Question

In the given figure PQ is a chord of length 16cm of a circle of radius 10cm. the tangents at P and Q intersect at a point T. Find the length of TP.

Answer 1

Given,$OR\perp PQ$,bisects $PQ$

$\therefore PR=8cm$ as $PQ=16cm$

In right angled triangle,$OR=\sqrt{{10}^2-8^2}=\sqrt{100-64}=\sqrt{36}=6cm$

$\angle TPR+\angle RPO={90}^{\circ }$

$\angle RPO=\angle PTR$

$\Rightarrow \angle TRP=\angle ORP={90}^{\circ }$

$\Rightarrow △TRP\sim △PRO$ by AAA similarity

or $\frac{TP}{PO}=\frac{RP}{RO}$

$\Rightarrow \frac{TP}{10}=\frac{8}{6}$

or $TP=\frac{80}{6}=\frac{40}{3}cm$

Hence length of $TP=\frac{40}{3}cm$