A point P is at a distance 13 cm from the centre C of a circle, and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.

The below diagram shows the circle and the tangent:

Given point P is 13 cm away from centre C, so CP = 13 cm.

PT = 12 cm

CT = radius of the circle.

Since the tangent at any point of a circle and the radius through the point are perpendicular to each other.

So, CT ⊥ PT

So, in right angled △CPT by pythagoras theorem,

$\Rightarrow CP^2 = CT^2 + PT^2 \\[1em] \Rightarrow 13^2 = CT^2 + 12^2 \\[1em] \Rightarrow 169 = CT^2 + 144 \\[1em] \Rightarrow CT^2 = 169 - 144 \\[1em] \Rightarrow CT^2 = 25 \\[1em] \Rightarrow CT = \sqrt{25} \text{ cm} \\[1em] \Rightarrow CT = 5 \text{ cm}.$

Hence, radius of circle = 5 cm.