Question

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

Answer 1

Given,

$OA=10$

$ON\perp AB$

$AN=NB=\frac{16}{2}=8$

In $△ONA,$

$O{N}^2+N{A}^2=O{A}^2$

$O{N}^2+8^2={10}^2$

$ON=6$

$\tan \angle AON=\frac{AN}{ON}=\frac{8}{6}=\frac{4}{3}$

$△OAP$

$\tan \angle AON=\frac{PA}{OA}$

$\frac{4}{3}=\frac{PA}{10}$

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