Question

P, Q, R and S are respectively the midpoints of the sides AB, BC, CD and DA of a quadrilateral ABCD. Show that

(i) PQ || AC and PQ = ½ AC

(ii) PQ || SR

(iii) PQRS is a parallelogram.

Answer 1

(i) Consider △ ABC

We know that P and Q are the mid points of AB and BC.

From the figure we know that PQ || AC

So we get PQ = ½ AC …… (1)

Therefore, it is proved that PQ || AC and PQ = ½ AC

(ii) Consider △ ADC

We know that R and S are the mid points of CD and AD

From the figure we know that SR || AC

So we get SR = ½ AC …….. (2)

Considering equations (1) and (2)

We get

PQ = SR and PQ || SR

Therefore, it is proved that PQ || SR.

(iii) We know that one pair of opposite sides are equal and parallel in quadrilateral PQRS.

Therefore, PQRS is a parallelogram.