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. Here’s a step-by-step solution:

### Step 1: Find the circumference of the circular wire
The diameter of the wire is given as 10 cm. The formula for the circumference \( C \) of a circle is:

\[ C = \pi \times d \]

where \( d \) is the diameter.

Substituting the diameter of the wire:

\[ C = \pi \times 10 \text{ cm} = 10\pi \text{ cm} \]

### Step 2: Convert the diameter of the larger circle to centimeters
The diameter of the larger circle is given as 1 meter. To convert this to centimeters:

\[ 1 \text{ meter} = 100 \text{ cm} \]

### Step 3: Find the radius of the larger circle
The radius \( R \) of the larger circle is half of its diameter:

\[ R = \frac{100 \text{ cm}}{2} = 50 \text{ cm} \]

### Step 4: Use the formula for the angle subtended at the center
The angle \( \theta \) subtended by an arc at the center of a circle can be calculated using the formula:

\[ \theta = \frac{\text{Arc Length}}{\text{Radius}} \]

In this case, the arc length is the circumference of the wire, which we found to be \( 10\pi \) cm, and the radius of the larger circle is 50 cm.

Substituting these values into the formula:

\[ \theta = \frac{10\pi \text{ cm}}{50 \text{ cm}} = \frac{10\pi}{50} = \frac{\pi}{5} \text{ radians} \]

### Conclusion
The angle subtended by the wire at the center of the larger circle is:

\[ \theta = \frac{\pi}{5} \text{ radians} \]

Thus, the correct answer is \( \frac{\pi}{5} \) radians.

---