If the piece of wire 25 cm long is bent into an arc of a circle subtending an angle of `75^(@)` at the centre , then the radius of the circle ( in cm ) is :

Options

A. `(pi)/(120)`
B. `(60)/(pi)`
C. `60 pi `
D. none of these

Answer

To find the radius of the circle formed by bending a piece of wire 25 cm long into an arc that subtends an angle of 75 degrees at the center, we can use the formula for the length of an arc.

### Step-by-Step Solution:

1. **Understand the formula for the length of an arc**:
The length of an arc (L) of a circle is given by the formula:
\[ L = \frac{\theta}{360} \times 2\pi r \]
where:
- \(L\) is the length of the arc,
- \(\theta\) is the angle subtended at the center (in degrees),
- \(r\) is the radius of the circle.

2. **Substitute the known values**:
Here, we know:
- \(L = 25 \, \text{cm}\)
- \(\theta = 75^\circ\)

Substitute these values into the formula:
\[ 25 = \frac{75}{360} \times 2\pi r \]

3. **Simplify the equation**:
To simplify, first calculate \(\frac{75}{360}\):
\[ \frac{75}{360} = \frac{75 \div 15}{360 \div 15} = \frac{5}{24} \]
Now substitute this back into the equation:
\[ 25 = \frac{5}{24} \times 2\pi r \]

4. **Multiply both sides by \(\frac{24}{5}\)** to isolate \(r\):
\[ 25 \times \frac{24}{5} = 2\pi r \]
Simplifying the left side:
\[ 25 \times \frac{24}{5} = 5 \times 24 = 120 \]
So we have:
\[ 120 = 2\pi r \]

5. **Divide both sides by \(2\pi\)** to solve for \(r\):
\[ r = \frac{120}{2\pi} = \frac{60}{\pi} \]

6. **Final answer**:
Therefore, the radius of the circle is:
\[ r = \frac{60}{\pi} \, \text{cm} \]