Let's solve the problem step by step.

We are given:

• A square with an area of $121 \, \text{in}^2$.
• A rectangle with dimensions $3x$ and $x - 2$.
• The perimeters of the square and rectangle are the same.

Step 1: Find the side length of the square.

The area of the square is given by: $\text{Area of square} = \text{side}^2 = 121 \, \text{in}^2$ So, to find the side length: $\text{side} = \sqrt{121} = 11 \, \text{in}$

Step 2: Find the perimeter of the square.

The perimeter of a square is $4$ times the side length: $\text{Perimeter of square} = 4 \times 11 = 44 \, \text{in}$

Step 3: Write an equation for the perimeter of the rectangle.

The perimeter $P$ of a rectangle is given by: $P = 2(\text{length} + \text{width})$ For this rectangle, the length is $3x$ and the width is $x - 2$. So, the perimeter is: $P = 2\left(3x + (x - 2)\right) = 2(3x + x - 2) = 2(4x - 2) = 8x - 4$

Step 4: Set up the equation.

Since the perimeters of the square and the rectangle are the same: $8x - 4 = 44$

Step 5: Solve for $x$.

$8x - 4 = 44$ Add $4$ to both sides: $8x = 48$ Divide both sides by $8$: $x = 6$

Final Answer:

The value of $x$ is $6$.

Would you like further details on any part of the solution?

Here are some related questions you might consider:

1. What is the perimeter of a square with an area of $225 \, \text{in}^2$?
2. How does the formula for perimeter differ for squares and rectangles?
3. If the width of the rectangle were $x + 3$, how would the equation change?
4. Can you find the area of the rectangle for $x = 6$?
5. How would the equation change if the perimeters were not equal?

Tip: When solving for unknowns in geometric problems, always start by writing the formula for each shape’s perimeter or area.