To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to first calculate the lengths of the sides of the triangle. We can do this using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will compute the lengths of the three sides:
- Distance between \((1, 5)\) and \((4, 1)\):
\[ d_1 = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Distance between \((4, 1)\) and \((-4, -5)\):
\[ d_2 = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
- Distance between \((-4, -5)\) and \((1, 5)\):
\[ d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \]
Now we can calculate \(5\sqrt{5}\) to a decimal value: \[ 5\sqrt{5} \approx 5 \times 2.236 = 11.180 \]
Now, we add up these distances to find the perimeter \(P\):
\[ P = d_1 + d_2 + d_3 = 5 + 10 + 5\sqrt{5} \]
So,
\[ P \approx 5 + 10 + 11.180 = 26.180 \]
Rounding to the nearest tenth, we get:
\[ P \approx 26.2 \]
Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is \(\boxed{26.2}\).