Question

ABCD is a quadrilateral in which P.Q.Rand S are midpoints of the sides AB, BC, CD and DA respectively. Show that PQRS is a parallelogram.​

Answer 1

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The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

(i) In △DAC , S is the mid point of DA and R is the mid point of DC. Therefore, SR∥AC and SR= ½AC.By mid-point theorem.

(ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQ∥AC and PQ= ½ AC.By mid-point theorem. But from (i) SR= ½SR= ½AC therefore PQ=SR

(iii) PQ∥AC & SR∥AC therefore PQ∥SR and PQ=SR. Hence, a quadrilateral with opposite sides equal and paralle is a parallelogram. Therefore PQRS is a parallelogram.

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