O is the circumcentre of the `triangle`ABC. If `angle BAC =50^@`, then `angleOBC` is equal to

Options

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

Answer

To solve the problem step by step, we will analyze the triangle ABC and the circumcenter O.

### Step 1: Identify the Given Information
We are given that O is the circumcenter of triangle ABC and that angle BAC = 50 degrees.

### Step 2: Understand the Properties of the Circumcenter
The circumcenter O is the point where the perpendicular bisectors of the sides of the triangle intersect. The segments OB and OC are equal because they are both radii of the circumcircle.

### Step 3: Establish Angles in Triangle OBC
Since OB = OC (radii of the circumcircle), triangle OBC is an isosceles triangle. Therefore, the angles opposite to the equal sides are equal:
- Let angle OBC = angle OCB = θ.

### Step 4: Relate Angle BOC to Angle BAC
The angle BOC at the circumcenter O is related to angle BAC at the circumference. The angle subtended by an arc at the center (angle BOC) is twice the angle subtended at the circumference (angle BAC):
- Therefore, angle BOC = 2 * angle BAC = 2 * 50 degrees = 100 degrees.

### Step 5: Use the Triangle Sum Property
In triangle OBC, the sum of the angles is 180 degrees:
- angle OBC + angle OCB + angle BOC = 180 degrees
- θ + θ + 100 degrees = 180 degrees
- 2θ + 100 degrees = 180 degrees.

### Step 6: Solve for θ
Now, we can solve for θ:
- 2θ = 180 degrees - 100 degrees
- 2θ = 80 degrees
- θ = 80 degrees / 2
- θ = 40 degrees.

### Conclusion
Thus, angle OBC = 40 degrees.

### Final Answer
Angle OBC is equal to 40 degrees.

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