ABC is an equilateral triangle inscribed in a circle. D is any point on the arc BC. What is `angleADB` equal to?

Options

A. `90^@`
B. `60^@`
C. `45^@`
D. None of these

Answer

To find the value of angle ADB in the given configuration, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have an equilateral triangle ABC inscribed in a circle.
- D is a point on the arc BC of the circle, not including points B and C.

2. **Properties of the Equilateral Triangle**:
- In an equilateral triangle, all angles are equal to 60 degrees.
- Therefore, angle ABC = angle BCA = angle CAB = 60 degrees.

3. **Identifying the Angles**:
- We need to find angle ADB.
- Since D lies on the arc BC, we can use the property of angles subtended by the same arc.

4. **Using the Inscribed Angle Theorem**:
- The inscribed angle theorem states that the angle subtended by an arc at the center of the circle is twice the angle subtended at any point on the circumference.
- Therefore, angle BDC (the angle subtended by arc BC at point D) is equal to angle BAC (which is 60 degrees).

5. **Finding Angle ADB**:
- Since angle BDC = angle BAC = 60 degrees, and angle ADB is the same as angle BDC due to the inscribed angle theorem, we conclude that:
- Angle ADB = 60 degrees.

### Final Answer:
Thus, the value of angle ADB is **60 degrees**.

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