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, the value of $\angle$POA is

• A. 25⁰
• B. 30⁰
• C. 45⁰
• D. 90⁰

Answer 1

The correct option is B

30⁰

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 = ${120}^{\circ }$

A line through P passing through O bisects $\angle$APB

So, $\angle$ APO = ${60}^{\circ }$---------(2)

In APO,

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

From (1) and (2)

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