The area of two sectors of two sectors of two different circles with equal corresponding arc lengths are equal. Is this statement trus ? Why ?

Explanation

To determine whether the statement "The area of two sectors of two different circles with equal corresponding arc lengths are equal" is true or false, we can analyze the relationship between the arc length, the radius of the circles, and the area of the sectors.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let Circle 1 have radius \( R_1 \) and the angle subtended by the arc be \( \theta_1 \).
- Let Circle 2 have radius \( R_2 \) and the angle subtended by the arc be \( \theta_2 \).
- We are given that the arc lengths \( L_1 \) and \( L_2 \) are equal, i.e., \( L_1 = L_2 \).

2. **Express Arc Lengths:**
- The arc length of a sector is given by the formula:
\[ L = \frac{\theta}{360} \times 2\pi R \]
- For Circle 1:
\[ L_1 = \frac{\theta_1}{360} \times 2\pi R_1 \]
- For Circle 2:
\[ L_2 = \frac{\theta_2}{360} \times 2\pi R_2 \]
- Since \( L_1 = L_2 \), we can set the equations equal to each other:
\[ \frac{\theta_1}{360} \times 2\pi R_1 = \frac{\theta_2}{360} \times 2\pi R_2 \]

3. **Simplify the Equation:**
- Cancel \( 360 \) and \( 2\pi \) from both sides:
\[ \theta_1 R_1 = \theta_2 R_2 \]
- Rearranging gives us:
\[ \frac{\theta_1}{\theta_2} = \frac{R_2}{R_1} \quad \text{(Equation 1)} \]

4. **Calculate the Area of the Sectors:**
- The area of a sector is given by:
\[ \text{Area} = \frac{\theta}{360} \times \pi R^2 \]
- For Circle 1:
\[ \text{Area}_1 = \frac{\theta_1}{360} \times \pi R_1^2 \]
- For Circle 2:
\[ \text{Area}_2 = \frac{\theta_2}{360} \times \pi R_2^2 \]
- Since we want to check if these areas are equal:
\[ \frac{\theta_1}{360} \times \pi R_1^2 = \frac{\theta_2}{360} \times \pi R_2^2 \]

5. **Simplify the Area Equation:**
- Cancel \( 360 \) and \( \pi \) from both sides:
\[ \theta_1 R_1^2 = \theta_2 R_2^2 \]
- Rearranging gives us:
\[ \frac{\theta_1}{\theta_2} = \frac{R_2^2}{R_1^2} \quad \text{(Equation 2)} \]

6. **Compare Equations 1 and 2:**
- From Equation 1, we have:
\[ \frac{\theta_1}{\theta_2} = \frac{R_2}{R_1} \]
- From Equation 2, we have:
\[ \frac{\theta_1}{\theta_2} = \frac{R_2^2}{R_1^2} \]
- Since \( \frac{R_2}{R_1} \) is not equal to \( \frac{R_2^2}{R_1^2} \) unless \( R_2 = R_1 \) or one of the radii is 1, we conclude that the two equations are not equal.

7. **Conclusion:**
- Therefore, the statement "The area of two sectors of two different circles with equal corresponding arc lengths are equal" is **false**.