Let’s solve the problem step-by-step. The goal is to find the value of $x$ in cm, given that the rectangle and the square have the same perimeter.

Step 1: Write the formula for the perimeter of each shape

1. Rectangle
   The perimeter of a rectangle is given by: $P = 2 \times (\text{Length} + \text{Width})$ Here, the length is $x + 10$ cm, and the width is $4$ cm. Therefore: $P_{\text{rectangle}} = 2 \times \left((x + 10) + 4\right) = 2 \times (x + 14)$

2. Square
   The perimeter of a square is given by: $P = 4 \times \text{Side length}$ Here, the side length is $x + 4$ cm. Therefore: $P_{\text{square}} = 4 \times (x + 4)$

Step 2: Set the perimeters equal to each other

We are told that the perimeters are the same: $2 \times (x + 14) = 4 \times (x + 4)$

Step 3: Solve the equation

1. Expand both sides of the equation: $2(x + 14) = 4(x + 4)$ $2x + 28 = 4x + 16$

2. Simplify and collect like terms: $28 - 16 = 4x - 2x$ $12 = 2x$

3. Solve for $x$: $x = \frac{12}{2} = 6$

Step 4: Verify the solution

1. For the rectangle: $P_{\text{rectangle}} = 2 \times (x + 14) = 2 \times (6 + 14) = 2 \times 20 = 40 \, \text{cm}$

2. For the square: $P_{\text{square}} = 4 \times (x + 4) = 4 \times (6 + 4) = 4 \times 10 = 40 \, \text{cm}$

The perimeters match, so $x = 6 \, \text{cm}$ is correct.

Final Answer:

$\boxed{x = 6 \, \text{cm}}$

Would you like a detailed breakdown of any step or further clarifications?

5 Related Questions:

1. What is the area of the rectangle when $x = 6$?
2. What is the area of the square when $x = 6$?
3. How does the perimeter formula differ for other polygons, like triangles or circles?
4. How do you solve equations with multiple variables like this one?
5. What happens if the perimeter of the rectangle is greater than the square?

1 Math Tip:

When solving word problems, always write the formula first, substitute values, and carefully solve step by step. It helps minimize errors and ensures clarity.