Question

Prove that by joining the midpoints of any quadrilateral, we get a parallelogram.

Answer 1

Given: A quadrilateral ABCD with P, Q, R and S as the respective midpoints of sides DA, AB, BC and CD

In ΔABD, P and Q are the midpoints of sides AD and AB respectively.

⇒ PQ || DB and PQ = ...(1)

Similarly, in ΔCBD, R and S are the midpoints of sides CB and CD respectively.

⇒ RS || BD and RS = …(2)

From equation (1) and (2), we have:

PQ || SR and PQ = SR

Similarly, PS || QR and PS = QR

Thus, the opposite sides of the quadrilateral PQRS are equal and parallel.

Hence, PQRS is a parallelogram.

Hence, we can say that by joining the midpoints of the sides of any quadrilateral, we get a parallelogram.