Kai
KO&>"<
._Kai
o!&"$#&

N<._Kai
N"$#&W(_Kai
 
( ) ( )
SH ABCD SH CDNM
⊥ ⇒ ⊥
N"$#&W(_Kaif
(
F
(
u
S CDNM CDNM
V SH S=

uSH a
=
K<._Kai

( )
CDNM ABCD AMN BMC
S S S S
∆ ∆
= − +

X X X X X
F F
( (
X X
AB AM AN BM BC
 

= − +
 ÷
 

X X X
X
d
• e •
a a a
a
 
= − + =
 ÷
 

u
(
F d u
(
u Xe
S CDNM CDNM
a
V SH S
= =
Y[
*#_#&W(q8_#2# GW:
nT09L2&.q8_12
d  |AB AC a BC a= = =

c a

< ∈
¡
(
XX
S
A
B
C
D
N
M
A
B
C
D
H
M
N
H
j#U:1YW8_[: 1b
c
|c
(N"$#&W(q8_+
Hướng dẫn Hình vẽ
9:;7
T09L2&
ABC
∆

o!&"$#&

N<.q8_
Wk3f

( ) ( ) ( )
S p p a p b p c
= − − −
{.#UYW8_[Yq8_[
N09W

Wk3f
S pr=
N"$#&W(q8_
 
( )
SH ABC⊥
N"$#&W(q8_f
(
F
(
u
S ABC ABC
V SH S
=
ak.q8_f
( )
F
•
X
p AB BC CA a
= + + =

t
• d up AB a a a− = − =

• d up AC a a a
− = − =
t
• | Xp BC a a a− = − =
K<.q8_f
( ) ( ) ( )
X
• (u (u (X FXS p p AB p AC p BC a a a a a
= − − − = =
jia@A052# 
8_q8q_
HM BC
⇒ ⊥
HM HN HP r
⇒ = = =
Y1.09L2&
.q8_[
( )
( )
SH ABC SH BC
BC SHM SM BC
HM BC
⊥ ⇒ ⊥


⇒ ⊥ ⇒ ⊥


⊥



( ) ( )
( )
( )


SBC ABC BC
SM SBC SM BC
HM ABC HM BC
∩ =

⊂ ⊥ ⇒


⊂ ⊥

#U:&>
YW8_[Yq8_[
·
c
|cSMH
=

N#f

X
FX u

(
• X
ABC
ABC
S
a a
S pr p HM HM
p a
∆
∆
= = ⇒ = = =
N.Wi
c
u u
( |c
X
a
SH HM
= =

u
(
F
( | u
u
S ABC ABC
V SH S a= =
Y[
*#8_#&W(q8_# q8_.8
XAC a

=

c a
< ∈
¡

·
c
ucACB
=
2# GW: " 
q_
XSH a
=
(N"$#&W(q8_+
Xu
S
H
A
B
C
M
N
P

M
H
A
B
C

P
N
r
r
r
(
60
0
(Trích đề thi minh họa – kỳ thi THPT Quốc gia năm 2015 )

Hướng dẫn Hình vẽ
9:;7
2# GW
: " q_
KsC.0509 
$#&"
$#&(
m"< ,.
E1.FX
o!&"$#&
N<.q8_

F
(
X
ABC
S AB BC
∆
=


c
F
( % uc
X
ABC
S AC BC
∆
=
N"$#&W(q8_
 
N#f
( )
t XSH ABC SH a⊥ =
N"$#&W(q8_f
(
F
(
u
S ABC ABC
V SH S
=
N.q8_8#f

·
c c
% % uc (%uc
AB
ACB AB AC a
AC
= = ⇒ = =



·
c c
% %uc (%uc u
BC
ACB BC AC a
AC
= = ⇒ = =
K<.q8_f
X
F u
(
X X
ABC
a
S AB BC
∆
= =

u
(
F |
(
u |
S ABC ABC
a
V SH S
= =
Y[

*#< _#&W(q8_# q8_.1b 
a

c a
< ∈
¡
(:1YWq8[YWq_[n#: W8:
 #1b
c
uc
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Hướng dẫn Hình vẽ
Nếu hai mặt phẳng (P) và (Q) cắt nhau
theo giao tuyến d và cùng vuông góc với
mặt phẳng (R) thì
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5. Bài tập áp dụng.
Đề bài Đáp án
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