#### Q : 3       ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Given: ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively.

To prove: the quadrilateral PQRS is a rhombus.

Proof :

In ACD,

S is the midpoint of DA.                (Given)

R  is the midpoint of DC.               (Given)

By midpoint theorem,

and   ...................................1

In ABC,

P is the midpoint of AB.                (Given)

Q  is the midpoint of BC.               (Given)

By midpoint theorem,

and   .................................2

From 1 and 2, we get

and

Thus,      and

So, the quadrilateral PQRS is a parallelogram.

Similarly, in BCD,

Q is the midpoint of BC.                (Given)

R  is the midpoint of DC.               (Given)

By midpoint theorem,

and      ...................5

AC = BD.......................6(diagonals )

From 2,  5 and 6, we get

PQ=QR

Thus, a parallelogram whose adjacent sides are equal is a rhombus. Hence, PQRS is a rhombus.