Question

In the figure, M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = `6sqrt(3)`, then find

1. Radius of the circle.
2. Measures of ∠K and ∠M.

Answer 1

i. Line KL is the tangent to the circle at point L and seg ML is the radius.   ...[Given]

∴ ∠MLK = 90°   ...(i) [Tangent theorem]

In ∆MLK, ∠MLK = 90°

∴ MK² = ML² + KL²    ...[Pythagoras theorem]

∴ 12² = ML² + `(6sqrt(3))^2`

∴ 144 = ML² + 108

∴ ML² = 144 – 108

∴ ML² = 36

∴ ML = `sqrt(36)`

∴ ML = 6 units   ...[Taking square root of both sides]

∴ Radius of the circle is 6 units.

ii. We know that,

ML = `1/2` MK,

∴ ∠K = 30°   ...(ii) [Converse of 30° – 60° – 90° theorem]

In ∆MLK,

∠L = 90°   ...[From (i)]

∠K = 30°   ...[From (ii)]

∴ ∠M = 60°   ...[Remaining angle of ∆MLK]