ABC is an equilateral triangle and O is its circumcentre, then the `/_AOC `is

Options

A. `100^@`
B. `110^@`
C. `120^@`
D. `130^@`

Answer

To find the angle \( \angle AOC \) in an equilateral triangle \( ABC \) with circumcentre \( O \), 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 \( \angle A \), \( \angle B \), and \( \angle C \) measures \( 60^\circ \).

### Step 2: Identify the relationship between the angles
The circumcentre \( O \) of triangle \( ABC \) is the point where the perpendicular bisectors of the sides intersect. The angle \( \angle AOC \) is formed by the lines \( OA \) and \( OC \).

### Step 3: Use the theorem about angles and arcs
According to the theorem regarding angles and arcs in a circle, the angle at the center of the circle (which is \( \angle AOC \)) is twice the angle at the circumference (which is \( \angle ABC \)). Therefore, we can express this relationship as:
\[ \angle AOC = 2 \times \angle ABC \]

### Step 4: Substitute the known angle
Since we know that \( \angle ABC = 60^\circ \) (because triangle \( ABC \) is equilateral), we can substitute this value into the equation:
\[ \angle AOC = 2 \times 60^\circ \]

### Step 5: Calculate the angle
Now, performing the multiplication gives us:
\[ \angle AOC = 120^\circ \]

### Final Answer
Thus, the measure of angle \( AOC \) is \( 120^\circ \).

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