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Question
In the following figure, ABCD and AEFD are two parallelograms. Prove that ar (PEA) = ar (QFD). [Hint: Join PD].
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Solution
Given: ABCD and AEFD are two parallelograms.
To prove: ar (APEA) = ar (AQFD)
Proof: In quadrilateral PQDA,
AP || DQ ...[Since, in parallelogram ABCD, AB || CD]
And PQ || AD ...[Since, in parallelogram AEFD, FE || AD]
Then, quadrilateral PQDA is a parallelogram.
Also, parallelogram PQDA and AEFD are on the same base AD and between the same parallels AD and EQ.
ar (parallelogram PQDA) = ar (parallelogram AEFD)
On subtracting ar (quadrilateral APFD) from both sides, we get
ar (parallelogram PQDA) – ar (quadrilateral APFD)
= ar (parallelogram AEFD) – ar (quadrilateral APFD)
⇒ ar (AQFD) = ar (APEA)
Hence proved.
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