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Byju's Answer
Standard XII
Mathematics
External Division
In Δ ABC, D...
Question
In
Δ
A
B
C
, D, E and F are the mid-points of BC, CA and AB respectively
(
A
D
+
B
E
+
C
F
)
(
A
B
+
B
C
+
C
A
)
is ,
A
>
3
4
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B
<
1
2
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C
<
3
5
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D
<
3
4
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Solution
The correct option is
D
>
3
4
Let
G
be the centroid, hence it divides
A
D
,
B
E
,
C
F
in the ratio
2
:
1
⟹
B
G
=
2
3
B
E
and
C
G
=
2
3
C
F
In
△
B
G
C
,
B
G
+
G
C
>
B
C
⟹
2
3
B
E
+
2
3
C
F
>
B
C
2
B
E
+
2
C
F
>
3
B
C
or
3
B
C
<
2
B
E
+
2
C
F
--------(i)
Similarly,
3
C
A
<
2
C
F
+
2
A
D
--------(ii)
3
A
B
<
2
A
D
+
2
B
E
--------(iii)
(i)+(ii)+(iii) we get,
3
(
A
B
+
B
C
+
C
A
)
<
4
(
A
D
+
B
E
+
C
F
)
(
A
D
+
B
E
+
C
F
)
(
A
B
+
B
C
+
C
A
)
>
3
4
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∆
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then
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→
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