Question

The angle between two tangents drawn from an external point to a circle is ______ to the angle subtended by the line segments joining the points of contact at the centre.

Answer 1

The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.

Explanation:

\[ \begin{array}{r l} \textbf{Given:} & \text{Tangents } PA \text{ and } PB \text{ drawn from an external point } P \text{ to a circle with centre } O, \text{ touching it at } A \text{ and } B. \\[4pt] \textbf{To Find:} & \text{The relation between } \angle APB \text{ and } \angle AOB. \\[4pt] \textbf{Solution:} & \text{The radius through the point of contact is perpendicular to the tangent, hence } \angle OAP = \angle OBP = 90^\circ. \\[4pt] & \text{In the quadrilateral } OAPB, \text{ by the angle sum property of a quadrilateral:} \\[4pt] & \begin{aligned} \angle AOB + \angle OAP + \angle APB + \angle OBP &= 360^\circ \\[4pt] \angle AOB + 90^\circ + \angle APB + 90^\circ &= 360^\circ \\[4pt] \angle AOB + \angle APB &= 360^\circ - 180^\circ \\[4pt] &= 180^\circ \\[4pt] &= 180.00^\circ \end{aligned} \\[4pt] \textbf{Answer:} & \text{The angle between the two tangents is } \textbf{supplementary} \text{ to the angle subtended at the centre, since } \angle APB + \angle AOB = 180^\circ = 180.00^\circ. \end{array} \]