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 whether the statement is true or false, we need to analyze the relationship between the lengths of the arcs and the angles subtended by those arcs in two circles with different radii.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have two circles: Circle 1 with radius \( r \) and Circle 2 with radius \( 2r \).
- The length of the arc in Circle 1 is equal to the length of the arc in Circle 2.

2. **Use the Formula for Arc Length:**
- The formula for the length of an arc \( L \) is given by:
\[ L = r \theta \]
where \( \theta \) is the angle in radians subtended by the arc at the center of the circle.

3. **Set Up the Equations for Both Circles:**
- For Circle 1 (radius \( r \)):
\[ L_1 = r \theta_1 \]
- For Circle 2 (radius \( 2r \)):
\[ L_2 = 2r \theta_2 \]

4. **Equate the Lengths of the Arcs:**
- Since the lengths of the arcs are equal:
\[ r \theta_1 = 2r \theta_2 \]

5. **Simplify the Equation:**
- Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
\[ \theta_1 = 2 \theta_2 \]

6. **Interpret the Result:**
- This result shows that the angle \( \theta_1 \) subtended by the arc in Circle 1 is indeed double that of the angle \( \theta_2 \) in Circle 2.

7. **Conclusion:**
- The statement "the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle" is **true**.