If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding of the other circle. Is this statement false ? Why ?

Explanation

To determine if the statement is true or false, we will analyze the relationship between the lengths of the arcs and the angles subtended by those arcs in two circles of different radii.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two circles. Circle 1 has a radius \( r \) and Circle 2 has a radius \( 2r \). We are given that the lengths of the arcs of both circles are equal.

2. **Formula for Arc Length**:
The length of an arc \( L \) in a circle can be calculated using the formula:
\[ L = \frac{\theta}{360} \times 2\pi r \]
where \( \theta \) is the angle in degrees subtended by the arc at the center of the circle.

3. **Length of Arc in Circle 1**:
For Circle 1 (radius \( r \)):
\[ L_1 = \frac{\theta_1}{360} \times 2\pi r \]

4. **Length of Arc in Circle 2**:
For Circle 2 (radius \( 2r \)):
\[ L_2 = \frac{\theta_2}{360} \times 2\pi (2r) = \frac{\theta_2}{360} \times 4\pi r \]

5. **Setting the Lengths Equal**:
According to the problem, the lengths of the arcs are equal:
\[ L_1 = L_2 \]
Therefore, we can write:
\[ \frac{\theta_1}{360} \times 2\pi r = \frac{\theta_2}{360} \times 4\pi r \]

6. **Canceling Common Terms**:
We can cancel \( \frac{2\pi r}{360} \) from both sides (as long as \( r \neq 0 \)):
\[ \theta_1 = 2\theta_2 \]

7. **Conclusion**:
The equation \( \theta_1 = 2\theta_2 \) indicates that the angle subtended by the arc in Circle 1 is double that of the angle subtended by the arc in Circle 2. Therefore, the statement is **true**.