Question

Tangents from a point P are drawn onto a circle with angle between them as ${120}^{\circ }$ . The tangents from P meet the circle at A and B. If a line is drawn from point P through the center of the circle O, then find the measure of $\angle$POA.

• A. ${25}^{\circ }$
• B. ${30}^{\circ }$
• C. ${45}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B

${30}^{\circ }$

By Theorem- The tangent at any point of a circle is perpendicular to the radius through the point of contact.

So, $\angle$ PAO = ${90}^{\circ }$ ...(1)

Given $\angle$ APB = ${120}^{\circ }$

By Theorem- Centre of a circle lies on the bisector of the angle between two tangents.
So, line OP bisects $\angle$APB

$\therefore$ $\angle$ APO = ${60}^{\circ }$ ...(2)

In $△$ APO,

$\angle$ APO + $\angle$ PAO + $\angle$POA = ${180}^{\circ }$

$\Rightarrow {60}^{\circ }+{90}^{\circ }+\angle POA={180}^{\circ }$
From (1) and (2)

$\Rightarrow \angle POA={30}^{\circ }$