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

**Solution:**

Given, ABCD is a __quadrilateral__

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 __rectangle__.

Since AC ⊥ BD

∠AOD = ∠AOB = ∠BOC = ∠COD = 90°

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)

Considering quadrilateral EOFR,

OE || FR

OF || ER

∠EOF = ∠ERF = 90°

Therefore, PQRS is a rectangle.

**✦ Try This: **In △ABC, ∠A=50° and ∠B=60°. Determine the longest and the shortest sides of the triangle.

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

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 4**

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

**Summary:**

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

**☛ Related Questions:**

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- A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus
- P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP . . . .

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