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P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.

Solution:

The points P, Q, R and S are the midpoints of the sides AB, BC, CD and AD.

AC = BD

We have to prove that PQRS is a rhombus.

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus

The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

Considering triangle ADC,

S and R are the midpoints of AD and DC

By midpoint theorem,

SR || AC

SR = 1/2 AC --------------------- (1)

Considering triangle ABC,

P and Q are the midpoints of AB and BC

By midpoint theorem,

PQ || AC

PQ = 1/2 AC -------------------- (2)

Comparing (1) and (2),

SR = PQ = 1/2 AC ----------- (3)

Considering triangle BCD,

By midpoint theorem,

RQ || BD

RQ = 1/2 BD ------------------ (4)

Considering triangle BAD,

SP || BD

By midpoint theorem,

SP = 1/2 BD ----------------- (5)

Comparing (4) and (5),

SP = RQ = 1/2 BD ----------- (6)

Given, AC = BD

Dividing by 2 on both sides,

1/2 AC = 1/2 BD

From (3) and (6),

SR = PQ = SP = RQ

This implies all the sides of the quadrilateral are equal.

Therefore, PQRS is a rhombus.

✦ Try This: In △ABC, ∠A=90°. What is the longest side?

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

NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 3

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.

Summary:

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. It is proven that PQRS is a rhombus

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