To find the perimeter of the triangle formed by the points \((1,5)\), \((4,1)\), and \((-4,-5)\), we first need to calculate the lengths of each side of the triangle using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the distance between points \((1,5)\) and \((4,1)\)
Let \((x_1, y_1) = (1, 5)\) and \((x_2, y_2) = (4, 1)\).
\[ d_{12} = \sqrt{(4 - 1)^2 + (1 - 5)^2} \] \[ = \sqrt{(3)^2 + (-4)^2} \] \[ = \sqrt{9 + 16} \] \[ = \sqrt{25} \] \[ = 5 \]
Step 2: Calculate the distance between points \((4,1)\) and \((-4,-5)\)
Let \((x_1, y_1) = (4, 1)\) and \((x_2, y_2) = (-4, -5)\).
\[ d_{23} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} \] \[ = \sqrt{(-8)^2 + (-6)^2} \] \[ = \sqrt{64 + 36} \] \[ = \sqrt{100} \] \[ = 10 \]
Step 3: Calculate the distance between points \((-4,-5)\) and \((1,5)\)
Let \((x_1, y_1) = (-4, -5)\) and \((x_2, y_2) = (1, 5)\).
\[ d_{31} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} \] \[ = \sqrt{(5)^2 + (10)^2} \] \[ = \sqrt{25 + 100} \] \[ = \sqrt{125} \] \[ = 5\sqrt{5} \]
To get an approximate value for \(5\sqrt{5}\):
\[ \sqrt{5} \approx 2.236 \quad \text{(using a calculator or square root table)} \] \[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
Step 4: Calculate the perimeter of the triangle
The perimeter \(P\) is the sum of the lengths of the sides:
\[ P = d_{12} + d_{23} + d_{31} = 5 + 10 + 5\sqrt{5} \]
Substituting the approximate value:
\[ P \approx 5 + 10 + 11.18 = 26.18 \]
Step 5: Round the perimeter to the nearest tenth
Rounding \(26.18\) to the nearest tenth gives:
\[ P \approx 26.2 \]
Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is:
\[ \boxed{26.2} \]