An equilateral `DeltaABC` is inscribed in circle with centre O. Then, `angleBOC` is equal to

Options

A. `120^(@)`
B. `75^(@)`
C. `180^(@)`
D. `60^(@)`

Answer

To find the measure of angle \( \angle BOC \) in an equilateral triangle \( \Delta ABC \) inscribed in a circle with center \( O \), we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all three angles are equal. Therefore, each angle measures:
\[ \angle A = \angle B = \angle C = 60^\circ \]

### Step 2: Identify the relationship between angles at the center and the circumference
According to the properties of circles, the angle subtended at the center of the circle (in this case, \( \angle BOC \)) is twice the angle subtended at any point on the circumference (in this case, \( \angle BAC \)):
\[ \angle BOC = 2 \times \angle BAC \]

### Step 3: Substitute the known angle
Since we know that \( \angle BAC = 60^\circ \) (from Step 1), we can substitute this value into the equation from Step 2:
\[ \angle BOC = 2 \times 60^\circ \]

### Step 4: Calculate the angle
Now, we can perform the multiplication:
\[ \angle BOC = 120^\circ \]

### Conclusion
Thus, the measure of angle \( \angle BOC \) is:
\[ \angle BOC = 120^\circ \]