The area of a rectangle is equal to the area of a square. If the area of the rectangle is `96 cm_(2)`, then what is the perimeter (in cm.) of the square ?

Options

A. 116
B. 56
C. 112
D. 42

Answer

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

### Step 1: Understand the relationship between the areas
Given that the area of the rectangle is equal to the area of the square, we can express this mathematically. If the area of the rectangle is \(96 \, \text{cm}^2\), then the area of the square is also \(96 \, \text{cm}^2\).

### Step 2: Write the formula for the area of a square
The area \(A\) of a square is given by the formula:
\[ A = s^2 \]
where \(s\) is the length of one side of the square.

### Step 3: Set up the equation
Since the area of the square is \(96 \, \text{cm}^2\), we can set up the equation:
\[ s^2 = 96 \]

### Step 4: Solve for \(s\)
To find the length of one side of the square, we take the square root of both sides:
\[ s = \sqrt{96} \]
Calculating the square root:
\[ s = \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \approx 9.798 \, \text{cm} \]

### Step 5: Calculate the perimeter of the square
The perimeter \(P\) of a square is given by the formula:
\[ P = 4s \]
Substituting the value of \(s\):
\[ P = 4 \times 4\sqrt{6} = 16\sqrt{6} \approx 16 \times 2.449 \approx 39.184 \, \text{cm} \]

### Step 6: Round the perimeter to the nearest whole number
The approximate value of the perimeter is \(39.184 \, \text{cm}\), which can be rounded to \(39 \, \text{cm}\). However, since this value does not match the options given, we can check if we made any miscalculations or if the options were incorrectly noted.

### Final Answer
The perimeter of the square is approximately \(39 \, \text{cm}\).

---