Area of rectangle whose breadth is 25% less than its length is 432 cm2, A square is drawn whose side is equal to diagonal of rectangle then find ratio of perimeter of square to that of rectangle?

Options

A. A)`9:7`
B. B)`48:35`
C. C)None of these
D. D)`12:7`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the dimensions of the rectangle
Let the length of the rectangle be \( L \) and the breadth be \( B \). According to the problem, the breadth is 25% less than the length. Therefore, we can express the breadth as:
\[ B = L - 0.25L = 0.75L \]

### Step 2: Set up the area equation
The area of the rectangle is given as 432 cm². The area of a rectangle is calculated as:
\[ \text{Area} = L \times B \]
Substituting the expression for breadth:
\[ 432 = L \times (0.75L) \]
This simplifies to:
\[ 432 = 0.75L^2 \]

### Step 3: Solve for \( L \)
To find \( L \), we rearrange the equation:
\[ L^2 = \frac{432}{0.75} \]
Calculating the right side:
\[ L^2 = 576 \]
Taking the square root of both sides gives:
\[ L = \sqrt{576} = 24 \text{ cm} \]

### Step 4: Calculate the breadth \( B \)
Now, we can find the breadth using the expression for \( B \):
\[ B = 0.75L = 0.75 \times 24 = 18 \text{ cm} \]

### Step 5: Calculate the diagonal of the rectangle
The diagonal \( D \) of the rectangle can be calculated using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} = \sqrt{24^2 + 18^2} \]
Calculating the squares:
\[ D = \sqrt{576 + 324} = \sqrt{900} = 30 \text{ cm} \]

### Step 6: Determine the side of the square
The side of the square \( a \) is equal to the diagonal of the rectangle:
\[ a = D = 30 \text{ cm} \]

### Step 7: Calculate the perimeter of the square
The perimeter \( P_s \) of the square is given by:
\[ P_s = 4 \times a = 4 \times 30 = 120 \text{ cm} \]

### Step 8: Calculate the perimeter of the rectangle
The perimeter \( P_r \) of the rectangle is given by:
\[ P_r = 2 \times (L + B) = 2 \times (24 + 18) = 2 \times 42 = 84 \text{ cm} \]

### Step 9: Find the ratio of the perimeter of the square to the perimeter of the rectangle
Now, we can find the ratio:
\[ \text{Ratio} = \frac{P_s}{P_r} = \frac{120}{84} \]
Simplifying this ratio:
\[ \text{Ratio} = \frac{120 \div 12}{84 \div 12} = \frac{10}{7} \]

### Final Answer
The ratio of the perimeter of the square to that of the rectangle is:
\[ \boxed{\frac{10}{7}} \]