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 the length of PT is

• A. 20 cm
• B. $\frac{20}{3}cm$
• C. 40 cm
• D. $\frac{40}{3}cm$

Answer 1

The correct option is B

$\frac{20}{3}cm$

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

$OR\perp PQ$ and so, OR bisects PQ. (The perpendicular line drawn from the center of a circle 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=3cm
In $\Delta TPO$ and $\Delta PRO,$
$\angle TOP=\angle ROP$ [common]
and $\angle TPO=\angle PRO$ [each ${90}^{\circ }]$
$\therefore \Delta TPOand\Delta PROaresimilar$ [by AAA Similarily]
$\Rightarrow \frac{TP}{PO}=\frac{RP}{RO}\Rightarrow \frac{TP}{5}=\frac{4}{3}\therefore TP=\frac{20}{3}cm$