In a `Delta ABC`, O is its circumcentre and `angle BAC = 50^@`. The measure of `angle OBC` is

Options

A. `60^@`
B. `30^@`
C. `40^@`
D. `50^@`

Answer

To solve for the measure of angle OBC in triangle ABC where O is the circumcenter and angle BAC is given as 50 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information:**
- We have triangle ABC with circumcenter O.
- Angle BAC = 50 degrees.

2. **Understand the Relationship of Angles:**
- The angle subtended by an arc at the center of the circle is twice the angle subtended at the circumference by the same arc. This means that angle BOC (the angle at the center) is twice the angle BAC.

3. **Calculate Angle BOC:**
- Since angle BAC = 50 degrees, we can find angle BOC as follows:
\[ \text{Angle BOC} = 2 \times \text{Angle BAC} = 2 \times 50 = 100 \text{ degrees} \]

4. **Use the Triangle Angle Sum Property:**
- In triangle BOC, the sum of the angles is 180 degrees. Therefore, we can write:
\[ \text{Angle BOC} + \text{Angle OBC} + \text{Angle OCB} = 180 \text{ degrees} \]
- Let angle OBC = θ and angle OCB = θ (since OB = OC as they are both radii of the circumcircle).

5. **Set Up the Equation:**
- Substituting the known values into the equation gives:
\[ 100 + θ + θ = 180 \]
- This simplifies to:
\[ 100 + 2θ = 180 \]

6. **Solve for θ:**
- Rearranging the equation:
\[ 2θ = 180 - 100 \]
\[ 2θ = 80 \]
\[ θ = \frac{80}{2} = 40 \text{ degrees} \]

7. **Conclusion:**
- Therefore, the measure of angle OBC is 40 degrees.

### Final Answer:
- The measure of angle OBC is **40 degrees**.

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