Let's define the lengths of the triangle here.
1. x = shortest side
2. 2x = the length of the second side (twice the length of the shortest side)
3. x + 12 = the length of the third side
Another given information in the problem is that perimeter of the triangle = 180 feet.
Perimeter is the total length of all sides. Hence, we can say that x + 2x + x + 12 = 180 feet. From this, we can solve for x.
[tex]\begin{gathered} x+2x+x+12=180 \\ \text{Add similar terms.} \\ 4x+12=180 \\ Subtract\text{ 12 on both sides of the equation.} \\ 4x+12-12=180-12 \\ 4x=168 \\ \text{Divide both sides by 4.} \\ x=42 \end{gathered}[/tex]
Hence, the length of the shortest side is 42 ft.
The length of the second side is 2 x 42 = 84 ft.
The length of the third side is 42 + 12 = 54 ft.
