Question

In the following figure, if tangents PA and PB drawn from a point P to a circle with centre O, are inclined to each other at an angle of 70°, then find the measure of ∠POA.

Answer 1

\[ \begin{array}{r l} \textbf{Given:} & \text{Tangents } PA \text{ and } PB \text{ drawn from an external point } P \text{ to a circle with centre } O, \text{ with } \angle APB = 70^\circ. \\[4pt] \textbf{To Find:} & \text{The measure of } \angle POA. \\[4pt] \textbf{Solution:} & \text{The radius through the point of contact is perpendicular to the tangent, hence } \angle OAP = 90^\circ. \\[4pt] & \text{The line joining the centre to an external point bisects the angle between the two tangents, hence } \angle APO = \tfrac{1}{2}\angle APB. \\[4pt] & \text{In the triangle } OAP, \text{ by the angle sum property of a triangle:} \\[4pt] & \begin{aligned} \angle APO &= \frac{1}{2} \times 70^\circ \\[4pt] &= 35^\circ \\[4pt] \angle POA &= 180^\circ - \angle OAP - \angle APO \\[4pt] &= 180^\circ - 90^\circ - 35^\circ \\[4pt] &= 55^\circ \end{aligned} \\[4pt] \textbf{Answer:} & \angle POA = 55^\circ. \end{array} \]