Question

In the given figure, PQ is a chord of length 8 cm of a circle with centre O and radius 5 cm. If the tangents to the circle at the points P and Q intersect at 'T', then find the length of PT.

Answer 1

In the given figure,
PQ = 8 cm and OP = 5 cm

$OR\perp PQ$ and so, OR bisects PQ. [$\because$ Perpendicular drawn from the center to the chord bisects the chord]
$\Rightarrow PR=RQ=4cm$

In $\Delta POR,$
$O{P}^2=O{R}^2+P{R}^2\Rightarrow 5^2=O{R}^2+4^2$
$\Rightarrow$ OR = 3 cm
In $\Delta TPO$ and $\Delta PRO,$
$\angle TOP=\angle ROP$ [common]
and $\angle TPO=\angle PRO$ [each ${90}^{\circ }]$
$\therefore \Delta TPOand\Delta PRO$ are similar. [by AAA Similarity]
$\Rightarrow \frac{TP}{PO}=\frac{RP}{RO}\Rightarrow \frac{TP}{5}=\frac{4}{3}\therefore TP=\frac{20}{3}cm$