In the given figure, $P A$ and $P B$ are two tangents drawn to the circle with centre $O$ and radius

Question

In the given figure, $P A$ and $P B$ are two tangents drawn to the circle with centre $O$ and radius 5 cm . If $\angle A P B=60^{\circ}$, then the length of $P A$ is :\

Professor PC Thomas Classes · Answer

(b) : Let us join $O P, O A$ and $O B$. In the given figure,\ $O A=O B=5 \mathrm{~cm}$ (Radii)\ In $	riangle O A P$ and $	riangle O B P$,\ $O A=O B$ (Radii)\  $A P=P B$ (Tangents from an external point)\ $\angle O A P=\angle O B P=90^{\circ}$ [Angle between radius and tangent to a circle at point of contact is always $90^{\circ}$ ]\ $\Rightarrow 	riangle \mathrm{OAP} \cong 	riangle \mathrm{OBP}$ (SAS) $\Rightarrow \angle \mathrm{OPA}=\angle \mathrm{OPB}=30^{\circ}$\ Now, in $	riangle O A P, \angle A=90^{\circ}$\ $	herefore \quad 	an 30^{\circ}=\frac{O A}{A P} \Rightarrow \frac{1}{\sqrt{3}}=\frac{5}{A P} \Rightarrow A P=5 \sqrt{3} \mathrm{~cm}$\