Maths-

General

Easy

Question

# The Quadrilateral PQRS has angles at S,Q right angles and the diagonals PR, QS are perpendicular. Prove that SR = QR.

Hint:

**Hint :-** We know that if a quadrilateral has a sum of opposite angles 180° then it is a cyclic quadrilateral . Now , find the relation between OS and OQ. By proving congruence of triangles, prove SR= QR .

## The correct answer is: Hence proved

### Solution :-

Aim :- Prove that SR = QR

Explanation(proof ) :-

We know that if a quadrilateral has a sum of opposite angles 180°

∠S + ∠ Q = 90 + 90 = 180°(S and Q are opposite angle)

Here PQRS is cyclic and ∠S = 90°

As the angle angle in a semicircle is 90° then PSR lies in a semi circle

and PR is diameter.

As PR passes through centre and perpendicular to chord SQ

The perpendicular drawn from centre to chord bisects the chord

So, OS = OQ as PR bisects chord SQ

In OQR and OSR

OS = OQ (side)

∠SOR = ∠QOR =90° (Angle)

OP =OP (common side )

By SAS rule OQR ≅ OSR

By congruence we get , SR= QR

Hence proved

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