The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal ? Why ?

Explanation

To determine whether the arc lengths of two sectors of different circles are equal when their areas are equal, we can analyze the relationship between the area of a sector and its arc length.

### Step-by-Step Solution:

1. **Understanding the Area of a Sector**:
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.

2. **Setting Up the Problem**:
Let’s consider two circles:
- Circle A with radius \( r_1 \) and angle \( \theta_1 \)
- Circle B with radius \( r_2 \) and angle \( \theta_2 \)

According to the problem, the areas of the two sectors are equal:
\[ \frac{\theta_1}{360} \times \pi r_1^2 = \frac{\theta_2}{360} \times \pi r_2^2 \]

3. **Simplifying the Area Equation**:
We can simplify the equation by canceling out common terms:
\[ \theta_1 r_1^2 = \theta_2 r_2^2 \]
Rearranging gives us:
\[ \frac{\theta_1}{\theta_2} = \frac{r_2^2}{r_1^2} \quad \text{(Equation 1)} \]

4. **Finding the Arc Length**:
The arc length \( L \) of a sector can be calculated using the formula:
\[ L = \frac{\theta}{360} \times 2\pi r \]
Therefore, for Circle A:
\[ L_1 = \frac{\theta_1}{360} \times 2\pi r_1 \]
And for Circle B:
\[ L_2 = \frac{\theta_2}{360} \times 2\pi r_2 \]

5. **Setting Up the Arc Length Equation**:
We can express the ratio of the arc lengths:
\[ \frac{L_1}{L_2} = \frac{\frac{\theta_1}{360} \times 2\pi r_1}{\frac{\theta_2}{360} \times 2\pi r_2} \]
This simplifies to:
\[ \frac{L_1}{L_2} = \frac{\theta_1 r_1}{\theta_2 r_2} \quad \text{(Equation 2)} \]

6. **Comparing Equations**:
From Equation 1, we have:
\[ \frac{\theta_1}{\theta_2} = \frac{r_2^2}{r_1^2} \]
Substituting this into Equation 2:
\[ \frac{L_1}{L_2} = \frac{r_1^2}{r_2^2} \times \frac{r_1}{r_2} = \frac{r_1^3}{r_2^3} \]
This shows that \( L_1 \) and \( L_2 \) depend on the radii of the circles and are not necessarily equal.

7. **Conclusion**:
Since the ratio of the arc lengths depends on the radii of the circles, it is not necessary for the arc lengths to be equal even if the areas of the sectors are equal. Therefore, the statement is incorrect.