If the perimeters of a rectangle and a square are equal and the ratio of two adjacent sides of the rectangle is 1:2 then the ratio of area of the rectangle and that of the square is

Options

A. `1:1`
B. `1:2`
C. `2:3`
D. `8:9`

Answer

To solve the problem, we need to find the ratio of the area of a rectangle to that of a square, given that their perimeters are equal and the ratio of the rectangle's sides is 1:2.

### Step-by-Step Solution:

1. **Define the sides of the rectangle**:
Let the length of the rectangle be \( l = x \) and the breadth be \( b = 2x \) (since the ratio of the sides is 1:2).

2. **Calculate the perimeter of the rectangle**:
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + b) = 2(x + 2x) = 2(3x) = 6x \]

3. **Set the perimeter of the square equal to that of the rectangle**:
Let the side of the square be \( s \). The perimeter of the square is given by:
\[ P = 4s \]
Since the perimeters are equal, we have:
\[ 4s = 6x \]

4. **Solve for the side of the square**:
Rearranging the equation gives:
\[ s = \frac{6x}{4} = \frac{3x}{2} \]

5. **Calculate the area of the rectangle**:
The area \( A_r \) of the rectangle is given by:
\[ A_r = l \times b = x \times 2x = 2x^2 \]

6. **Calculate the area of the square**:
The area \( A_s \) of the square is given by:
\[ A_s = s^2 = \left(\frac{3x}{2}\right)^2 = \frac{9x^2}{4} \]

7. **Find the ratio of the areas**:
The ratio of the area of the rectangle to the area of the square is:
\[ \text{Ratio} = \frac{A_r}{A_s} = \frac{2x^2}{\frac{9x^2}{4}} = \frac{2x^2 \times 4}{9x^2} = \frac{8}{9} \]

### Final Answer:
The ratio of the area of the rectangle to that of the square is \( \frac{8}{9} \).