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 + (6√3)2
∴ 144 = ML2 + 108
∴ ML2 = 144 – 108
∴ ML2 = 36
∴ ML = √36 = 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]