In the quadrilateral given below, AD = BC, P, Q, R and S are mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.

It is given that,

In △ABD,

P and Q are mid-points of AB and BD,

PQ || AD and PQ = $\dfrac{1}{2}$AD = $\dfrac{1}{2}$BC …….(i)

In △ACD,

R and S are mid-points of CD and AC,

RS || AD and RS = $\dfrac{1}{2}$AD = $\dfrac{1}{2}$BC …….(ii)

In △BCD,

R and Q are mid-points of CD and BD,

RQ || BC and RQ = $\dfrac{1}{2}$BC …….(iii)

In △ABC,

P and S are mid-points of AB and AC,

PS || BC and PS = $\dfrac{1}{2}$BC …….(iv)

From (i) and (ii) we get,

PQ || RS

From (iii) and (iv) we get,

RQ || PS

From (i), (ii), (iii) and (iv) we get,

PQ = RS = PS = RQ.

Since, all sides are equal and opposite sides are parallel.

Hence, proved that PQRS is a rhombus.