Question

In the adjoining figure, AC is the diameter of the larger circle with centre O. AB is tangent to the smaller circle with centre O. If OD = r, then BC is equal to:

Options

• r

• `(3r)/2`

• 2r

• 4r

Answer 1

2r

Explanation:

OD ⊥ AB   ....(∵ AB is tangent to the smaller circle at D.)

∠ ADO = 90°

∠ ABC = 90°   ....(∵ Angle in a semicircle subtended by diameter AC.)

Also, OD ⟂ AB and O is centre.

∴ D is the midpoint of AB = AD = DB

Let AD = x and AB = 2x

In ΔADO and ΔABC,

∠ADO = ∠ABC   ....(Each 90°)

∠A = ∠A   ....(Common)

By AA similarity criterion,

ΔADO ∼ ΔABC

`(AD)/(AB) = (DO)/(BC)`

`x/(2x) = r/(BC)`

BC = 2r