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.