Question

If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ______.

Answer 1

If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is 5 cm.

Explanation:

\[ \begin{array}{r l} \textbf{Given:} & \text{Tangents } PA \text{ and } PB \text{ drawn from an external point } P \text{ to a circle, with } PA = 5 \text{ cm and } \angle APB = 60^\circ. \\[4pt] \textbf{To Find:} & \text{The length of the chord } AB. \\[4pt] \textbf{Solution:} & \text{The lengths of tangents drawn from an external point to a circle are equal, hence } PB = PA = 5 \text{ cm and the triangle } PAB \text{ is isosceles.} \\[4pt] & \text{In an isosceles triangle the angles opposite the equal sides are equal, hence } \angle PAB = \angle PBA. \\[4pt] & \text{In the triangle } PAB, \text{ by the angle sum property of a triangle:} \\[4pt] & \begin{aligned} \angle PAB + \angle PBA &= 180^\circ - 60^\circ \\[4pt] &= 120^\circ \\[4pt] \angle PAB = \angle PBA &= \frac{120^\circ}{2} \\[4pt] &= 60^\circ \\[4pt] \angle APB = \angle PAB = \angle PBA &= 60^\circ \\[4pt] AB &= PA \\[4pt] &= 5 \\[4pt] &= 5.00 \end{aligned} \\[4pt] \textbf{Answer:} & \text{The triangle } PAB \text{ is equilateral, hence } AB = 5 \text{ cm} = 5.00 \text{ cm.} \end{array} \]