#### Q : 2      ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Given: ABCD is a rhombus in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA . AC, BD are  diagonals.

To prove: the quadrilateral PQRS is a rectangle.

Proof: In ACD,

S is midpoint of DA.                (Given)

R  is midpoint of DC.               (Given)

By midpoint theorem,

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

In ABC,

P is midpoint of AB.                (Given)

Q  is mid point of BC.               (Given)

By mid point theorem,

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

From 1 and 2,we get

and

Thus,      and

So,the quadrilateral PQRS is a parallelogram.

Similarly, in BCD,

Q is mid point of BC.                (Given)

R  is mid point of DC.               (Given)

By mid point theorem,

So,  QN || LM ...........5

LQ || MN ..........6  (Since, PQ || AC)

From 5 and 6, we get

LMPQ is a parallelogram.

Hence, LMN=LQN   (opposite angles of the parallelogram)

But, LMN= 90     (Diagonals of a rhombus are perpendicular)

so,   LQN=90

Thus, a parallelogram whose one angle is right angle,ia a rectangle.Hence,PQRS is a rectangle.