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
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)
