**ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Figure). AC is a diagonal. Show that :**

**(i) and ****(ii) PQ = SR****(iii) PQRS is a parallelogram.**

**Solution (i)**

In , S is the midpoint of DA and R is the midpoint of DC.

Therefore, **by midpoint theorem**, we have and

**Solution (ii)**

In , P is the midpoint of AB and Q is the midpoint of BC.

Therefore, **by midpoint theorem**, we have and **(1)**

We have already proved above in **solution (i)** that and . **(2)**

From **(1)** and **(2)**, we can say that

**Solution (iii)**

From **(1)** and **(2)** above, we can say that and

is a parallelogram.

**(A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.)**