To find the perimeter of the rectangle defined by the points (0, 0), (4, 4), (5, 3), and (1, -1), we first need to determine the lengths of the sides of the rectangle.
We will find the distances between pairs of vertices to identify the lengths of the sides of the rectangle.
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Calculate the 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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Calculate the 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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Calculate the 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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Lastly, calculate the 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, we have the lengths of two pairs of opposite sides of the rectangle:
- The longer sides are approximately \( 5.7 \) units, and there are two of them.
- The shorter sides are approximately \( 1.4 \) units, and there are also two of them.
Now, we can calculate the perimeter \( P \): \[ P = 2(\text{length} + \text{width}) = 2(5.7 + 1.4) = 2(7.1) = 14.2 \]
Thus, the perimeter of the rectangle is approximately 14.2 units.
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