Question

$TP$ and $TQ$ are tangents from $T$ to the circle with centre $O.$ Then is it possible to draw a circle through the points $P,O,Q$and $T?$

• A. No
• B. Yes
• C. Cannot say
• D. Data insufficient

Answer 1

The correct option is B Yes

In quadrilateral, OPTQ
$\angle OPT=\angle OQT={90}^{\circ }$ (Angle between radius and the tangent)
$\angle OPT+\angle OQT=180$
These are also opposite angles.
Sum of angles = 360
$\angle OPT+\angle OQT+\angle POQ+\angle PTQ=360$
$90+90+\angle POQ+\angle PTQ=360$
$\angle POQ+\angle PTQ=180$
Thus, sum of opposite angles $={180}^{\circ }$
Since, sum of both pair of opposite angles is $180$. Hence, it will form a cyclic quadrilateral.