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.

Answer 1

From figure

$\angle BAC={120}^0$ (given) & $AB=AC$ (tangent property)

$OB=OC$ (radius)

$\Rightarrow \angle BAO=\angle CAO={60}^0$ {$\angle$ bisector}

In $\Delta ABO$, $\cos A=\frac{AB}{OA}$........$(\because △ABOisart\angle d△asAB\perp BO)$

$\Rightarrow {\cos 60}^0=\frac{AB}{OA}$

$\Rightarrow OA=2AB$