# The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if

a. PQRS is a rhombus

b. PQRS is a parallelogram

c. diagonals of PQRS are perpendicular

d. diagonals of PQRS are equal.

**Solution:**

Join AC and BD

In triangle ABC

P and Q are the midpoints of AB and AC

PQ is parallel to AC and PQ = 1/2 AC …. (i)

In triangle ADC

SR is parallel to AC and SR = 1/2 AC …. (ii)

So PQ || SR and PQ = SR

In quadrilateral PQRS one pair of side is equal and parallel to each parallelogram

PQ || QR and PS = PR …. (iii)

In triangle BCD

Q and R are the mid points of BC and CD

So QR is parallel to BD and QR = 1/2 BD ….. (iv)

The diagonal of rectangle is equal using equations (i), (ii), (iii) and (iv)

So PQRS is a rhombus

Therefore, the diagonals of PQRS are perpendicular.

**✦ Try This: **The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if

a. ABCD is a rhombus

b. ABCD is a parallelogram

c. diagonals of ABCD are perpendicular

d. diagonals of ABCD are equal.

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

**NCERT Exemplar Class 9 Maths Exercise 8.1 Problem 5**

## The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if , a. PQRS is a rhombus, b. PQRS is a parallelogram, c. diagonals of PQRS are perpendicular, d. diagonals of PQRS are equal

**Summary:**

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if diagonals of PQRS are perpendicular

**☛ Related Questions:**

- If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3:7:6:4, then AB . . . .
- If bisectors of ∠A and ∠B of a quadrilateral ABCD intersect each other at P, of ∠B and ∠C at Q, of ∠ . . . .
- If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form , . . . .

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