To calculate the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we'll follow these steps:
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Calculate 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:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculate the lengths of the three sides of the triangle:
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Side 1: Between \((1, 5)\) and \((4, 1)\): \[ d_1 = \sqrt{(4 - 1)^2 + (1 - 5)^2} \] \[ = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
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Side 2: Between \((4, 1)\) and \((-4, -5)\): \[ d_2 = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} \] \[ = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
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Side 3: Between \((-4, -5)\) and \((1, 5)\): \[ d_3 = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \] \[ \approx 5 \times 2.236 \approx 11.180 (for calculation purposes, you can round at the end as needed) \]
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Sum the lengths of the sides to find the perimeter: \[ P = d_1 + d_2 + d_3 = 5 + 10 + 5\sqrt{5} \] Approximating \(5\sqrt{5}\), where \(\sqrt{5} \approx 2.236\): \[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.180 \] Thus, \[ P \approx 5 + 10 + 11.180 = 26.180 \]
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Round the perimeter to the nearest tenth: \[ P \approx 26.2 \]
So, the perimeter of the triangle rounded to the nearest tenth is approximately 26.2 units.
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