Question

$TP$ and $TQ$ are two tangents to a circle $C(0,r)$ at points $P$ and $Q$ respectively. If $\angle POQ={70}^{\circ }$. Then find $\angle PTQ.$

• A. $90$ ${}^{\circ }$
• B. $70$ ${}^{\circ }$
• C. $60$ ${}^{\circ }$
• D. $110$ ${}^{\circ }$

Answer 1

The correct option is D $110$ ${}^{\circ }$

In figure TP and TQ are tangents at points P and Q respectively.
$\therefore \angle TPO={90}^{\circ }$,
$\angle TQO={90}^{\circ }$
also $\angle POQ={70}^{\circ }$
Now in quadrilateral OPTQ,
we have $\angle OPT+\angle PTQ+\angle TQO+\angle POQ={360}^{\circ }$
$\Rightarrow {90}^{\circ }+\angle PTQ+{90}^{\circ }+{70}^{\circ }={360}^{\circ }$
$\therefore \angle PTQ={110}^{\circ }$