Question

In figure, if O is the centre of a circle, PQ is a chord and the tangent PR makes an angle of ${50}^{\circ }$ with PQ, then $\angle POQ$ is equal to:

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
$\therefore$ $\angle OPR={90}^{\circ }$
$\Rightarrow$ $\angle OPQ+\angle QPR={90}^{\circ }$ [From figure]
$\Rightarrow$ $\angle OPQ={90}^{\circ }-{50}^{\circ }={40}^{\circ }$ $[\because \angle QPR={50}^{\circ }]$
Now, OP = OQ = Radius of circle
$\therefore$ $\angle OQP=\angle OPQ={40}^{\circ }$
[Since, angles opposite to equal sides are equal]
In $△OPQ,\angle O+\angle P+\angle Q={180}^{\circ }$
[Since, angles opposite to equal sides are equal]
$\Rightarrow$ $\angle O={180}^{\circ }-({40}^{\circ }-{40}^{\circ })$ $[\because \angle P={40}^{\circ }=\angle Q]$
= ${180}^{\circ }-{80}^{\circ }={100}^{\circ }$