Question

The diagonals of a quadrilateral ABCD are equal. Prove that the quadrilateral formed by joining the midpoints of its sides is a rhombus.

Answer 1

Consider △ ABC

We know that P and Q are the midpoints of AB and BC

So we get PQ || AC and

PQ = ½ AC …… (1)

Consider △ BCD

We know that Q and R are the midpoints of BC and CD

So we get QR || BD and

QR = ½ BD ……. (2)

Consider △ ADC

We know that S and R are the midpoints of AD and CD

So we get RS || AC and

RS = ½ AC ……. (3)

Consider △ ABD

We know that P and S are the midpoints of AB and AD

So we get SP || BD and

SP = ½ BD ……. (4)

Using all the equations

PQ || RS and QR || SP

Thus, PQRS is a parallelogram

It is given AC = BD

We can write it as

½ AC = ½ BD

From the equations we get

PQ = QR = RS = SP

PQRS is a rhombus

Therefore, it is proved that the quadrilateral formed by joining the midpoints of its sides is a rhombus.