Question

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of 30° with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA.

Answer 1

∠OAB = 30°

∠OAP = 90°  ...[Angle between the tangent and the radius at the point of contact]

∠PAB = 90° – 30° = 60°

AP = BP  ...[Tangents to a circle from an external point]

∠PAB = ∠PBA  ...[Angles opposite to equal sides of a triangle]

In ΔABP, ∠PAB + ∠PBA + ∠APB = 180°  ...[Angle Sum Property]

60° + 60° + ∠APB = 180°

∠APB = 60°

∴ ΔABP is an equilateral triangle, where AP = BP = AB.

PA = 6 cm

In Right ΔOAP, ∠OPA = 30°

tan 30° = `(OA)/(PA)`

`1/sqrt(3) = (OA)/6`

OA = `6/sqrt(3) = 2sqrt(3)cm`