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In the fig two tangents AB and AC are drawn to a circle O such that ∠BAC = 120°. Prove that OA = 2AB.

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#### Solution

Consider Centre O for given circle

∠BAC = 120°

AB and AC are tangents

From the fig.

In ΔOBA, ∠OBA = 90° [radius perpendicular to tangent at point of contact]

∠OAB = ∠OAC =`1/2`∠𝐵𝐴𝐶 =`1/2`× 120° = 60°

[Line joining Centre to external point from where tangents are drawn bisects angle formed by tangents at that external point1]

In ΔOBA, cos 60° =`(AB)/(OA)`

`1/2=(AB)/(OA)`⇒ 𝑂𝐴 = 2𝐴𝐵

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