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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:

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 $straight capital delta$ OQR and $straight capital delta$ OSR
OS = OQ (side)
∠SOR = ∠QOR =90° (Angle)
OP =OP (common side )
By SAS rule $straight capital delta$ OQR ≅ $straight capital delta$ OSR
By congruence we get , SR= QR
Hence proved

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