To find the perimeter of the rectangle formed by the given vertices \((0,0)\), \((4,4)\), \((5,3)\), and \((1,-1)\), we first need to determine if these points truly form a rectangle.
Next, we'll calculate the distances between each pair of adjacent vertices.
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Distance between \((0,0)\) and \((4,4)\): \[ d = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.7 \]
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Distance between \((4,4)\) and \((5,3)\): \[ d = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
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Distance between \((5,3)\) and \((1,-1)\): \[ d = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.7 \]
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Distance between \((1,-1)\) and \((0,0)\): \[ d = \sqrt{(0 - 1)^2 + (0 - (-1))^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
Now that we have calculated the lengths of all four sides, we can compute the perimeter \(P\): \[ P = 2 \times (5.7 + 1.4) = 2 \times 7.1 = 14.2 \]
Therefore, the perimeter of the rectangle is approximately \( \boxed{14.2} \) units.
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