To find the perimeter of the rectangle with vertices at (0,0), (4,4), (5,3), and (1,-1), we need to first confirm that these points actually form a rectangle and then calculate its perimeter.
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Check if the four points form a rectangle: We need to ensure opposite sides are equal in length, and the angles between the sides are right angles.
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Calculate the distances between each pair of points:
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Distance from (0,0) to (4,4): \[ d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance from (4,4) to (5,3): \[ d_2 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance from (5,3) to (1,-1): \[ d_3 = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance from (1,-1) to (0,0): \[ d_4 = \sqrt{(0 - 1)^2 + (0 + 1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Now we see that opposite sides are equal, resulting in:
- Side 1: \(d_1 = 4\sqrt{2}\)
- Side 2: \(d_2 = \sqrt{2}\)
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Calculate the perimeter: A rectangle's perimeter \(P\) can be calculated using the formula: \[ P = 2 \times (\text{length} + \text{width}) \]
Here, we can take the lengths we calculated:
- Length = \(4\sqrt{2}\)
- Width = \(\sqrt{2}\)
The perimeter can now be calculated as: \[ P = 2 \times (4\sqrt{2} + \sqrt{2}) = 2 \times (5\sqrt{2}) = 10\sqrt{2} \]
Now we need to approximate \(10\sqrt{2}\) using \(\sqrt{2} \approx 1.414\): \[ 10\sqrt{2} \approx 10 \times 1.414 \approx 14.14 \]
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Final answer: Rounding to the nearest tenth, the perimeter is approximately \(14.1\) units.
Thus, the perimeter of the rectangle is approximately 14.1 units.