1. **State the problem:** We have a triangle with side lengths $2x$, $4x - 2$, and $2(x + 7)$ yards, and a square with side length $2.5x$ yards. The perimeter of the triangle equals the perimeter of the square. We need to find the value of $x$. 2. **Write the formula for perimeter:** - Perimeter of triangle $P_{triangle} = 2x + (4x - 2) + 2(x + 7)$ - Perimeter of square $P_{square} = 4 \times 2.5x = 10x$ 3. **Set the perimeters equal:** $$2x + (4x - 2) + 2(x + 7) = 10x$$ 4. **Simplify the left side:** $$2x + 4x - 2 + 2x + 14 = 10x$$ $$ (2x + 4x + 2x) + (-2 + 14) = 10x$$ $$8x + 12 = 10x$$ 5. **Isolate $x$:** $$8x + 12 = 10x$$ $$12 = 10x - 8x$$ $$12 = 2x$$ 6. **Solve for $x$:** $$x = \frac{12}{2} = 6$$ 7. **Check the answer:** - Triangle perimeter: $2(6) + (4(6) - 2) + 2(6 + 7) = 12 + 22 + 26 = 60$ - Square perimeter: $4 \times 2.5(6) = 4 \times 15 = 60$ Both perimeters are equal, so $x=6$ is correct. **Final answer:** $x = 6$