Calculate the angle subtended by the chord at the point on the major arc if the radius and the chord of the circle are equal in length

Options

A. A) `30^@`
B. B) `120^@`
C. C) `200^@`
D. D) `300^@`

Answer

To solve the problem of calculating the angle subtended by the chord at the point on the major arc when the radius and the chord of the circle are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle and Points**:
- Let O be the center of the circle.
- Let A and B be the endpoints of the chord.

2. **Establish Relationships**:
- Given that the radius (OA and OB) and the chord (AB) are equal in length, we can denote their length as 'r'. Thus, OA = OB = AB = r.

3. **Form Triangle AOB**:
- Since OA, OB, and AB are all equal, triangle AOB is an equilateral triangle.

4. **Calculate Angles in Triangle AOB**:
- In an equilateral triangle, all angles are equal. Therefore, each angle in triangle AOB is 60 degrees.
- Thus, ∠AOB = 60 degrees.

5. **Use the Angle Subtended at the Center**:
- The angle subtended by the chord AB at the center O is ∠AOB, which we found to be 60 degrees.

6. **Apply the Angle Subtended at the Circle**:
- The angle subtended by the chord AB at any point D on the major arc (angle ADB) is half of the angle subtended at the center (angle AOB).
- Therefore, ∠ADB = 1/2 * ∠AOB = 1/2 * 60 degrees = 30 degrees.

7. **Conclusion**:
- The angle subtended by the chord AB at the point on the major arc is 30 degrees.

### Final Answer:
The angle subtended by the chord at the point on the major arc is **30 degrees**.

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