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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.

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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!

The rectangle and the square below both have the same perimeter. What is the value of x in cm?

Learn how to find the value of x when a rectangle and a square have the same perimeter. This geometry problem involves the concepts of perimeter, algebra, and solving linear equations. Suitable for students in Grades 6-8.

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