Question

1. the side lengths in yards of a triangle and a square are shown in the diagram. the perimeter of the triangle is equal to the perimeter of the square. what is the value of x?

Explanation:

Step1: Find perimeter of square

The side - length of the square is $2.5x$ yards. The perimeter formula for a square is $P = 4s$ (where $s$ is the side - length). So the perimeter of the square $P_{square}=4\times2.5x=10x$ yards.

Step2: Find perimeter of triangle

The side - lengths of the triangle are $2x$, $2(x + 7)$ and $(4x-2)$ yards. First, expand $2(x + 7)=2x+14$. Then the perimeter of the triangle $P_{triangle}=2x+(2x + 14)+(4x-2)$. Combine like terms: $P_{triangle}=(2x+2x + 4x)+(14 - 2)=8x + 12$ yards.

Step3: Set perimeters equal and solve for x

Since $P_{square}=P_{triangle}$, we have the equation $10x=8x + 12$. Subtract $8x$ from both sides: $10x-8x=8x + 12-8x$, which simplifies to $2x=12$. Divide both sides by 2: $\frac{2x}{2}=\frac{12}{2}$, so $x = 6$.

Answer:

$x = 6$