Question

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = ______.

Answer 1

If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = 2a.

Explanation:

 
\[ \begin{array}{r l} \textbf{Given:} & \text{Tangents } PA \text{ and } PB \text{ drawn from an external point } P \text{ to a circle with centre } O \text{ and radius } a, \text{ with } \angle APB = 60^\circ. \\[4pt] \textbf{To Find:} & \text{The length } OP. \\[4pt] \textbf{Solution:} & \text{The radius through the point of contact is perpendicular to the tangent, hence } \angle OAP = 90^\circ. \\[4pt] & \text{The line joining the centre to an external point bisects the angle between the two tangents, hence } \angle APO = \tfrac{1}{2}\angle APB. \\[4pt] & \text{In the right triangle } OAP, \text{ taking the trigonometric ratio of } \angle APO : \\[4pt] & \begin{aligned} \angle APO &= \frac{1}{2} \times 60^\circ \\[4pt] &= 30^\circ \\[4pt] \sin 30^\circ &= \frac{OA}{OP} \\[4pt] \frac{1}{2} &= \frac{a}{OP} \\[4pt] OP &= 2a \\[4pt] &= 2.00\,a \end{aligned} \\[4pt] \textbf{Answer:} & OP = 2a = 2.00\,a. \end{array} \]