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

(Q 23 – 30/2/2 – CBSE 2026 Question Paper)

Ans:

Since in a cyclic Quadrilateral, sum of opposite angles is 180 ⁰

∴ if we need to prove that PQOR is a cyclic Quadrilateral,

then we need to prove: ∠ RPQ + ∠ ROQ = 180 ⁰

Step 1: We know that the tangents on a circle are perpendicular to the radius at point of contact.

Here, at point R, tangent PR is Ʇ radius OR.

∴ ∠ ORP = 90 ⁰

Similarly, at point Q, tangent PQ is Ʇ radius OQ.

∴ ∠ OQP = 90 ⁰

Step 2: We know that the sum of all angels of a quadrilateral is 360

∴ ∠ RPQ + ∠ ORP + ∠ ROQ + ∠ OQP = 360 ⁰

∴ ∠ RPQ + 90 ⁰ + ∠ ROQ + 90 ⁰ = 360 ⁰ (values taken from step 1)

∴ ∠ RPQ + ∠ ROQ + 180 ⁰ = 360 ⁰

∴ ∠ RPQ + ∠ ROQ = 180 ⁰         

Hence Proved!

Therefore, PQOR is a cyclic Quadrilateral.

Please press the “Heart” button if you like the solution.