ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

Given: AD = BC

P, Q, R and S are the midpoints of AB, AC, CD and BD respectively.

So in ∆ABC, if P and Q are midpoints of AB and A respectively ⇒ PQ ∥ BC

And PQ = (1/2)BC …(i)

Similarly in ∆ADC,

QR = (1/2)AD …(ii)

In ∆BCD,

SR = (1/2)BC …(iii)

In ∆ABD,

PS = (1/2)AD = (1/2)BC [∵ AD = BC]

Using equations (i), (ii), (iii) & (iv), we get

PQ = QR = SR = PS

All these sides are equal.

⇒ PQRS is a rhombus.

Hence, shown that PQRS is a rhombus.