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

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