The circular wire of diameter 10 cm is cut and placed along the circumference of a circle of diameter 1 meter. The angle subtended by the wire at the centre of circle is equal to

Options

A. `(pi)/(4)` radian
B. `(pi)/(3)` radian
C. `(pi)/(3)` radian
D. `(pi)/(10)` radian

Explanation

To solve the problem, we need to find the angle subtended by a circular wire of diameter 10 cm when it is placed along the circumference of a larger circle with a diameter of 1 meter (100 cm).

### Step-by-Step Solution:

1. **Calculate the circumference of the wire:**
The diameter of the wire is given as 10 cm. The circumference \( C \) of a circle is given by the formula:
\[ C = \pi \times d \]
where \( d \) is the diameter. Therefore, for the wire:
\[ C_{\text{wire}} = \pi \times 10 = 10\pi \text{ cm} \]

2. **Convert the diameter of the larger circle to radius:**
The diameter of the larger circle is given as 1 meter, which is equal to 100 cm. The radius \( r \) is half of the diameter:
\[ r = \frac{100}{2} = 50 \text{ cm} \]

3. **Use the relationship between arc length, radius, and angle:**
The angle \( \theta \) in radians subtended by an arc at the center of a circle can be calculated using the formula:
\[ L = r \theta \]
where \( L \) is the length of the arc (which is the circumference of the wire), and \( r \) is the radius of the circle.

4. **Substitute the known values into the formula:**
Here, \( L = 10\pi \) cm and \( r = 50 \) cm. Therefore:
\[ 10\pi = 50\theta \]

5. **Solve for \( \theta \):**
Rearranging the equation to find \( \theta \):
\[ \theta = \frac{10\pi}{50} = \frac{\pi}{5} \text{ radians} \]

6. **Convert the angle to degrees (if needed):**
To convert radians to degrees, use the conversion factor \( \frac{180}{\pi} \):
\[ \theta_{\text{degrees}} = \frac{\pi}{5} \times \frac{180}{\pi} = 36^\circ \]

Thus, the angle subtended by the wire at the center of the circle is \( \frac{\pi}{5} \) radians or \( 36^\circ \).

### Final Answer:
The angle subtended by the wire at the center of the circle is \( \frac{\pi}{5} \) radians.