Question

If two tangents inclined at angle of 60° are drawn to a circle of radius 3 cm, then find the length of each tangents.

Answer 1

\[ \begin{array}{r l} \textbf{Given:} & \text{Two tangents drawn from an external point } P \text{ to a circle with centre } O \text{ and radius } 3 \text{ cm, inclined at } 60^\circ. \\[4pt] \textbf{To Find:} & \text{The length of each tangent.} \\[4pt] \textbf{Solution:} & \text{Let } PA \text{ and } PB \text{ be the tangents with points of contact } A \text{ and } B, \text{ so that } \angle APB = 60^\circ \text{ and } OA = 3 \text{ cm.} \\[4pt] & \text{The line joining the centre to an external point bisects the angle between the two tangents, hence } \angle APO = \tfrac{1}{2}\angle APB. \\[4pt] & \text{The radius through the point of contact is perpendicular to the tangent, hence } \angle OAP = 90^\circ. \\[4pt] & \text{In the right triangle } OAP, \text{ taking the trigonometric ratio of } \angle APO : \\[4pt] & \begin{aligned} \angle APO &= \frac{1}{2} \times 60^\circ \\[4pt] &= 30^\circ \\[4pt] \tan 30^\circ &= \frac{OA}{PA} \\[4pt] \frac{1}{\sqrt{3}} &= \frac{3}{PA} \\[4pt] PA &= 3\sqrt{3} \\[4pt] &= 5.196 \\[4pt] &\approx 5.20 \end{aligned} \\[4pt] \textbf{Answer:} & \text{The length of each tangent is } 3\sqrt{3} \text{ cm} \approx 5.20 \text{ cm.} \end{array} \]