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 step by step, we will follow the information provided in the question and the video transcript.

### Step 1: Define the sides of the square and rectangle
Let the side of the square be \( a \). Therefore, the perimeter of the square is given by:
\[ \text{Perimeter of square} = 4a \]

For the rectangle, let the lengths of the adjacent sides be \( x \) and \( 2x \) (since the ratio of the sides is 1:2). The perimeter of the rectangle is given by:
\[ \text{Perimeter of rectangle} = 2(x + 2x) = 2(3x) = 6x \]

### Step 2: Set the perimeters equal
According to the problem, the perimeters of the rectangle and the square are equal. Therefore, we can set the two perimeter equations equal to each other:
\[ 4a = 6x \]

### Step 3: Solve for the ratio of \( a \) and \( x \)
From the equation \( 4a = 6x \), we can simplify to find the ratio of \( a \) to \( x \):
\[ \frac{a}{x} = \frac{6}{4} = \frac{3}{2} \]

### Step 4: Express \( a \) in terms of \( x \)
From the ratio, we can express \( a \) in terms of \( x \):
\[ a = \frac{3}{2}x \]

### Step 5: Calculate the areas of the rectangle and the square
Now, we can calculate the areas:
- The area of the square is:
\[ \text{Area of square} = a^2 = \left(\frac{3}{2}x\right)^2 = \frac{9}{4}x^2 \]

- The area of the rectangle is:
\[ \text{Area of rectangle} = x \times 2x = 2x^2 \]

### Step 6: Find the ratio of the areas
Now we need to find the ratio of the area of the rectangle to the area of the square:
\[ \text{Ratio of area of rectangle to area of square} = \frac{\text{Area of rectangle}}{\text{Area of square}} = \frac{2x^2}{\frac{9}{4}x^2} \]

### Step 7: Simplify the ratio
Simplifying the ratio:
\[ \frac{2x^2}{\frac{9}{4}x^2} = \frac{2}{\frac{9}{4}} = 2 \times \frac{4}{9} = \frac{8}{9} \]

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