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Question

In a triangle ABC, E is the mid-point of median AD. Show that ar(BED)=14ar(ABC).


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Solution

Prove that, the area of triangle ABC is four-time the area of triangle BED.

Compute the area of triangle ABC as follows:

ar(ABC)=ar(ABD)+ar(ACD)ar(ABC)=12BD(AD)+12CD(AD)

Since, AD is a median. So, BD=CD,

Therefore,

ar(ABC)=12BD(AD)+12BD(AD)ar(ABC)=212BD(AD)

Since, AD=AE+ED.

Therefore, ar(ABC)=212BD(AE+ED)

It is given that E is the mid-point of median AD.

So,

ar(ABC)=212BD(DE+DE)ar(ABC)=212·2BD(DE)ar(ABC)=412BD(DE)ar(ABC)=4ar(BED)

Hence prove, ar(BED)=14ar(ABC)


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