If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the length of each tangent.
Generated by claude-sonnet-4-6 · 2026-06-15 10:23 · grounding rag
Model Answer
Let the external point be P and centre be O. The angle between the two tangents = 60°, so ∠TPO = 30° (since OP bisects the angle between tangents).
Radius OT = 3 cm, ∠OTP = 90° (radius ⊥ tangent).
In right △OTP:
$$\tan 30° = \frac{OT}{TP} \Rightarrow \frac{1}{\sqrt{3}} = \frac{3}{TP}$$
$$TP = 3\sqrt{3} \text{ cm}$$
The length of each tangent is $3\sqrt{3}$ cm.
Source: Chapter 10, Circles
---
Explanation
- The key property used: radius is perpendicular to tangent at point of contact (Theorem 10.1), and OP bisects the angle between the two tangents (since the two tangents from an external point are equal).
- The full angle between tangents is 60°, so the half-angle at P in the right triangle is 30°. Use tan 30° = opposite/adjacent = radius/tangent length.
- Examiners expect you to show the right triangle formation and the trigonometric step clearly.