From the circumcentre O of traingle ABC , perpendicular OD is drawn on BC.If angle BAC =60 , Then the value of angle BOD

Options

A. `75^@`
B. `60^@`
C. `45^@`
D. `80^@`

Answer

To solve the problem, we need to find the value of angle BOD given that angle BAC = 60 degrees in triangle ABC, with O being the circumcenter and OD being perpendicular to BC.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have triangle ABC.
- O is the circumcenter of triangle ABC.
- OD is perpendicular to BC.
- Angle BAC = 60 degrees.

2. **Draw the Triangle and Circumcircle:**
- Draw triangle ABC and its circumcircle with center O.
- Mark points A, B, and C on the triangle.
- Draw a line segment BC and drop a perpendicular from O to BC, marking the foot of the perpendicular as point D.

3. **Understand the Relationship Between Angles:**
- According to the properties of a circumcircle, the angle at the center (angle BOC) is twice the angle at the circumference (angle BAC).
- Therefore, angle BOC = 2 * angle BAC = 2 * 60 degrees = 120 degrees.

4. **Use the Perpendicular Bisector Property:**
- Since OD is perpendicular to BC, it also acts as the angle bisector of angle BOC.
- This means that angle BOD = angle COD.

5. **Calculate Angle BOD:**
- Since OD is the angle bisector of angle BOC, we have:
\[ \text{Angle BOD} = \frac{1}{2} \text{Angle BOC} \]
- Substituting the value of angle BOC:
\[ \text{Angle BOD} = \frac{1}{2} \times 120^\circ = 60^\circ \]

6. **Conclusion:**
- Therefore, the value of angle BOD is 60 degrees.

### Final Answer:
Angle BOD = 60 degrees.