Question

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Answer 1

Proof:

In the given circle with centre O, PQ and PR are tangents at points Q and R respectively.

We know that the radius is perpendicular to the tangent at the point of contact.

∠OQP = 90°

∠ORP = 90° 

In quadrilateral PQOR, the sum of all interior angles is 360°.

∠QPR + ∠OQP + ∠QOR + ∠ORP = 360°

∠QPR + 90° + ∠QOR + 90° = 360°

∠QPR + ∠QOR + 180° = 360°

∠QPR + ∠QOR = 180°

Since the sum of the opposite angles of quadrilateral PQOR is 180°, it is a cyclic quadrilateral.