Question

In the given figure, $A,B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC={30}^{\circ }$ and $\angle AOB={60}^{\circ }$. If $D$ is a point on the circle other than the arc $ABC,$ find $\angle ADC$.

• A. $\frac{1}{2}\angle AOC$
  
• B. ${45}^{\circ }$
• C. $\frac{1}{2}\angle AOB$
• D. ${30}^{\circ }$

Answer 1

Answer: (A), (B)

The correct options are
A

$\frac{1}{2}\angle AOC$

B

${45}^{\circ }$

It can be observed that $\angle AOC=\angle AOB+\angle BOC={60}^{\circ }+{30}^{\circ }={90}^{\circ }$

We know that angle subtended by an arc at the centre is double the angle subtended by it any point on the remaining part of the circle.

So, $\angle ADC=\frac{1}{2}\angle AOC=\frac{1}{2}\times {90}^{\circ }={45}^{\circ }$

$\Rightarrow \angle ADC={45}^{\circ }$.