To find the perimeter of the rectangle, we first need to determine the lengths of the sides of the rectangle formed by the given vertices: \((0,0)\), \((4,4)\), \((5,3)\), and \((1,-1)\).
However, we need to confirm whether these points really make a rectangle. A rectangle has opposite sides that are equal in length and angles that are 90 degrees.
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Calculate the distances between each pair of vertices:
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Distance between \((0,0)\) and \((4,4)\): \[ d_{1} = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.7 \]
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Distance between \((0,0)\) and \((5,3)\): \[ d_{2} = \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.8 \]
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Distance between \((0,0)\) and \((1,-1)\): \[ d_{3} = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
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Distance between \((4,4)\) and \((5,3)\): \[ d_{4} = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
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Distance between \((4,4)\) and \((1,-1)\): \[ d_{5} = \sqrt{(1-4)^2 + (-1-4)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.8 \]
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Distance between \((5,3)\) and \((1,-1)\): \[ d_{6} = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.7 \]
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Check for pairs of opposite sides: A rectangle has two pairs of equal-length sides. Here,
- \(d_1\) and \(d_6\) are approximately \(5.7\),
- \(d_2\) and \(d_5\) are approximately \(5.8\),
- \(d_3\) and \(d_4\) are approximately \(1.4\).
Since we have matched sides, and they are paired accordingly, it appears that these vertices form a quadrilateral, but we should check whether they form a rectangle by verifying right angles.
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Positioning:
After checking, the coordinates do not represent a regular rectangle due to lack of systematic right angle segmentation in Cartesian space.
Since the distances have been calculated correctly, and confirmed they are equal or fit within context of dimensions typical of a rectangle.
In terms of perimeter calculation: Given the dimensions (length and width).
Perimeter is calculated by the formula: \[ \text{Perimeter} = 2(\text{length} + \text{width}) = 2(5.7 + 1.4) \approx 2 \times 7.1 \approx 14.2 \]
Final calculated perimeter (to nearest tenth) is:
\[ \text{Perimeter} \approx \boxed{14.2} \text{ units} \]