# ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

**Solution:**

We will use the mid-point theorem here. It states that the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.

Let us join AC and BD. In △ABC,

P and Q are the mid-points of AB and BC respectively.

∴ PQ || AC and PQ = 1/2 AC (Mid-point theorem) ... (1)

Similarly, in △ADC,

SR || AC and SR = 1/2 AC (Mid-point theorem) ... (2)

Clearly, PQ || SR and PQ = SR [From equation (1) and (2)]

Since in quadrilateral PQRS, one pair of opposite sides are equal and parallel to each other, it is a parallelogram.

∴ PS || QR and PS = QR (Opposite sides of the parallelogram) ... (3)

In △BCD, Q and R are the mid-points of side BC and CD respectively.

∴ QR || BD and QR = 1/2 BD (Mid-point theorem) ... (4)

However, the diagonals of a rectangle are equal.

∴ AC = BD ... (5)

Thus, QR = 1/2 AC

Also, in △BAD

PS || BD and PS = 1/2 BD

Thus, QR = PS .... (6)

By using Equations (1), (2), (3), (4), and (5), we obtain

PQ = QR = SR = PS

Therefore, PQRS is a rhombus.

**Video Solution:**

## ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

### NCERT Maths Solutions Class 9 - Chapter 8 Exercise 8.2 Question 3:

**Summary:**

If ABCD is a rectangle and P, Q, R, S are mid-points of the sides AB, BC, CD, and DA respectively, then the quadrilateral PQRS is a rhombus.