Question

In figure $AB$ is a chord of length $16cm$ of a circle of radius $10cm$. The tangents at $A$ and$B$ intersect at a point $P$. Find the length of $PA$.

Answer 1

given Radius of circle $=r=BO=10cm$

$AB=16cm$

$\angle PBO={90}^{\circ }$ ......Property of Tangent to circle

Property : A pair of tangents drawn from external point to circle are Equal and subtend ${90}^{\circ }$ at he centre.

$\Rightarrow PA=PB$

$OP$ perpendicularly bisects $AB$ at $M$ Please mark on figure .

$BM=MA=8cm$

Considering the triangle $\Delta BOM$

$sin\left(\angle MOB\right)=sin\left(\angle POB\right)=\frac{8}{10}$

$\Rightarrow tan\left(\angle POB\right)=\frac{8}{6}$

Considering the triangle $\Delta POB$

$tan\left(\angle POB\right)=\frac{PB}{BO}=\frac{8}{6}$

$\frac{PB}{10}=\frac{8}{6}$

$PA=PB=\frac{40}{3}cm$