Perimeter of square is two times of perimeter of a rectangle and length of rectangle is 6 cm more than that of breadth. If side of square is 75% more than length of rectangle then, find area of square?

Options

A. A)1764 `cm^(2)`
B. B)1798 `cm^(2)`
C. C)1864 `cm^(2)`
D. D)1824 `cm^(2)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the relationships given in the problem
We know that:
1. The perimeter of the square (P_square) is twice the perimeter of the rectangle (P_rectangle).
2. The length (L) of the rectangle is 6 cm more than its breadth (B).
3. The side (A) of the square is 75% more than the length of the rectangle.

### Step 2: Write the formulas for the perimeters
The perimeter of the square is given by:
\[ P_{\text{square}} = 4 \times A \]

The perimeter of the rectangle is given by:
\[ P_{\text{rectangle}} = 2 \times (L + B) \]

According to the problem, we have:
\[ P_{\text{square}} = 2 \times P_{\text{rectangle}} \]
Thus,
\[ 4A = 2 \times 2(L + B) \]
This simplifies to:
\[ 4A = 4(L + B) \]
Cancelling 4 from both sides gives us:
\[ A = L + B \]

### Step 3: Express breadth in terms of length
From the problem, we know:
\[ L = B + 6 \]
Substituting this into the equation \( A = L + B \):
\[ A = (B + 6) + B \]
This simplifies to:
\[ A = 2B + 6 \]

### Step 4: Relate side of the square to the length of the rectangle
The problem states that the side of the square is 75% more than the length of the rectangle:
\[ A = L + 0.75L = 1.75L \]

### Step 5: Set the two expressions for A equal to each other
We have two expressions for A:
1. \( A = 2B + 6 \)
2. \( A = 1.75L \)

Setting them equal gives:
\[ 2B + 6 = 1.75L \]

### Step 6: Substitute L in terms of B
Since \( L = B + 6 \), we can substitute this into the equation:
\[ 2B + 6 = 1.75(B + 6) \]
Expanding the right side:
\[ 2B + 6 = 1.75B + 10.5 \]

### Step 7: Solve for B
Rearranging gives:
\[ 2B - 1.75B = 10.5 - 6 \]
\[ 0.25B = 4.5 \]
Dividing both sides by 0.25:
\[ B = 18 \text{ cm} \]

### Step 8: Find the length of the rectangle
Using \( L = B + 6 \):
\[ L = 18 + 6 = 24 \text{ cm} \]

### Step 9: Find the side of the square
Using \( A = L + B \):
\[ A = 24 + 18 = 42 \text{ cm} \]

### Step 10: Calculate the area of the square
The area of the square is given by:
\[ \text{Area} = A^2 = 42^2 = 1764 \text{ cm}^2 \]

Thus, the area of the square is **1764 cm²**.

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