To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we'll first find the lengths of each side of the triangle using the distance formula. The distance \(d\) 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 denote the points as follows:
- \(A(1, 5)\)
- \(B(4, 1)\)
- \(C(-4, -5)\)
We'll calculate the lengths of sides \(AB\), \(BC\), and \(CA\).
Step 1: Calculate the length of side \(AB\)
Using the distance formula: \[ d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} \] Calculating the differences: \[ = \sqrt{(3)^2 + (-4)^2} \] \[ = \sqrt{9 + 16} \] \[ = \sqrt{25} \] \[ = 5 \]
Step 2: Calculate the length of side \(BC\)
Now for side \(BC\): \[ d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} \] Calculating the differences: \[ = \sqrt{(-8)^2 + (-6)^2} \] \[ = \sqrt{64 + 36} \] \[ = \sqrt{100} \] \[ = 10 \]
Step 3: Calculate the length of side \(CA\)
Finally for side \(CA\): \[ d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] Calculating the differences: \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} \] \[ = \sqrt{(5)^2 + (10)^2} \] \[ = \sqrt{25 + 100} \] \[ = \sqrt{125} \] \[ = 5\sqrt{5} \approx 11.18 \quad (\text{Using } \sqrt{5} \approx 2.236) \]
Step 4: Calculate the perimeter
Now, we can add these lengths together to find the perimeter \(P\):
\[ P = d_{AB} + d_{BC} + d_{CA} \] \[ = 5 + 10 + 5\sqrt{5} \] \[ \approx 5 + 10 + 11.18 \approx 26.18 \]
Step 5: Round to the nearest tenth
Rounding \(26.18\) to the nearest tenth gives us:
\[ \boxed{26.2} \]
Thus, the perimeter of the triangle is approximately \(26.2\) units.