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Question

ABCD is a quadrilateral in which P, Q, R, and S are mid-points of the lines AB, BC, CD and DA ( see in fig ). AC is a diagonal. Show that:
(i) SR || AC and SR=12AC
(ii) PQ= SR
(iii) PQRS is a parallelogram.
1033765_e18a888e4ea74c2192154c5d60ce6976.png

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Solution

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, SRAC and SR=12AC.By mid-point theorem.

(ii) In BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQAC and PQ=12AC.By mid-point theorem. But from (i) SR=12AC therefore PQ=SR

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

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