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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.

971340_a2973bc640b046dfb8b4ab3023636aab.png


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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