Question

Abcd is a parallelogram. e is the midpoint of cd. prove that: ar(abed) = 3ar(bec).

Answer 1

Hi friend
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Your answer
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Given that : - ABCD is a parallelogram. E is the mid point of CD.

To prove : - ar(ABED) = 3ar(∆BEC)

Now,
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Construction : - Join BE and BD.

Then,
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In ∆BCD, we have,

ar(∆BED) = ar(∆BEC) [As E is the midpoint , BE is a median and a median divides a triangle into two triangles of equal area.

Again,
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ar(∆ABD) = ar(∆BCD) [Diagonal of a parallelogram divides it into two triangles of equal area.]

Then,
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ar(ADB) = ar(∆BCD)

=> ar(∆ADB) = ar(∆BED) + ar(∆BEC)

So, ar(∆ADB) = 2ar(∆BEC) [Because ar(∆BED) = ar(∆BEC)]

Then,
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ar(∆ADB) + ar(∆BED) = 2ar(∆BEC) + ar(∆BEC)

=> ar(ABED) = 3ar(∆BEC) [ Because ar(ABED) = ar(∆ABD) + ar(∆BED)]

Hence, proved.

HOPE IT HELPS