To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to calculate the lengths of the three sides of the triangle using the distance formula.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Let's calculate each side:
- Length between \((1, 5)\) and \((4, 1)\):
\[ d_1 = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Length between \((4, 1)\) and \((-4, -5)\):
\[ d_2 = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
- Length between \((-4, -5)\) and \((1, 5)\):
\[ d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.2 \]
Now we can find the perimeter of the triangle by adding the lengths of the three sides:
\[ \text{Perimeter} = d_1 + d_2 + d_3 = 5 + 10 + 11.2 \approx 26.2 \]
Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is:
\(\boxed{26.2}\) units.