To solve this problem, we need to calculate the perimeter of both shapes, set them equal to each other, and solve for $x$.

1. Perimeter of the triangle

The three sides of the triangle are given as:

• $3x + 4$
• $2x + 5$
• $5x + 1$

The perimeter of the triangle is the sum of all three sides: $P_{\text{triangle}} = (3x + 4) + (2x + 5) + (5x + 1)$ Simplify: $P_{\text{triangle}} = 3x + 2x + 5x + 4 + 5 + 1$ $P_{\text{triangle}} = 10x + 10$

2. Perimeter of the rectangle

The lengths of the rectangle are:

• Width: $2x$
• Length: $x + 13$

The perimeter of a rectangle is calculated as: $P_{\text{rectangle}} = 2 \times \text{Length} + 2 \times \text{Width}$ Substitute the given values: $P_{\text{rectangle}} = 2(x + 13) + 2(2x)$ Simplify: $P_{\text{rectangle}} = 2x + 26 + 4x$ $P_{\text{rectangle}} = 6x + 26$

3. Equating the perimeters

Since the problem states the figures have the same perimeter, set the two perimeters equal to each other: $10x + 10 = 6x + 26$

Solve for $x$: $10x - 6x = 26 - 10$ $4x = 16$ $x = 4$

4. Verification

• Substituting $x = 4$ into the triangle's perimeter: $P_{\text{triangle}} = 10(4) + 10 = 40 + 10 = 50$
• Substituting $x = 4$ into the rectangle's perimeter: $P_{\text{rectangle}} = 6(4) + 26 = 24 + 26 = 50$

Both perimeters are $50$, confirming $x = 4$ is correct.

Final Answer:

$\boxed{x = 4}$

Do you want a detailed breakdown of each calculation step or further clarification?

Related Questions:

1. How do you calculate the perimeter of any triangle?
2. What is the formula for the perimeter of a rectangle?
3. How can algebraic equations help solve geometric problems?
4. How do you check your solution to an equation?
5. What happens when you simplify algebraic expressions?

Tip: Always verify your solution by substituting back into the original problem to ensure consistency.