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 using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the lengths of the sides.
- 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):
\[ d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.18 \]
Step 2: Calculate the perimeter.
Now we can find the perimeter \(P\) by adding the lengths of the sides:
\[ P = d_1 + d_2 + d_3 = 5 + 10 + 5\sqrt{5} \]
Using \(\sqrt{5} \approx 2.236\):
\[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
So,
\[ P \approx 5 + 10 + 11.18 \approx 26.18 \]
Rounding to the nearest tenth, the perimeter of the triangle is approximately:
\[ \text{Perimeter} \approx 26.2 \]
Thus, the answer is 26.2.