In figure (i) given below, O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the 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, from figure,

OB ⊥ AB

In right angled △OBA,

$OB^2 + AB^2 = OA^2 \\[1em] r^2 + 15^2 = (r + 7.5)^2 \\[1em] r^2 + 225 = r^2 + 56.25 + 15r \\[1em] r^2 - r^2 + 225 - 56.25 = 15r \\[1em] 15r = 168.75 \\[1em] r = \dfrac{168.75}{15} \\[1em] r = 11.25 \text{ cm}.$

Hence, radius of the circle = 11.25 cm.