In the figure, two tangents $ A B $ and $ A C $ are drawn to a circle with centre $ O $ such that $ \angle B A C=120^{\circ} . $ Prove that $ O A=2 A B $."

Given:

In the figure, two tangents \( A B \) and \( A C \) are drawn to a circle with centre \( O \) such that \( \angle B A C=120^{\circ} . \)

To do:

We have to prove that \( O A=2 A B \).

Solution:

In $\triangle OAB$ and $\triangle OAC$,
$\angle OBA = \angle OCA - 90^o$    ($OB$ and $OC$ are radii)

$OA = OA$    (Common side)

$OB = OC$     (Radii of the circle)

$\triangle OAB\ \sim\ \triangle OAC$

$\angle OAB = \angle OAC = 60^o$

In right angled triangle $OAB$,

$\cos 60^{\circ}=\frac{AB}{OA}$

$\Rightarrow \frac{1}{2}=\frac{AB}{OA}$

$\Rightarrow OA=2AB$

Hence proved.

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Updated on: 10-Oct-2022

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