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# 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.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 8

**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.

**☛ Related Questions:**

- ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC
- In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.
- Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
- ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show thati) D is the mid-point of ACii) MD ⊥ ACiii) CM = MA = 1/2 AB

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