To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to determine the lengths of the sides of the triangle. The perimeter is the sum of these lengths.
Step 1: Determine the lengths of the sides
- Let \(A(1, 5)\), \(B(4, 1)\), and \(C(-4, -5)\). We will calculate the distances \(AB\), \(BC\), and \(CA\).
Length of AB: Using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(A(1, 5)\) and \(B(4, 1)\): \[ AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Length of BC: Using the distance formula where \(B(4, 1)\) and \(C(-4, -5)\): \[ BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
Length of CA: Using the distance formula where \(C(-4, -5)\) and \(A(1, 5)\): \[ CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \] Calculating \(5\sqrt{5}\): \[ \sqrt{5} \approx 2.236 \implies 5\sqrt{5} \approx 5 \times 2.236 = 11.18 \]
Step 2: Calculate the perimeter
Now, we add the lengths of the sides: \[ \text{Perimeter} = AB + BC + CA = 5 + 10 + 5\sqrt{5} \] To get the numerical value of the perimeter: \[ \text{Perimeter} \approx 5 + 10 + 11.18 = 26.18 \]
Step 3: Round to the nearest tenth
Rounding \(26.18\) to the nearest tenth, we get: \[ 26.2 \]
Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is \(\boxed{26.2}\).
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