To find the perimeter of the rectangle with vertices at the coordinates (0,0), (4,4), (5,3), and (1,-1), we first check whether these points actually form a rectangle.
We will calculate the lengths of the sides formed by these points.
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Calculate the distances between the points:
- Distance between (0,0) and (4,4): \[ d = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Distance between (4,4) and (5,3): \[ d = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Distance between (5,3) and (1,-1): \[ d = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Distance between (1,-1) and (0,0): \[ d = \sqrt{(0-1)^2 + (0-(-1))^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Check if opposite sides are equal: We see that:
- The distances (0,0) to (4,4) and (5,3) to (1,-1) are both \(4\sqrt{2}\).
- The distances (4,4) to (5,3) and (1,-1) to (0,0) are both \(\sqrt{2}\).
This shows that opposite sides are equal which is typical of a rectangle. Now we can compute the perimeter.
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Calculate the perimeter: The perimeter \(P\) of a rectangle is given by the formula: \[ P = 2(\text{length} + \text{width}) \] Here, we take:
- Length = \(4\sqrt{2}\)
- Width = \(\sqrt{2}\)
Thus: \[ P = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2} \]
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Calculate \(10\sqrt{2}\) numerically: Using \(\sqrt{2} \approx 1.414\): \[ 10\sqrt{2} \approx 10 \times 1.414 \approx 14.14 \]
Finally, rounding to the nearest tenth, we find:
\[ \text{Perimeter} \approx 14.1 \]
Therefore, the perimeter of the rectangle is approximately \(\boxed{14.1}\).
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