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.
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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, SR∥AC 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, PQ∥AC and PQ=12AC.By mid-point theorem. But from (i)SR=12AC 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.