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

**Solution:**

Given, ABCD is a __quadrilateral__

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

AC = BD

AC ⊥ BD

We have to prove that PQRS is a __square__.

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 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.

Considering quadrilateral OERF,

OE || FR

OF || ER

∠EOF = ∠ERF = 90°

Sinc AC ⊥ BD

∠DOC = ∠EOF = 90°

∠QRS = 90°

∠RQS = 90°

Therefore, PQRS is a square.

**✦ Try This: **In △ABC, ∠A=100° and ∠C=50°. Which is its shortest side?

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

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 5**

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

**Summary:**

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

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