The sum of the length and breadth of a rectangle is 6 cm. A square is construted such that one of its sides is equal to a diagonal of the rectangle. If the ratio of areas of the square and rectangle is 5 :2, the area of the square in `cm^(2)` is

Options

A. 20
B. 10
C. `4sqrt(5)`
D. `10sqrt(2)`

Answer

To solve the problem step by step, we will follow the information given in the question and apply the necessary mathematical concepts.

### Step 1: Define Variables
Let the length of the rectangle be \( l \) cm and the breadth be \( b \) cm. According to the problem, we have:
\[ l + b = 6 \quad \text{(1)} \]

### Step 2: Express Area of Rectangle
The area \( A_r \) of the rectangle can be expressed as:
\[ A_r = l \times b \quad \text{(2)} \]

### Step 3: Diagonal of the Rectangle
Using the Pythagorean theorem, the diagonal \( d \) of the rectangle can be expressed as:
\[ d = \sqrt{l^2 + b^2} \quad \text{(3)} \]

### Step 4: Relate Diagonal to Square
The side of the square \( s \) is equal to the diagonal of the rectangle, so:
\[ s = d = \sqrt{l^2 + b^2} \quad \text{(4)} \]

### Step 5: Area of the Square
The area \( A_s \) of the square can be expressed as:
\[ A_s = s^2 = d^2 = l^2 + b^2 \quad \text{(5)} \]

### Step 6: Use the Ratio of Areas
According to the problem, the ratio of the areas of the square and rectangle is given as:
\[ \frac{A_s}{A_r} = \frac{5}{2} \]
Substituting the expressions for \( A_s \) and \( A_r \):
\[ \frac{l^2 + b^2}{l \times b} = \frac{5}{2} \quad \text{(6)} \]

### Step 7: Substitute \( l + b \)
From equation (1), we can express \( b \) in terms of \( l \):
\[ b = 6 - l \]
Now substitute \( b \) in equation (6):
\[ \frac{l^2 + (6 - l)^2}{l(6 - l)} = \frac{5}{2} \]

### Step 8: Expand and Simplify
Expanding \( (6 - l)^2 \):
\[ (6 - l)^2 = 36 - 12l + l^2 \]
So,
\[ l^2 + 36 - 12l + l^2 = 2l^2 - 12l + 36 \]
Now substituting back into equation (6):
\[ \frac{2l^2 - 12l + 36}{l(6 - l)} = \frac{5}{2} \]

### Step 9: Cross Multiply
Cross multiplying gives:
\[ 2(2l^2 - 12l + 36) = 5l(6 - l) \]
Expanding both sides:
\[ 4l^2 - 24l + 72 = 30l - 5l^2 \]
Rearranging gives:
\[ 9l^2 - 54l + 72 = 0 \]

### Step 10: Solve the Quadratic Equation
Dividing the entire equation by 9:
\[ l^2 - 6l + 8 = 0 \]
Factoring gives:
\[ (l - 2)(l - 4) = 0 \]
Thus, \( l = 2 \) or \( l = 4 \).

### Step 11: Find Corresponding \( b \)
If \( l = 2 \), then \( b = 6 - 2 = 4 \).
If \( l = 4 \), then \( b = 6 - 4 = 2 \).

### Step 12: Calculate Area of Rectangle
Using \( l = 2 \) and \( b = 4 \):
\[ A_r = l \times b = 2 \times 4 = 8 \text{ cm}^2 \]

### Step 13: Calculate Area of Square
Using \( l^2 + b^2 \):
\[ A_s = l^2 + b^2 = 2^2 + 4^2 = 4 + 16 = 20 \text{ cm}^2 \]

### Final Answer
The area of the square is:
\[ \boxed{20 \text{ cm}^2} \]