Question

In the figure below, two tangents are drawn from a point P to a circle meeting it at points A and C.
If $\angle AOC={120}^{\circ }$, what is the value of $\angle APC$?

• A. ${120}^{\circ }$
• B. ${30}^{\circ }$
• C. ${60}^{\circ }$
• D. ${80}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

AOCP forms a quadrilateral.
Sum of angles of a quadrilateral = ${360}^{\circ }$
$\angle OAP={90}^{\circ }$ (Angle between radius and tangent at the point of contact is a right angle)
Similarly, $\angle OCP={90}^{\circ }$

$\angle CPA+\angle OCP+\angle OAP+\angle AOC={360}^{\circ }$
$\angle CPA+{90}^{\circ }+{90}^{\circ }+{120}^{\circ }={360}^{\circ }$
$\angle CPA+{300}^{\circ }={360}^{\circ }$
$\angle CPA={360}^{\circ }-{300}^{\circ }$
$\angle CPA={60}^{\circ }$