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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.
Solution
∠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`
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