Question

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

Answer 1

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.