Q1. [1] § 10.1 Introduction § 10.2 Tangent to a Circle § 10.4 Summary
If PQ and PR are tangents to the circle with centre O and radius 4 cm such that $\angle QPR = 90^\circ$, then the length OP is
- A $4$ cm
- B $4\sqrt{2}$ cm
- C $8$ cm
- D $2\sqrt{2}$ cm
Previously asked in CBSE board exam
2026 30/4/1 Q4
Generated by claude-sonnet-4-6 · 2026-06-15 10:21 · grounding rag
Model Answer
Option B: $4\sqrt{2}$ cm
Since OQ ⊥ PQ (radius ⊥ tangent) and ∠QPR = 90°, so ∠QPO = 45°. In right △OQP: $OP = \dfrac{OQ}{\sin 45°} = \dfrac{4}{1/\sqrt{2}} = 4\sqrt{2}$ cm.
Explanation
Since ∠QPR = 90° and OP bisects ∠QPR (centre lies on angle bisector of the two tangents), ∠QPO = 45°. With OQ = 4 cm (radius) and ∠OQP = 90°, use sin 45° = OQ/OP to get OP = 4√2 cm. Key theorems used: radius ⊥ tangent, and OP bisects the angle between the two tangents.
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