Question

In a triangle ABC , E is the mid-point of median AD. show that ar(BED) =1/4 ar( ABC )

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Answer 1

GIVEN:-

E is the mid-point of median AD.

TO PROVE:-

$\small\sf{A(∆BED)=A(∆ABC)}$

PROOF:-

$\small\sf{In\:∆ABC,AD\:is\:median}$

$\small\sf{Hence,A(∆ABD)=A(∆ACD)}$

$\therefore$$\small\sf{A(∆ABD)=\frac{1}{2}A(∆ABC)}$......(1)

(A median of a ∆ divides it into two equal triangles on same area)

$\large\sf\underline\red{Similarly,}$

$\small\sf{In\:∆ABD,BE\:is\:median}$

$\large\sf\underline\blue{Hence,}$

$\small\sf{A(∆BED)=\frac{1}{2}A(∆ABD)}$

$\small\sf{A(∆BED)=\frac{1}{2}(\frac{1}{2}A(∆ABC))}$ from........(1)

$\small\sf{A(∆BED)=\frac{1}{4}A(∆ABC)}$

$\small\sf{A(∆BED)=\frac{1}{4}A(∆ABC)}$