# ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. (see Fig.).

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ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. (see Fig.). AC is a diagonal. Show that :

(i) SR || AC and SR = 1 2 AC

(ii) PQ = SR

(iii) PQRS is a parallelogram

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Given : ABCD is a quadrilateral in which P, Q, R and S are mid-points of AB, BC, CD and DA. AC is a diagonal.

To Prove :

(i) SR || AC and SR = 1 2 AC

(ii) PQ = SR

(iii) PQRS is a parallelogram

Proof :

(i) In ∆ABC, P is the mid-point of AB and Q is the mid-point of BC.

∴ PQ || AC and PQ = 1/ 2 AC …(1) [Mid-point theorem]

In ∆ADC, R is the mid-point of CD and S is the mid-point of AD

∴ SR || AC and SR = 1 2 AC …(2) [Mid-point theorem]

(ii) From (1) and (2), we get PQ || SR and PQ = SR

(iii) Now in quadrilateral PQRS, its one pair of opposite sides PQ and SR is equal and parallel.

∴ PQRS is a parallelogram. Proved.