Mathematics — Standard (041) — CBSE Class 10 board question

Since OQ ⊥ PQ and OR ⊥ PR (radius is perpendicular to tangent at point of contact),

∠OQP = 90° and ∠ORP = 90°

In quadrilateral PQOR:
∠QOR + ∠QPR + ∠OQP + ∠ORP = 360°
∠QOR + ∠QPR + 90° + 90° = 360°
∠QOR + ∠QPR = 180°

Since opposite angles are supplementary, quadrilateral PQOR is cyclic. Hence proved.

Source: Chapter 10, Section 10.2 (Theorem 10.1)

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• The key property used is Theorem 10.1: radius ⊥ tangent at point of contact, giving two 90° angles.
• A quadrilateral is cyclic if and only if its opposite angles are supplementary (sum = 180°). Show that ∠QOR + ∠QPR = 180°, which follows directly from the angle sum of the quadrilateral being 360°.
• Write the angle-sum step clearly — examiners award marks for each logical step.