The area of a square and a rectangle are equal. The length of the rectangle is greater than the side of square by 9 cm and its breadth is less than the side of square by 6 cm. What will be the perimeterof the rectangle?

Options

A. 84 cm
B. 74 cm
C. 78 cm
D. 76 cm

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the variables
Let the side of the square be \( x \) cm.

### Step 2: Write the area of the square
The area of the square is given by:
\[ \text{Area of square} = x^2 \]

### Step 3: Define the dimensions of the rectangle
According to the problem:
- The length of the rectangle is \( x + 9 \) cm (since it is greater than the side of the square by 9 cm).
- The breadth of the rectangle is \( x - 6 \) cm (since it is less than the side of the square by 6 cm).

### Step 4: Write the area of the rectangle
The area of the rectangle is given by:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = (x + 9)(x - 6) \]

### Step 5: Set the areas equal
Since the areas of the square and rectangle are equal, we can set up the equation:
\[ x^2 = (x + 9)(x - 6) \]

### Step 6: Expand the right-hand side
Expanding the right-hand side:
\[ x^2 = x^2 - 6x + 9x - 54 \]
\[ x^2 = x^2 + 3x - 54 \]

### Step 7: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 0 = 3x - 54 \]

### Step 8: Solve for \( x \)
Rearranging gives:
\[ 3x = 54 \]
\[ x = 18 \]

### Step 9: Find the dimensions of the rectangle
Now that we have \( x = 18 \):
- Length of the rectangle:
\[ \text{Length} = x + 9 = 18 + 9 = 27 \text{ cm} \]
- Breadth of the rectangle:
\[ \text{Breadth} = x - 6 = 18 - 6 = 12 \text{ cm} \]

### Step 10: Calculate the perimeter of the rectangle
The perimeter \( P \) of the rectangle is given by:
\[ P = 2 \times (\text{Length} + \text{Breadth}) = 2 \times (27 + 12) = 2 \times 39 = 78 \text{ cm} \]

### Final Answer
The perimeter of the rectangle is \( 78 \) cm.

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