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