Q1. [1]
In the given figure, TA is a tangent to the circle with centre O such that $OT = 4$ cm, $\angle OTA = 30^\circ$, then the length of TA is :
- (a) $2\sqrt{3}$ cm
- (b) $2$ cm
- (c) $2\sqrt{2}$ cm
- (d) $\sqrt{3}$ cm
Previously asked in CBSE board exam
2023 30/4/1 Q15
Generated by claude-sonnet-4-6 · 2026-06-15 10:24 · grounding rag
Model Answer
(a) $2\sqrt{3}$ cm
Since OA ⊥ TA (radius ⊥ tangent), in right △OTA: $\cos 30° = \dfrac{TA}{OT}$, so $TA = OT \cos 30° = 4 \times \dfrac{\sqrt{3}}{2} = 2\sqrt{3}$ cm.
Source: Chapter 10, Theorem 10.1
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Explanation
- The tangent is perpendicular to the radius at the point of contact, so ∠OAT = 90°.
- In right △OTA, the right angle is at A, so TA is the side adjacent to ∠OTA and OT is the hypotenuse.
- Use $\cos(\angle OTA) = \dfrac{TA}{OT}$ → $TA = 4\cos 30° = 2\sqrt{3}$.
- Examiners expect you to state "radius ⊥ tangent" as justification before applying trigonometry.
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