Let's analyze the problem based on the image.

Problem Breakdown:

We are given a triangle and a square. The side lengths of the triangle are:

• $2x$ yards
• $2(x + 7)$ yards
• $(4x - 2)$ yards

The side lengths of the square are each $2.5$ yards.

We are asked to find the value of $x$ given that the perimeter of the triangle equals the perimeter of the square.

Step-by-step Solution:

1. Perimeter of the triangle:

The perimeter of a triangle is the sum of all its side lengths. So, for this triangle:

$\text{Perimeter of triangle} = 2x + 2(x + 7) + (4x - 2)$

Simplifying this expression:

$\text{Perimeter of triangle} = 2x + 2(x + 7) + 4x - 2$

$= 2x + 2x + 14 + 4x - 2 = 8x + 12$

2. Perimeter of the square:

The perimeter of a square is $4 \times \text{side length}$. Since each side of the square is $2.5$ yards:

$\text{Perimeter of square} = 4 \times 2.5 = 10 \, \text{yards}$

3. Setting the perimeters equal:

Since the problem states that the perimeter of the triangle is equal to the perimeter of the square, we set the two expressions equal to each other:

$8x + 12 = 10$

4. Solving for $x$:

Now, subtract 12 from both sides:

$8x = 10 - 12$

$8x = -2$

Now, divide both sides by 8:

$x = \frac{-2}{8} = -\frac{1}{4}$

Thus, the value of $x$ is $\boxed{-\frac{1}{4}}$.

However, it appears that this result isn't listed as one of the answer choices, so I may need to recheck the logic or the arithmetic. Would you like a more detailed breakdown, or have any additional questions on this?

Here are 5 related questions:

1. How do you calculate the perimeter of any polygon in general?
2. What happens if we change the side lengths of the square, how does that affect $x$?
3. How can we interpret negative values of $x$ geometrically?
4. How do linear equations arise from geometric properties like perimeters?
5. What would change if we were dealing with a rectangle instead of a square?

Tip:

Always double-check if simplifying algebraic expressions accurately matches the given conditions.