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. A)`1:1`
B. B)`2:1`
C. C)`3:1`
D. D)`sqrt(2):1`

Answer

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

### Step-by-Step Solution:

1. **Identify the Circle and Chord**:
Let the chord be \(AB\) and the angle subtended at the circumference by the chord \(AB\) be \(30^\circ\).

2. **Use the Circle Theorem**:
According to the circle theorem, the angle subtended at the center of the circle by the same chord is double the angle subtended at the circumference. Therefore, the angle subtended at the center \(O\) will be:
\[ \angle AOB = 2 \times 30^\circ = 60^\circ \]

3. **Draw the Triangle**:
Now, we can visualize triangle \(OAB\) where \(O\) is the center of the circle, and \(OA\) and \(OB\) are the radii of the circle. Since \(OA\) and \(OB\) are radii, we have:
\[ OA = OB = r \quad \text{(where \(r\) is the radius of the circle)} \]

4. **Determine the Angles in Triangle \(OAB\)**:
In triangle \(OAB\):
- \(\angle AOB = 60^\circ\)
- Since \(OA = OB\), triangle \(OAB\) is isosceles. The other two angles, \(\angle OAB\) and \(\angle OBA\), can be calculated as follows:
\[ \angle OAB = \angle OBA = \frac{180^\circ - 60^\circ}{2} = 60^\circ \]
Thus, triangle \(OAB\) is actually an equilateral triangle.

5. **Calculate the Length of the Chord \(AB\)**:
In an equilateral triangle, all sides are equal. Therefore, the length of the chord \(AB\) is equal to the radius \(r\):
\[ AB = r \]

6. **Find the Ratio of Radius to Length of Chord**:
We need to find the ratio of the radius \(OA\) to the length of the chord \(AB\):
\[ \text{Ratio} = \frac{OA}{AB} = \frac{r}{r} = 1 \]

### Final Answer:
The ratio of the radius of the circle to the length of the chord is:
\[ 1:1 \]