Question

$\text{In the figure given below,}\text{O is the center of the circle, CA is}\text{tangent at A and CB is tangent at B}\text{drawn to the circle.}$

$\text{If}\angle ACB={75}^{\circ },\text{then}\angle AOB=$ ______.

• A. ${75}^{\circ }$​
• B. ${85}^{\circ }$​
• C. ${95}^{\circ }$​
• D. ${105}^{\circ }$​

Answer 1

The correct option is D

${105}^{\circ }$​

$\angle OAC=\angle OBC={90}^{\circ }$​
$(\because \text{The tangent at any point on a circle}$
$\text{is perpendicular to the radius through}$$\text{that point.})$

$\angle OAC+\angle OBC+\angle ACB+\angle AOB={360}^{\circ }$​
$(\because \text{Sum of angles of a quadrilateral})$

${90}^{\circ }+{90}^{\circ }+{75}^{\circ }+\angle AOB={360}^{\circ }$

$\angle AOB={360}^{\circ }-{255}^{\circ }={105}^{\circ }$​​
$\angle AOB={105}^{\circ }$