In `Delta` ABC, O is the circum centre of the circle. If `angle OBC = 37^(@)` then `angle BAC` is equal to :

Options

A. `74^(@)`
B. `106^(@)`
C. `53^(@)`
D. `37^(@)`

Answer

To solve the problem, we need to find the measure of angle BAC in triangle ABC, given that angle OBC is 37 degrees and O is the circumcenter of the circle.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We know that O is the circumcenter of triangle ABC.
- Angle OBC = 37 degrees.

2. **Understand the Properties of the Circumcenter:**
- Since O is the circumcenter, OA, OB, and OC are all radii of the circumcircle. Therefore, OB = OC = OA.

3. **Use the Isosceles Triangle Property:**
- In triangle OBC, since OB = OC, the angles opposite to these sides are equal. Thus, angle OCB = angle OBC = 37 degrees.

4. **Calculate Angle BOC:**
- The sum of angles in triangle OBC is 180 degrees. Therefore:
\[ \text{Angle BOC} + \text{Angle OBC} + \text{Angle OCB} = 180^\circ \]
\[ \text{Angle BOC} + 37^\circ + 37^\circ = 180^\circ \]
\[ \text{Angle BOC} + 74^\circ = 180^\circ \]
\[ \text{Angle BOC} = 180^\circ - 74^\circ = 106^\circ \]

5. **Relate Angle BOC to Angle BAC:**
- According to the property of circles, the angle at the center (angle BOC) is twice the angle at the circumference (angle BAC) that subtends the same arc. Therefore:
\[ \text{Angle BOC} = 2 \times \text{Angle BAC} \]
\[ 106^\circ = 2 \times \text{Angle BAC} \]

6. **Solve for Angle BAC:**
- Dividing both sides by 2 gives:
\[ \text{Angle BAC} = \frac{106^\circ}{2} = 53^\circ \]

### Final Answer:
- Angle BAC = 53 degrees.