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Question

Let ABCD be a quadrilateral and let P, Q, R, S be the mid-points of AB, BC, CD, DA respectively. Prove that PQRS is a parallelogram. (What extra result you need to prove this)?

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Solution

It is given that AP = BP, BQ = CQ, DR = CR and AS = SD.

Join AC and BD

We know that in any triangle, the line joining the midpoints of two sides is parallel to the third side.

Therefore, in triangles ACD and ABC:

SR || AC and PQ || AC

SR || PQ

Similarly, it can be shown that SP || QR

Hence, PQRS is a parallelogram.


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