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8R3V
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CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
I
  [  [ [  
Qi(8T  iT(   ( 
I I I I I I [ d
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= = α = α = α
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Ru(Q8 Qi 8i (  
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α +
= = + = α + =
ϕ ϕ
∆

M
4Q8T5 R3 T Q 4Q8T5
⊥ ⇒ =
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I
Q Q Q Q
Q8T Q8R Q8R Q8R
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⇒ = ∠ = =
∆ ∆ ∆ ∆
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⇔ α = α + ⇔ α =
ϕ
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QiF(3<'=4;5RN1 <'=4Q3N5(
0gvSg H %+ %B "w 
ϕ
1 
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 ! <
"$<%#P
/2/2
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H
A B
D C
S
M
N
E
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
M: D6EAB% S.8NR3(
QE 3N
QED 44QR35648R3N55
DE 3N
⊥


⇒ ⇒ ∠ = ∠ = ϕ

⊥

MNeNX
⊥
Q3(
02
RN Qi
RN 4Qi35
i3 RN
⊥

⇒ ⊥

⊥

RN Q3
4RNX5 Q3 4RNX5 4Q3N5 4 P 4RNX5 4;55
⇒ ⊥
⇒ ⊥ ⇒ ⊥ ≡
M: )F)
Q(8R3N
6)

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h72
 
Q(R3N
) )
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) I() I Q3
= =
02

I I I I I
I I
   
Q3 QE E3 Ei ( E3   
I I(
 QED 
= + = + = + = ϕ +
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ϕ
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⊥
Q3
RX Q3
⇒ ⊥
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I I I
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 
RX(Q3 QE(R3 RX(Q3 QE(R3
QR3
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= = ⇔ =
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+
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QP2
I
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−

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CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
./9234;5*!K8R3N1 8RF6R3F"(0$%9
=1 <'=48R3N5C % So.8R-P% SQ
QoF"(0$R3-PRDFL6$3N-PNEFP
4D R36 E 3N5
∈ ∈
0H  !L6P66"2
( ) ( )
( ) ( )
5
( QoD QDE
"( QoE QDE
⊥
⊥
I5;% S.QD6% S.oE(0% k %S
;F+4QD6oE5
:  2
5(: U2
( ) ( )
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⊥
(NeEDq
⊥
QD4Dq
∈
QD5(
02
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ED f 4QoE5
QD QoE QDE
⊥


⇒ ⊥

= ∩


ED f Qo
⇒ ⊥
D<#2Qo
( )
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⊥ ⇒ ⊥
N%2D
≡
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( )
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⇒ ⊥ ⇒ ⊥
)KP%S
( ) ( )
QoE QDE
⊥
' 
I I I
OM MN ON+ =
45

0RDFL
⇒
3DF"L
NEFP
⇒
NEFP
`
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
( ) ( )
I I I
I I I
I I
I
I
I I
I I I I I I I
I
4 5
\
I I8 \ I4 5 I
\ \
BN b a y
AN b y
MN b x a y
a
OM x
BN N AB b a y y a
ON
⇒ = + −
= +

= − + −
= +
+ − + − + −
⇒ = =
)KPx452
( ) ( )
I I I I I
I I
I
I I
\ I4 5 I
\ \
I I
b a y y a a
x b x a y
bx ay x a
+ − + −
= + + − + −
⇔ + = +
)KP% k %S4QoD54QDE52
I I
I Ibx ay x a+ = +
"5K'K$% k %S4QoE5
⊥
4QDE5oE
⊥
DE
V %2
I I I
IP I"  I"L [P

+ + = +
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; QD
; oE
⊥

⇔

⊥

45
02

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( )
I
I I I
I
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I I I I I
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o

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Q o Qo  "
`
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QD Qo oD  " L
\

 " L
\
o; Q;
\
+ − + −
= =
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= + = + +
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( ) ( )
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I I I I I
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; Q Q; o; o
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\ "  \L
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` ` d
I " I" IP L IP
= − = −
⇒ + = + =
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+ +
⇒ + + =
⇔ + + = +
j
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
)KP% k %S;# !oEQD2
I I I I I
I " I" IP L IP+ + = +
R n23'Q(8R3N1 %#P8R3NC%9
Q8F
 `
I
(: 4;5<'=81 Q36@#C
Q36QR6QNABC 3q6Rq6Nq(
0g !3qNq8N
:  2
M: 
X 3fNf 3N 3fX Q3
= ∩ ⇒ ⊥
NeNV
⊥
Q3

( )
V Q3∈

NV ? ?3fX
43fNf68N5 43fX68N5 4NV68N5 8NV
⇒
⇒ ∠ = ∠ = ∠ = ∠

Q8 3N
3N 4Q8N5 3N QN
3N 8N
⊥

⇒ ⊥ ⇒ ⊥

⊥


D
3fX Q3
⊥
$P2
QN3V
%y+C1 
X3f3V
3fX 3f3
QN 3N
⇒ =
45
83q

⊥
Q3
I I I
I I
I I
83 83 I I \
33f 
Q3 j
[
Q8 83
 I
I
⇒ = = = =
+
+
0]4523qXF
I I
I I
I \ [
  
QN(33f 33f Q8 8N I [a
j I

3N 3N  j
+
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= = =

I I I I
d Ip

X3 33f 3fX   I
j j
3N  NV 3V
4+ NV ? ?X3f5
X3 I X3f 33f
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3V 
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⇒ = + = + =
⇒ = = =
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d
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
Q38F
83  I I j
Q3 j
\


I
= =

I I I
I I
I
8V 3V 83 I3V(83(Q38
I \ I j
 I I( I( (
j j j
d

j
I \
8V 
j
⇒ = + −
= + −
=
⇒ =
M8NV
I I I
I I I
a d
  
N8 NV 8V I [a
j j
I(N8(NV [a
[a
I 

j
+ −
+ −
= = =
I [a
8NV 
[a
⇒ ∠ =
./0:13'Q(8R3N1 8R3N C

R8N `p∠ =
(: o % S.83RN6" &Qo
⊥
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QoF
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"(0g !4QR354Q8N5(
:  2
(o+e%9=+
⊥
8N@8N6R3ABC T6u(
MNeTi
⊥
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Qo 48R3N5
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Tu R3
⊥
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⇒ ⊥
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Ti R3
⇒ ⊥

Ti 4QR35 Ti QR⇒ ⊥ ⇒ ⊥
8N??R3
Ti 8N⇒ ⊥
)KPTiF+48N6QR5
n
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
NO-PoTFouF
[
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\
(NeY &.o$Qu6+O+P
%B2oYF
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\
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( ) ( )
Q8N QR3∩

( )
QT 8N QT +
QTu 8N 4+ + ? ?8N ? ?R35
Qu 8N Qu +
TQu 44Q8N564QR355
⊥ ⊥
 
⊥ ⇒ ⇒
 
⊥ ⊥
 
⇒ ∠ = ∠

0+O+%B2
TuFI(oTF
[
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I


I I I I
n [ [
QT Qu Qo oT   

` ` I
= = + = + =

QTu⇒V
%k

TQu `p⇒ ∠ =
)KP !4Q8N54QR35

TQu `p∠ =
EKL72z" #P6%Sg%+ # ! %C8NQR
>S2
M

R8N `p∠ = ⇒

8RNV
%kC

I
[
Q 
\
8RN
⇒ =
∆
Qo
48R3N5⊥
QP2)
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F
I [
8RN
  [ [ [
Qo(Q   
[ [ \ \ `
= =
V
45
MD<#2
)
Q(8RN
F

QR(8N(+48N6QR5(  48N6QR5
`
∠
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QRF
I I I I
n  [
oR Qo   
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+ = + =
Q3F
I I I I
n [ I
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+ = + =

Ip
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
8N??R3
⇒

48N6QR5 4R36QR5 QR3∠ = ∠ = ∠

I I I
I I I
[ I
  
QR R3 Q3 [
` `
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\
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[
+ −
+ −
= = =
⇒ =
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F
I
[

 (+4QR68N5
I
4I5
M0]454I5P%B2+48N6QR5F
[

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 " &%} " S*
./0013'Q(8R36%#P #C 8(8RF6R3FI(
NeQi1 48R35C i
38 `
3i 6 Qi 
[ [
= =
(: T6u
AB% S.R36Q8(: 
4 5β
<'=Ru
1 <'=4QiT5(
0g !
4 5β
48R35(
:  2
MNO-P38F
[


8R3 `p∠ =

: V% S.8i
QP28V

 [ 8V [
38   8RV 8RV [p
[ [ 8R [
= = ⇒ = = ⇒ ∠ =
45
M: E% S.QT(~'+,%m•Dxx #T3Q
"% SR6E6D2

RT D3 EQ  DQ
( (  ( (4 5 
I
R3 DQ ET D3
DQ
I
D3
= ⇒ − =
⇒ = −
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83Q"% S06D6u2
I
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
08 D3 uQ DQ
( (  4 I5( (4 5 
03 DQ uT D3
08 
I
03

= ⇒ − − =
⇒ =
N%28% S.03(
QP2
∆
R03^C 0

0R8 8R3 `p⇒ ∠ = ∠ =
(4I5
0]454I52
RV 0R
⇒ ⊥
4[5
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 &.Q8$48R35(
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⇒

uV 0R
⊥
4\5
M0]4[54\5%B24uRV5
⊥
0R
uRV 44 5648R355⇒ ∠ = ∠ β
0+O+g%B2
` I [ `
uV 6 RV  Ru 
` [ I

= = ⇒ =
⇒
uRVF
RV I I
Ru [
=
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88qRRq1 48R356+e%9.
8qRRq3(
:  2
0<'=48RRq8q5W%9=Rq61 8qR@
8RC N(
0]RWRV
⊥
3N4
V 3N
∈
5(
0]RWRi
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i RfV∈
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II
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
0]uW%9=1 Ri@8qRC T
02
3N RV
3N 4RfRV5

Ri Rf3
Ri 4RfN35
3N RRf
Ri NRf
Ri RfV

⊥

⇒ ⊥
⊥

 
⇒ ⊥ ⇒
⊥
 

⊥


⊥

DNRq??8qR$Ri
⊥
8qR
D<#Tu??Ri$
Ri Rf3
Ri 8fR
⊥



⊥

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6%9Q8F(Ne%9.RN6Q3sL#%m
^%9$#CQ3RN(0g%+ %C
%(
:  2
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VWRi
⊥
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( )
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uW%9=1 Ri@RNC T(
MN8R3NU, #%kC$RN
⊥
8R
VX 8R
VX 4Q8V5 VX Ri
VX Q8
⇒ ⊥

⇒ ⊥ ⇒ ⊥

⊥

M
Ri QV

Ri 4QVX5
Tu 4QVX5 Tu VX Tu RN 4+ RN ? ?VX5
Ri VX
Tu ? ? Ri

⊥

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 
⇒ ⊥ ⇒ ⊥ ⇒ ⊥
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

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M
Tu 4QVX5 Tu Q3
⊥ ⇒ ⊥
)KPTu%9.Q3RN(
I[
CHUYÊN ĐỀ HÌNH CHÓP_TỔ 4_LỚP 11A1
NO-P2
I I
  [ [  [
VR 6 V3 6 V8 6 VQ Q8 8V
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