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).

(i) In ΔADC, S and R are the mid-points of sides AD and CD respectively

In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it

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

(ii) In ΔABC, P and Q are mid-points of sides AB and BC respectively. Therefore, by using mid-point theorem

PQ || AC and PQ = 1/2AC (2)

Using equations (1) and (2), we obtain

PQ || SR and PQ = SR (3)

PQ = SR

(iii) From equation (3), we obtained

PQ || SR and

PQ = SR

Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal

Hence, PQRS is a parallelogram