Question

In the given figure, $AB$ is the diameter of a circle with $O$ and $AT$ is a tangent. If $\angle AOQ={58}^{\circ }$, then the value of $\angle ATQ$ is _________.

• A. ${52}^{\circ }$
• B. ${61}^{\circ }$
• C. ${46}^{\circ }$
• D. ${75}^{\circ }$

Answer 1

The correct option is B ${61}^{\circ }$

Angle subtended by an arc(here, $AQ$) at the centre is twice the angle subtended by it on the circumference on same side of the chord(here, $AQ$)

Given, $\angle AOQ={58}^{\circ }$

So, $\angle ABT=\angle ABQ={29}^{\circ }$

Since, radius of circle is perpendicular to the tangent at the point of contact, $\angle BAT={90}^{\circ }$

In $△ABT$, $\angle BAT+\angle ABT+\angle BTA={180}^{\circ }$

$\Rightarrow {90}^{\circ }+{29}^{\circ }+\angle BTA={180}^{\circ }$

$\Rightarrow \angle BTA={61}^{\circ }$

$\angle ATQ=\angle BTA={61}^{\circ }$

So, Option (B)