**Question:**

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

**Solution:**

Given:

ABCD is quadrilateral in which AD = BC and P, Q, R, S are the mid points of AB, AC, CD, BD respectively.

To Prove:

PQRS is a rhombus.

Proof:

In ∆ABC, P and Q are the midpoints of the sides AB and AC respectively.

By the Mid point theorem, we get

PQ || BC and PQ = 12BC ...(1)

In ∆ADC, Q and R are the midpoints of the sides AC and DC respectively.

By the Mid point theorem, we get

QR || AD and QR = 12AD = 12BC (Since AD = BC) ...(2)

Similarly, in ∆BCD, we have

RS || BC and RS = 12BC ...(3)

In ∆BAD, we have

PS || AD and PS = 12AD = 12BC (Since AD = BC) ...(4)

From the equations (1), (2), (3), (4), we get

PQ = QR = RS = RS

Thus, PQRS is a rhombus.