Question

In Fig. 2, ABCD is a quadrilateral in which P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively. Show that PQRS is a parallelogram.

Answer 1

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

To prove: PQRS is a parallelogram.

Construction: Join AC.


Proof:

In $∆$ACD,

As, R and S are mid-points of DC and DA, respectively.

So, by using the Mid-point theorem − The segment connecting the mid-points of two sides of a triangle is parallel to the third side and is half the length of the third side, we get

RS$\parallel$AC and RS = $\frac{1}{2}$AC .....(1)

Similarly, in $∆$ABC, by using the mid-point theorem, we get

PQ$\parallel$AC and PQ = $\frac{1}{2}$AC .....(2)

Therefore, from (1) and (2), we get

PQ$\parallel$RS and PQ = RS

But this is a pair of opposite sides of the quadrilateral PQRS.

Therefore, PQRS is a parallelogram.