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
(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.