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)*#+ ,,-!*./#012 
&3*4*56,2 /#)*78*
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?.7@(/-AB6B*7*CD
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E'H:*71I#0H!1!JF#
G.?1&
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LKOM ,0GGHB!#0!#1&'*7
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:75 &'05;
1<=>('2'?@A)<?5 2
;=B,CD1E0F7B,192G1.2H0:
&&'0'B4 7I':1>0J()=K1<1L
4/9M
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WX*72C HHG?6,:J
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N_S87!B?GA,?&
E Z/7!D`06,GCc*)*. S87
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1 ,80)*JGH.?1)*<B?&
E H*G1D'*_.?1  G ,
.B6dGH_)*K+K'V:2_&
6Q<+,
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!,&K?16*C/=* D/
J#1^KO#T6AB_# XH?&G* _
:?LN ,S*_M&h*#N1iT7j.
2/7?&
RST+,
%S87! ,2H^B? ,7?V:
2_1i4N 2/0J ,
*G?1 /?8
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TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
JUVUWXYZ
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[41E 7
  $41EDLYKOG1:5ACB/!
_HM

= = =
uuur uuur uuur uuuur
&

KJP KJf
6
TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
E~L
#OA OB
uuur uuur
M„Q
Q
J LJfM#6N
&OAOB
uuur uuur
& ‡OAOB
=
uuur uuuur
&
 7#:N*D
&OAOB
uuur uuur
& ‡OAOB
=
uuur uuuur
L•M
$41E*rLKB!_HM
EU%58*RKODY_HL•MFNG_GL
#OA OB

Những diễn biến tư duy của HS có thể xảy ra:
K?1",A2D
&OAOB
uuur uuur
L ‡ ‡ M & ‡ & ‡OA OB B B OAOB OA B B
= + = +
uuur uuuur uuuur uuur uuuur uuur uuuur
& ‡OAOB
=
uuur uuuur

L%Jb…b *_ †e
& ‡ QOA B B
=
uuur uuuur
M&
EU%7*4*?1AC*B*_HJ*&
$41EOrLY.^_HM
Cho đường tròn (O; R) và điểm M cố định. Một đường thẳng
∆
thay đổi luôn
đi qua M, cắt đường tròn đó tại hai điểm A và B. Chứng minh rằng:
&MA MB
uuur uuur
f f
MO R
= −
.

7

đi thẳng theo hai hướng tạo với nhau một góc
α
. Tàu B chạy với tốc độ 20 hải lí
một giờ, tàu C chạy với tốc độ 15 hải lí một giờ. Hỏi sau 2 giờ, 2 tàu cách nhau
bao nhiêu hải lí nếu? (hình vẽ)
a.
α
= 90
0
b.
α
= 60
0

[1<=op=oq5)1r
+HPA,!A:KO\ BB
HN#0!/&Y1i*7G$.JC^J6:
&~G)*7A,!,706,B6!_N
GHC<J,B*c60&

$41E*rLKBB/c60M
U%6,ˆ ,7*4*a!: 723
87&
8
TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
%F8D! 8_=
:Nt#9AG4&
": 723r
YebYbY„#eY„A#eb„ , &

=
#-7AC*c)*A ,z
& %7CAC*"
f
a
1*D
f f f
fa b c bcp= + ±
LPM#6,1^!
G_z
& K-7S‚S1^€HHLPM*B
α
,z
'<!_HAC*c)*A# ,
α
z
o&':
Q
nQ
α
<
-7AC*c)*A# ,
α
z~CGebY*&
v&':
α
ATGJ#-7!_HAC*c)*A# ,
α
z
Những diễn biến tư duy của HS có thể xảy ra:

f f f
fa b c bc
= + +
&
%7C!:
Q
nQ
α
>
JD
f f f
fa b c bcp
= + ±
LfM#6,1^!&
w [s*D+^* HL•M1^56,1
α
5
α
 J*
Q
nQ
α
=
#G *_eJ
Qp
=
9
f f f
a b c= +
&

α
<
JbY.,7A‚/*7
f f f
f 1a b c bcc
α
= + −
6,6X# JG
1 Qc
α
>

f 1 Qbcc
α
>
&
'k ebY*#GD

f f f f f f
P
f 1ˆQ f&&L M
f
o
a b c bcc a a a
= + − = + − =
&
%7?1/1VG?!D &
w[sOD'<G)*8*f ,oKOC!D
f f f
f 1ea b c bcc

⇔ < ⇔ + <
e?
f f f
1 QA b c a
⇔ > ⇔ + >
e *_
f f f
1 QA b c a
⇔ = ⇔ + =
$41EOrL%c60 <JC)*7#G‚G0 T5
GB/M
U)*7A,!*<4*A,&
KO",01*f:,*bvQ60#,*YoQ
60&[ebYeb„vQ#eY„oQ#eAˆQ

&
‰c601 ,ebYD

f f f
 „A w EfA1e
&
%7bY„
PoQQ oˆ
≈
&O*f:#,**Goˆ60&
$41EPrLKB.^c6XM
Cho tam giác ABC có các cạnh là a=7, b=24, c=23. Tính góc A?
+:C,7#?1", )*.c601C
G)*D


N?1G#GH<1iS87!c6X&
$41E*rLKB!c60M
11
TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
Exj,A,-J6
1b#1Y#-ACJAJ
6)*(6,z
Es‚:5A#€
ebY *_e#-7AC*c#A#)*x ,1e#1b#1Yz
Lx6,AG0:FebYM&
E'<?1ND
f
1 1 1
a b c
R
A B C
= = =
L•M
Những diễn biến tư duy của HS có thể xảy ra:
EK?1C/DZ6)*(6
A#1b #1Y&
E[! ,H *_-?V
T'KYO?1",D
 1
 1
 1
b B
c C
a A

Những diễn biến tư duy của HS có thể xảy ra:
*XG‹:G0be…J:e?
LJM5eLJAM?1CT7G1e„1e… , 
6:F66,G_)*G^ ?1&
12
TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
$41EOrL%c60 <JC)*7#G‚G0 T5
GB/M
U 7*4*?1)*7A,A4*5&
s‚ebY#DY„PuQ
Q
@LˆQ
Q
wvŠ
Q
M„pŠ
Q
&
‰c6X1 ,ebYJ
1 1
b c
B C
=
&
O*7   &%7G<eYSTSqˆG&
$41EPrLKB.^c60M
Từ hai vị trí A và B của một toà nhà, người ta quan sát đỉnh C của ngọn núi
(hình dưới). Biết rằng độ cao AB bằng 70m, phương nhìn AC tạo với phương nằm
ngang một góc 30

&
Y*^# J *_eYKYK^ oQ
Q
D

Pov#pL M
f
AC
CH m
= ≈
&%7?NGPoŠ&
%0vDDạy học công thức độ dài đường trung tuyến trong tam giác&
[41E 7
$41EDLUB/#!_HM
Cho tam giác ABC vuông tại A, M là trung
13
TrÇn Quang Huy
S¸ng kiÕn kinh nghiÖm
điểm của BC. Hãy biểu diễn AM
2
theo a, b, c?
Những diễn biến tư duy của HS có thể xảy raD
E%JebY *_?1"
,   , &
E'<?1C
f f f
f
f v
b c a
AM

+ −
= + −
f f f
f v
b c a+
= −

E*?1S*T<€H
P
L M
f
AM AB AC= +
uuuur uuur uuur
J D

f f f
P
L f & M
v
AM AB AC AB AC
= + +
uuur uuur
f f
P
L f & &1 M
v
b c b c A
= + +

f f f

$41EPrLKBGGH6)*c60M
Từ các công thức về độ dài đường trung tuyến, hãy tìm một hệ thức liên hệ
giữa m
a
, m
b
, m
c
với a, b, c?
 :_H B,:**7?1",G
G)*D
f f f f f f
o
L M
v
a b c
m m m a b c
+ + = + +
&
*-E237(95);>;u;os'41E 7
7(9
E%TS87! ,2HB? (
/7?0!6,B7*^)*7cC*
1*a7#8706,B6!C ! ,7&
EU :S*7A=t 1C 
6X*77?#0!#&&&L)* !?#!A=t ,)*
A=t:S*7*GJ#&&&M&
EY/1V T _7,774,C
H  S87! ,2HB?
(/7?0! ,!"&

P
@'KZ'K,%$`

L1^KO?oˆM]W^
HDPQY
f
@'KZ''4gYL1^KO?oˆM&
YT6G14*$.KOV6/^=*&
':,!D'PfŒfQPf ,QPŒfQPo&
6E'=
[!:?G 2#R T*XG.
?1 , 7!62H!&
+^ 6!#N_,2H6*7
?1/0 B?0!#1*-7fL
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