Question

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

Answer 1

we know that the line joining the outside point to centre is perpendicular bisector of line joining points of tangency from outside point to the circle.

$OT\perp PQ$

And since $TP$ is a tangent , $TP\perp OP.$

$OP=5cm$ and $PR=\frac{PQ}{2}=4cm$

$In\Delta OPQ-$

$\sin \theta =\frac{PR}{OP}=\frac{4}{5}$

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\sin \theta }{\sqrt{1-{\cos }^2\theta }}$

$⟹\tan \theta =\frac{4/5}{\sqrt{1-(4/5{)}^2}}=\frac{4}{3}$

In $\Delta OPT-$

$\tan \theta =\frac{TP}{OP}$

$⟹\frac{4}{3}=\frac{TP}{5}$

$⟹TP=\frac{20}{3}cm$