The areas of a square and a rectangle are equal. If the length of the rectangle is 8 meter more than the side of the square and its breadth is 6 meter less than the side of the square, what is the perimeter of the rectangle ? (in meter)

Options

A. 100
B. 90
C. 110
D. 80

Answer

To solve the problem step by step, let's denote the side of the square as \( a \) meters.

### Step 1: Define the dimensions of the rectangle
- The length of the rectangle is \( a + 8 \) meters (8 meters more than the side of the square).
- The breadth of the rectangle is \( a - 6 \) meters (6 meters less than the side of the square).

### Step 2: Set up the equation for the areas
Since the areas of the square and the rectangle are equal, we can write the equation:
\[ \text{Area of the square} = \text{Area of the rectangle} \]
This translates to:
\[ a^2 = (a + 8)(a - 6) \]

### Step 3: Expand the right side of the equation
Now, let's expand the right side:
\[ a^2 = a^2 - 6a + 8a - 48 \]
This simplifies to:
\[ a^2 = a^2 + 2a - 48 \]

### Step 4: Rearrange the equation
Now, we can subtract \( a^2 \) from both sides:
\[ 0 = 2a - 48 \]

### Step 5: Solve for \( a \)
Now, we can solve for \( a \):
\[ 2a = 48 \implies a = 24 \text{ meters} \]

### Step 6: Calculate the dimensions of the rectangle
Now that we have \( a \), we can find the dimensions of the rectangle:
- Length of the rectangle:
\[ a + 8 = 24 + 8 = 32 \text{ meters} \]
- Breadth of the rectangle:
\[ a - 6 = 24 - 6 = 18 \text{ meters} \]

### Step 7: Calculate the perimeter of the rectangle
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2 \times (\text{Length} + \text{Breadth}) \]
Substituting the values we found:
\[ P = 2 \times (32 + 18) = 2 \times 50 = 100 \text{ meters} \]

### Final Answer
The perimeter of the rectangle is \( 100 \) meters.

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