Question

In the given figure, O is the centre of the circle. If $\angle BOC={120}^{\circ },$ then $x$ =___.

• A. ${40}^{\circ }$​​​​​​​
• B. ${30}^{\circ }$​​​​​​​
• C. ${45}^{\circ }$​​​​​​​
• D. ${60}^{\circ }$

Answer 1

The correct option is B ${30}^{\circ }$​​​​​​​

$\angle AOC+\angle COB={180}^{\circ }$ (Linear pair)
$⟹\angle AOC={180}^{\circ }-\angle COB$
$={180}^{\circ }-{120}^{\circ }$
$={60}^{\circ }$
Since the angle subtended by an arc at the centre of a circle is double the angle subtended by it on any remaining point of the circle, we have
$\angle AOC=2\times ADC.$
$⟹\angle ADC=\frac{1}{2}\angle AOC=\frac{{60}^{\circ }}{2}={30}^{\circ }$