To find the perimeter of the rectangle defined by the vertices at coordinates (0,0), (4,4), (5,3), and (1,-1), we first verify if the given points form a rectangle.
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Calculate the distances between all pairs of points to check if the distances correspond to a rectangle's properties (two sets of equal lengths).
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Distance between (0,0) and (4,4): \[ d = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance between (0,0) and (5,3): \[ d = \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34} \]
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Distance between (0,0) and (1,-1): \[ d = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance between (4,4) and (5,3): \[ d = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance between (4,4) and (1,-1): \[ d = \sqrt{(1-4)^2 + (-1-4)^2} = \sqrt{9 + 25} = \sqrt{34} \]
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Distance between (5,3) and (1,-1): \[ d = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Summarizing the distances:
- \(d((0,0),(4,4)) = 4\sqrt{2}\)
- \(d((0,0),(5,3)) = \sqrt{34}\)
- \(d((0,0),(1,-1)) = \sqrt{2}\)
- \(d((4,4),(5,3)) = \sqrt{2}\)
- \(d((4,4),(1,-1)) = \sqrt{34}\)
- \(d((5,3),(1,-1)) = 4\sqrt{2}\)
From the computed distances, we find pairs of equal lengths:
- \(4\sqrt{2}\) appears twice.
- \(\sqrt{34}\) appears twice.
- \(\sqrt{2}\) appears twice.
Thus, the opposite sides are equal in length.
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Calculate the perimeter: The perimeter \(P\) of a rectangle is given by: \[ P = 2 \times (\text{length} + \text{width}) \] Here, we can take the lengths as \(4\sqrt{2}\) and \(\sqrt{34}\). Thus: \[ P = 2 \times (4\sqrt{2} + \sqrt{34}) \]
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Calculating the numerical value:
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First calculate \(4\sqrt{2}\): \[ \sqrt{2} \approx 1.414 \Rightarrow 4\sqrt{2} \approx 4 \times 1.414 \approx 5.656 \]
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Next calculate \(\sqrt{34}\): \[ \sqrt{34} \approx 5.831 \]
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Putting these together: \[ 4\sqrt{2} + \sqrt{34} \approx 5.656 + 5.831 \approx 11.487 \]
Thus, the perimeter \(P\) would be: \[ P \approx 2 \times 11.487 \approx 22.974 \]
- Rounding the answer to the nearest tenth: \[ P \approx 23.0 \]
The final answer for the perimeter of the rectangle is 23.0.
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