Question

In the given figure, two tangents AB and AC are drawn to a circle with center O such that $$\angle BAC = 120^{\circ}$$. Prove that OA = 2AB.

Solution

From figure$$\angle BAC={ 120 }^{ 0 }$$ (given)   &   $$AB=AC$$  (tangent property)$$OB=OC$$ (radius)$$\Rightarrow \angle BAO=\angle CAO={ 60 }^{ 0 }$$  {$$\angle$$ bisector}In $$\Delta ABO$$,  $$\cos A=\dfrac { AB }{ OA }$$........$$(\because \triangle ABO \ is \ a \ rt \angle d \ \triangle \ as \ AB \bot BO)$$$$\Rightarrow \quad { \cos60 }^{ 0 }=\dfrac { AB }{ OA }$$$$\Rightarrow \quad \boxed { OA=2AB }$$Maths

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