If a chord of a circle subtends an angle of `30^(@)` at the circumference of the circle, then what is the ratio of the radius of the circle and the length of the chord respectively?

Options

A. `1 : 1`
B. `2 : 1`
C. `3 : 1`
D. ` sqrt2 : 1`

Answer

To solve the problem, we need to find the ratio of the radius of a circle to the length of a chord that subtends an angle of \(30^\circ\) at the circumference.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the center of the circle be \(O\) and the endpoints of the chord be \(A\) and \(B\). The angle subtended by the chord \(AB\) at the circumference is \(30^\circ\).

2. **Finding the Angle at the Center**:
- According to the properties of circles, the angle subtended at the center \(O\) by the same chord \(AB\) is double that subtended at the circumference. Therefore, the angle \(AOB = 2 \times 30^\circ = 60^\circ\).

3. **Drawing a Perpendicular**:
- Draw a perpendicular from the center \(O\) to the chord \(AB\) at point \(D\). This divides the chord \(AB\) into two equal halves, making \(AD = DB\).

4. **Using Right Triangle Properties**:
- In triangle \(AOD\), we can use the sine function:
\[ \sin(30^\circ) = \frac{AD}{OA} \]
- Here, \(AD\) is half the length of the chord \(AB\) and \(OA\) is the radius \(R\) of the circle.

5. **Calculating the Length of \(AD\)**:
- We know that \(\sin(30^\circ) = \frac{1}{2}\). Thus,
\[ \frac{AD}{R} = \frac{1}{2} \]
- Rearranging gives us:
\[ AD = \frac{R}{2} \]

6. **Finding the Length of the Chord \(AB\)**:
- Since \(AD = DB\), the total length of the chord \(AB\) is:
\[ AB = AD + DB = 2 \times AD = 2 \times \frac{R}{2} = R \]

7. **Calculating the Ratio**:
- Now we have the radius \(R\) and the length of the chord \(AB = R\). Thus, the ratio of the radius to the length of the chord is:
\[ \text{Ratio} = \frac{R}{AB} = \frac{R}{R} = 1 \]

8. **Final Ratio**:
- Therefore, the ratio of the radius of the circle to the length of the chord is:
\[ \text{Ratio} = 1:1 \]