Question

In the figure $P$ is at a distance of $6cm$ from the centre of the circle. $PA$ and $PB$ are tangents from the point $P$. Find the radius of the circle and length of tangents.

Answer 1

Join $OP$
$m\angle AOB={120}^o$
$\Rightarrow m\angle APO={180}^o-{120}^o={60}^o$.... (opposite)
angles of a quadrilateral are supplementary)
$\Rightarrow m\angle APO=m\angle BPO={60}^o/2={30}^o$
.... (tangents drawn from an external point to a circle subtend equal angles at the centre)
Now, a tangent to a circle is perpendicular to the radius through the point of contact.
$\Rightarrow m\angle OAP={90}^o$
Also, Now in right-angled $△OAP=OP=6cm=$ Hypotenuse
$\sin {30}^o=\frac{OA}{OP}\Rightarrow \frac{1}{2}=\frac{OA}{6}\Rightarrow OA=\frac{6}{2}=3cm$
$\cos {30}^o=\frac{PA}{OP}\Rightarrow \frac{\sqrt{3}}{2}=\frac{PA}{6}\Rightarrow PA=\frac{6\times \sqrt{3}}{2}=3\sqrt{3}cm$
Thus radius of circle $=OA=OB=3cm$ and length of tangents $=PA=PB=3\sqrt{3}cm$