Question

If a pair of tangents 12cm long each is inclined to each other at the angle of 60° then find the radius of the circle

Answer 1

here two tangents are inclined at the angle 60° and they are 12 cm long...... ( given)

now,

let there be an angle bisector BD between the angle ABC........here AB and BC are tangents.

therefore,

angle ABD and angle CBD are congruent and measure 30°

also, let AD and CD be the radii.

now in ∆ ABD and ∆ CBD,

seg AB = seg BC........ tangents drawn from a common point outside the circle are congruent)... (1)

angle BAD= angle BCD..... both measure 90°.......(2)

and,

side BD= side DB..... common side.... (3)

therefore, from (1), (2) and (3)

∆ ABD and ∆ DBC... are congruent by SAS test

now in ∆ABD, side BD is Hypotenuse.

by property of 30°-60°-90° of right angled triangles,

side AB= √3/2 of hypotenuse...... (side opposite to 60°)

12×2

√3 = hypotenuse

24

√3 = hypotenuse

8√3 = hypotenuse BD.

now,

DA= radius= 1/2 of hypotenuse BD

DA= 1/2 × 8√3

DA= 4√3 cm

therefore, the radius of the circle = 4√3 cm