Question

In figure displayed below, if TP and TQ are the two tangents to a circle with centre O so that $\angle POQ={110}^{\circ },then\angle PTQ$ is equal to

• A. ${70}^{\circ }$
• B. ${50}^{\circ }$
• C. ${90}^{\circ }$
• D. ${80}^{\circ }$

Answer 1

OP and OQ are radii of the circle to the tangents TP and TQ respectively.
Since, the line drawn from the centre of the circle to the tangent is perpendicular to the tangent.
$\therefore$ OP $\perp$ TP and OQ $\perp$ TQ.
$\angle OPT=\angle OQT={90}^{\circ }$
In quadrilateral POQT,
Sum of all interior angles = ${360}^{\circ }$
$\angle PTQ+\angle OPT+\angle POQ+\angle OQT={360}^{\circ }\Rightarrow \angle PTQ+{90}^{\circ }+{110}^{\circ }+{90}^{\circ }={360}^{\circ }\Rightarrow \angle PTQ={70}^{\circ }$
$\angle PTQ$ is equal to option (A) ${70}^{\circ }$.