# The radii of two concentric circles are 12 cm and 13 cm AB is a diameter of the bigger circle BD is a tangent to a smaller circle touching it at D Find the length of AD

**Exam Target**

The radii of two concentric circles are 12 cm and 13 cm. AB is a diameter of the bigger circle. BD is a tangent to a smaller circle touching it at D. Find the length (in cm) of AD? (correct to one decimal place)

## Answer (Detailed Solution Below)

## Detailed Solution

Given:

Radius of the bigger circle = 13 cm

Radius of the smaller circle = 12 cm

AB is the diameter of the bigger circle.

BD is a tangent to a smaller circle.

Calculation:

By Pythagoras Theorem; in triangle ODB

⇒ OB2 = OD2 + BD2

⇒ 169 = 144 + BD2

⇒ BD = 5 cm

BD = DE = 1/2 × BE ∵ [Perpendicular drawn from the centre on a chord bisects it in two equal parts]

⇒ DE = BD = 5 cm

⇒ BE = 10 cm

In triangle ABE,

∠AEB = 90° ∵ [Angle made in semicircle]

⇒ AB2 = EB2 + AE2

⇒ 676 = 100 + AE2

⇒ AE = √576

⇒ AE = 24 cm

In triangle ADE

⇒ AD2 = ED2 + AE2

⇒ AD2 = 25 + 576

⇒ AD2 = 601

⇒ AD = √601

⇒ AD = 24.51 cm

∴ AD = 24.5 (Approximately)