To find the perimeter of the triangle formed by the points (1,5), (4,1), and (-4,-5), Nathan can follow these steps:
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Label the points:
- Let A = (1, 5)
- Let B = (4, 1)
- Let C = (-4, -5)
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Calculate the lengths of the sides using the distance formula: The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Length of side AB: \[ d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
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Length of side BC: \[ d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
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Length of side CA: \[ d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \]
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Calculate \(5\sqrt{5}\) numerically: Using an approximate value for \(\sqrt{5} \approx 2.236\): \[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
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Calculate the perimeter: The perimeter \(P\) of the triangle is the sum of the lengths of the sides: \[ P = d_{AB} + d_{BC} + d_{CA} = 5 + 10 + 5\sqrt{5} \approx 5 + 10 + 11.18 = 26.18 \]
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Round to the nearest tenth: Rounding \(26.18\) to the nearest tenth gives \(26.2\).
So, the perimeter of the triangle, rounded to the nearest tenth of a unit, is 26.2 units.
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