Question

In the given figure, AB is chord of the circle with centre O, BT is tangent to the circle. The values of x and y are

• A. ${52}^{\circ },{52}^{\circ }$
• B. ${58}^{\circ },{52}^{\circ }$
• C. ${58}^{\circ },{58}^{\circ }$
• D. ${60}^{\circ },{64}^{\circ }$

Answer 1

The correct option is C ${58}^{\circ },{58}^{\circ }$

Given AB is a chord of circle and BT is a tangent, $\angle BAO={32}^{\circ }$
Here, $\angle OBT={90}^{\circ }$
[$\because$ Tangent is $\perp$ to the radius at the point of contact]
OA = OB [Radii of the same circle]
$\therefore \angle OBA=\angle OAB={32}^{\circ }$
[Angles opposite to equal side are equal]
$\therefore \angle OBT=\angle OBA+\angle ABT={90}^{\circ }or{32}^{\circ }+x={90}^{\circ }\angle x={90}^{\circ }-{32}^{\circ }={58}^{\circ }Also,\angle AOB={180}^{\circ }-\angle OAB-\angle OBA={180}^{\circ }-{32}^{\circ }-{32}^{\circ }={116}^{\circ }$
Now $Y=\frac{1}{2}\angle AOB$
[Angle formed at the center of a circle is double the angle formed in the remaining part of the circle]
$=\frac{1}{2}\times {116}^{\circ }={58}^{\circ }$