If tangents $P A$ and $P B$ drawn from an external point $P$ to the circle with centre $O$ are incli

Question

If tangents $P A$ and $P B$ drawn from an external point $P$ to the circle with centre $O$ are inclined to each other at an angle of $80^{\circ}$ as shown in the given figure, then the measure of $\angle P O A$ is\

Professor PC Thomas Classes · Answer

(b): Given, $P A$ and $P B$ are two tangents drawn from external point $P$. Also, given $\angle A P B=80^{\circ}$. We know that tangents are perpendicular to the radius through the point of contact. $$ 	herefore \quad \angle O A P=\angle O B P=90^{\circ} $$ Since, $O P$ is angle bisector of $\angle A P B$. $$ 	herefore \quad \angle A P O=\frac{1}{2} 	imes 80^{\circ}=40^{\circ} $$ In $	riangle P A O$, we have $\angle A P O+\angle O A P+\angle P O A=180^{\circ}$\ $\Rightarrow 40^{\circ}+90^{\circ}+\angle P O A=180^{\circ}$ $$ \begin{aligned} & \Rightarrow \quad 130^{\circ}+\angle P O A=180^{\circ} \ & \Rightarrow \quad \angle P O A=180^{\circ}-130^{\circ}=50^{\circ} \end{aligned} $$