Question

In the given figure, AB is the diameter of the circle, with centre O, and AT is the tangent calculate the numerical value of x.

• A. ${58}^{\circ }$
• B. ${98}^{\circ }$
• C. ${28}^{\circ }$
• D. ${50}^{\circ }$

Answer 1

The correct option is A

${58}^{\circ }$

In $\Delta OBC,OB=OC$ (radii of same circle)
$\therefore \angle OBC=\angle OCB$ $=2\angle OBC$
$\Rightarrow {64}^{\circ }=2\angle OBC$
$\therefore \angle OBC=\frac{{64}^{\circ }}{2}={32}^{\circ }$

Now in $\Delta ABT$,
$\angle BAT={90}^{\circ }(OA\perp AT)$
$\angle OBC$ or $\angle ABT={32}^{\circ }$
$\therefore \angle BAT+\angle ABT+x^{\circ }={180}^{\circ }$
$\Rightarrow {90}^{\circ }+{32}^{\circ }+x^{\circ }={180}^{\circ }$
$\Rightarrow {122}^{\circ }+x^{\circ }={180}^{\circ }$
$x^{\circ }={180}^{\circ }-{122}^{\circ }={58}^{\circ }$