Question

In the given figure, $O$ is the centre of the circle. $A$ is any point on minor arc $BC$. Find the value of $\angle BAC-\angle OBC$.

• A. $90°$
• B. $120°$
• C. $60°$
• D. $45°$

Answer 1

The correct option is A

$90°$

Step 1: Given data

From the data given in the question,

Here $OB=OC=$radius

As $∆OBC$ is isosceles

Step 2: Theorem related to the circle

Angle made by same segments are equal.

So, $\angle OBC=\angle OCB=y$

Step 3: Applying the angle sum property

Now in $∆OBC$,

$$\begin{gathered} \angle OBC+\angle OCB+\angle BOC=180° \\ y+y+t=180° \\ 2y+t=180° \\ 2y+360°-Reflex\angle z=180° \end{gathered}$$

Step 4: Applying the degree measure theorem

By degree measure theorem,

Reflex$\angle z=2x$

$$\begin{gathered} \Rightarrow 2y+360°-2x=180° \\ \Rightarrow 2y-2x=180°-360° \\ \Rightarrow 2y-2x=-180° \\ \Rightarrow y-x=-90° \\ \Rightarrow x-y=90° \end{gathered}$$

Hence,$\angle BAC-\angle OBC=90°.$

Hence, option (A) is the correct choice.