Question

$PQ$ is a chord of length $8cm$ of a circle of radius $5cm$. The tangent at $P$ and $Q$ intersect at a point $T$ (see Fig). find the length $TP$.

Answer 1

As OT is perpendicular to PQ, so it divides PQ in two equal parts, then, $PR=4cm$.

In $\Delta POR$,

${\left(OR\right)}^2=5^2-4^2$

${\left(OR\right)}^2=9$

$OR=3cm$

Let $TP=x$ and $\angle POT=\theta$.

In $\Delta POR,$

$\tan \theta =\frac{4}{3}$

In $\Delta POT$,

$\tan \theta =\frac{x}{5}$

$\frac{4}{3}=\frac{x}{5}$

$x=\frac{20}{3}$

Therefore, $TP=\frac{20}{3}cm$.