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Question

In ΔABC, D, E and F are the mid-points of BC, CA and AB respectively (AD+BE+CF)(AB+BC+CA) is ,

A
>34
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B
<12
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C
<35
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D
<34
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Solution

The correct option is D >34
Let G be the centroid, hence it divides AD, BE, CF in the ratio 2:1

BG=23 BE and CG=23 CF

In BGC,
BG+GC>BC

23 BE+23 CF>BC

2 BE+2 CF>3 BC or 3 BC<2 BE+2 CF --------(i)

Similarly, 3 CA<2 CF+2 AD --------(ii)
3 AB<2 AD+2 BE --------(iii)

(i)+(ii)+(iii) we get,

3(AB+BC+CA)<4(AD+BE+CF)

(AD+BE+CF)(AB+BC+CA)>34

938433_394627_ans_7df67067512f42a4a41d926d932c137b.JPG

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