The angle subtended by a chord at its centre is `60^@`, then the ra tio between chord and radius is

Options

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

Answer

To solve the problem of finding the ratio between the chord and the radius when the angle subtended by the chord at the center is \(60^\circ\), we can follow these steps:

### Step 1: Understand the Relationship
The angle subtended by a chord at the center of a circle is related to the chord length and the radius of the circle. If we denote the radius of the circle as \(r\) and the length of the chord as \(c\), we can use the formula that relates these quantities.

### Step 2: Use the Chord Length Formula
The length of the chord \(c\) can be calculated using the formula:
\[ c = 2r \sin\left(\frac{\theta}{2}\right) \]
where \(\theta\) is the angle subtended by the chord at the center. In this case, \(\theta = 60^\circ\).

### Step 3: Substitute the Values
Substituting \(\theta = 60^\circ\) into the formula, we have:
\[ c = 2r \sin\left(\frac{60^\circ}{2}\right) = 2r \sin(30^\circ) \]

### Step 4: Calculate \(\sin(30^\circ)\)
We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
Thus, substituting this value into the equation gives:
\[ c = 2r \cdot \frac{1}{2} = r \]

### Step 5: Find the Ratio
Now, we can find the ratio of the chord length \(c\) to the radius \(r\):
\[ \text{Ratio} = \frac{c}{r} = \frac{r}{r} = 1 \]

### Final Answer
The ratio between the chord and the radius is \(1:1\).

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