We are given a rectangle and a square with the same perimeter, and we are asked to find the value of $x$ in cm.

Step 1: Define the dimensions

• For the rectangle:
  • Length: $x + 5$ cm
  • Width: 4 cm
• For the square:
  • Side length: $x + 2$ cm

Step 2: Calculate the perimeter of each shape

• Perimeter of the rectangle: The formula for the perimeter of a rectangle is: $P_{\text{rectangle}} = 2(\text{length} + \text{width})$ Substituting the values: $P_{\text{rectangle}} = 2((x + 5) + 4) = 2(x + 9)$

• Perimeter of the square: The formula for the perimeter of a square is: $P_{\text{square}} = 4 \times \text{side length}$ Substituting the value: $P_{\text{square}} = 4(x + 2)$

Step 3: Set up the equation

Since the perimeters are equal, we can set the two expressions equal to each other: $2(x + 9) = 4(x + 2)$

Step 4: Solve for $x$

Expand both sides: $2x + 18 = 4x + 8$ Now, move all terms involving $x$ to one side and constant terms to the other: $18 - 8 = 4x - 2x$ $10 = 2x$ Divide by 2: $x = 5$

Final Answer:

The value of $x$ is $\boxed{5}$ cm.

Would you like further details on the solution, or have any questions?

Here are 5 related questions to explore:

1. How would the answer change if the perimeter of the rectangle was larger than the square's?
2. What is the area of the rectangle and the square when $x = 5$?
3. How can the formula for perimeter be used for other polygons like triangles or pentagons?
4. What if the square had a side length of $x + 3$ cm instead? How would this affect the solution?
5. How does the perimeter of the rectangle compare to that of the square when the shapes are scaled up proportionally?

Tip: When solving geometry problems, always double-check if the formulas for area and perimeter are applied correctly based on the given shape!