ABCD is a quadrilateral . P ,Q ,R and S are the mid points of AB, BC, CD
and AD respectively . Prove that PQRS is a IIgram and perimeter of
PQRS = AC + BD
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Step-by-step explanation:
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= 1/2 AC.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= 1/2 AC.By mid-point theorem. But from (i) SR= 1/2 AC 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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