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

Answer

To determine whether the statement "If the length of an arc of a circle of radius a is equal to that of an arc of a circle of radius 2a, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle" is false, we can follow these steps:

### Step 1: Understand the formula for the length of an arc
The length of an arc \( L \) of a circle can be calculated using the formula:
\[ L = \frac{\theta}{360} \times 2\pi r \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.

### Step 2: Set up the equations for both circles
Let’s denote:
- \( r_1 = a \) (radius of the first circle)
- \( r_2 = 2a \) (radius of the second circle)
- \( \theta_1 \) = angle of the sector of the first circle
- \( \theta_2 \) = angle of the sector of the second circle

Using the formula for the length of an arc, we can write:
1. For the first circle:
\[ L_1 = \frac{\theta_1}{360} \times 2\pi a \]
2. For the second circle:
\[ L_2 = \frac{\theta_2}{360} \times 2\pi (2a) = \frac{\theta_2}{360} \times 4\pi a \]

### Step 3: Set the lengths of the arcs equal
According to the problem, the lengths of the arcs are equal:
\[ L_1 = L_2 \]
Substituting the expressions we found:
\[ \frac{\theta_1}{360} \times 2\pi a = \frac{\theta_2}{360} \times 4\pi a \]

### Step 4: Simplify the equation
We can cancel \( 2\pi a \) from both sides (assuming \( a \neq 0 \)):
\[ \frac{\theta_1}{360} = \frac{2\theta_2}{360} \]
This simplifies to:
\[ \theta_1 = 2\theta_2 \]

### Step 5: Analyze the result
The equation \( \theta_1 = 2\theta_2 \) indicates that the angle of the sector of the first circle (radius \( a \)) is indeed double the angle of the sector of the second circle (radius \( 2a \)).

### Conclusion
Thus, 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**.

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