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We are given two shapes: a rectangle and a square, which both have the same perimeter. We need to find the value of \( x \) in centimeters.

### Perimeter Formulas:
1. **Rectangle**: The length is \( x + 8 \) cm, and the width is \( 10 \) cm. The perimeter of a rectangle is given by:
   \[
   P_{\text{rectangle}} = 2 \times (\text{length} + \text{width}) = 2 \times ((x + 8) + 10)
   \]
   Simplifying:
   \[
   P_{\text{rectangle}} = 2 \times (x + 18) = 2x + 36
   \]

2. **Square**: Each side of the square is \( x + 6 \) cm. The perimeter of a square is given by:
   \[
   P_{\text{square}} = 4 \times \text{side length} = 4 \times (x + 6)
   \]
   Simplifying:
   \[
   P_{\text{square}} = 4x + 24
   \]

### Since the perimeters are equal:
\[
2x + 36 = 4x + 24
\]

### Solving for \( x \):
\[
2x + 36 = 4x + 24
\]
Subtract \( 2x \) from both sides:
\[
36 = 2x + 24
\]
Subtract 24 from both sides:
\[
12 = 2x
\]
Divide by 2:
\[
x = 6
\]

Thus, the value of \( x \) is **6 cm**.

Would you like more details or clarifications on any step?

Here are 5 follow-up questions to deepen your understanding:
1. What would the perimeter of the square be if \( x = 6 \) cm?
2. How would the solution change if the rectangle's width were doubled?
3. Can you verify the perimeters for both shapes when \( x = 6 \) cm?
4. What would the new perimeter be if the square's side length were increased by 2 cm?
5. How would you set up the equation if the shapes were a circle and a rectangle?

**Tip**: Always double-check whether units and dimensions are consistent when solving geometry problems.

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

Solve a geometry problem where the perimeters of a rectangle and a square are equal. Learn how to use algebra to calculate the value of x in this step-by-step solution.

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