Question

In the given figure, $PA$ and $PB$ are two tangents to the circle with centre O. If $\angle APB={60}^{\circ }$ then find the measure of $\angle OAB$.

Answer 1

From given figure,

$PA$ and $PB$ are the two tangents to the circle.

O be the centre of circle. $OA$ and $AB$ are joined

$\angle APB={60}^{\circ }$

To Find : $\angle OAB$

Since $PA$ and $PB$ are tangents to the circle from P

$PA=PB$

$\angle PAB=\angle PBA$

But $\angle APB={60}^{\circ }$

$\angle PAB+\angle PBA={180}^{\circ }-{60}^{\circ }={120}^{\circ }$

$2\angle PAB={120}^{\circ }$

$\angle PBA={60}^{\circ }$

$OA4$ is radius and $PA$ is tangent.

$OA\perp PA$

$\angle OAP={90}^{\circ }$

$\angle OAB+\angle PAB={90}^{\circ }$

$\angle OAB+{60}^{\circ }={90}^{\circ }$

$\angle OAB={90}^{\circ }-{60}^{\circ }={30}^{\circ }$

Therefore, the meansure of $\angle OAB$ is ${30}^{\circ }$.