A steel frame is welded in the shape of a triangle ABC…

Step 1. The squared sides. $AB^2 = 4 + 1 + 4 = 9$. $BC^2 = (k - 1)^2 + 4 + 4$. $AC^2 = (k + 1)^2 + 1 + 0$.

Step 2. The right angle at $B$. $9 + (k - 1)^2 + 8 = (k + 1)^2 + 1$. The left side is $k^2 - 2k + 18$ and the right side is $k^2 + 2k + 2$. So $4k = 16$ and $k = 4$.

Step 3. The longest side. It is $AC$, opposite the right angle. $AC^2 = 25 + 1 = 26$, so $AC = \sqrt{26} \approx 5.10\ \text{m}$.