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

Solution:

Consider ABCD as a quadrilateral

P, Q, R and S are the midpoints of AB, BC, CD and AD

Let us now join AC

In triangle ABC,

P is the midpoint of AB

Q is the midpoint of BC

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

In triangle ADC,

R is the midpoint of CD

S is the midpoint of AD

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

Using equation (i) and (ii)

RS || PQ and RS = PQ

So PQRS is a parallelogram

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 rectangle, if
a. ABCD is a rectangle
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 4

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

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