The perimeters of a square and a rectangle are equal. If their area be 'A' `m^(2)` and 'B' `m^(2)` respectively, then correct statement is

Options

A. `A lt B`
B. `A le B`
C. `A gt B`
D. `A ge B`

Answer

To solve the problem, we need to establish the relationship between the areas of a square and a rectangle when their perimeters are equal.

### Step-by-Step Solution:

1. **Understanding the Perimeter of a Square**:
The formula for the perimeter of a square is given by:
\[ P_{\text{square}} = 4 \times \text{side} \]
Let the side of the square be \( s \). Therefore, we can express the perimeter as:
\[ P_{\text{square}} = 4s \]

2. **Understanding the Perimeter of a Rectangle**:
The formula for the perimeter of a rectangle is:
\[ P_{\text{rectangle}} = 2 \times (L + B) \]
where \( L \) is the length and \( B \) is the breadth of the rectangle.

3. **Setting the Perimeters Equal**:
Since the perimeters of the square and rectangle are equal, we can set them equal to each other:
\[ 4s = 2(L + B) \]
Simplifying this gives:
\[ 2s = L + B \quad \text{(1)} \]

4. **Calculating the Area of the Square**:
The area \( A \) of the square is given by:
\[ A = s^2 \]

5. **Calculating the Area of the Rectangle**:
The area \( B \) of the rectangle is given by:
\[ B = L \times B \]

6. **Expressing Length in Terms of Breadth**:
From equation (1), we can express \( L \) in terms of \( B \):
\[ L = 2s - B \]
Substituting this into the area formula for the rectangle gives:
\[ B = (2s - B) \times B \]
Rearranging this leads to:
\[ B = 2sB - B^2 \]
Rearranging further gives us a quadratic equation:
\[ B^2 - 2sB + B = 0 \]

7. **Comparing Areas**:
To compare the areas \( A \) and \( B \), we need to analyze the relationship. From the previous calculations, we can see that for any given perimeter, the area of the square will always be greater than the area of the rectangle, provided the rectangle is not a square.

### Conclusion:
Thus, the correct statement is that the area of the square \( A \) is greater than the area of the rectangle \( B \):
\[ A > B \]