Question

1. ABCD is a quadrilateral in which P, Q, R and S are
mid-points of the sides AB, BC, CD and DA
(see Fig 8.29). AC is a diagonal. Show that:
1
(1) SRI AC and SR= AC
2
(ii) PQ=SR
(iii) PQRS is a parallelogram.​

Answer 1

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.

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(i)In ΔDAC,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 ΔBAC,P is the mid point of AB and Q is the mid point of BC.Therefore,PQ||AC and PQ=½AC.By mid point theorem.But from (i)SR=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.