To find the perimeter of the rectangle with the given vertices at coordinates \((0,0)\), \((4,4)\), \((5,3)\), and \((1,-1)\), we first need to verify if these points indeed form a rectangle and then calculate the lengths of the sides.
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Calculate the lengths of the sides:
- We can find the distance between pairs of points using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculate distance for each pair of points:
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Between \((0,0)\) and \((4,4)\): \[ d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Between \((0,0)\) and \((1,-1)\): \[ d_2 = \sqrt{(1 - 0)^2 + (-1 - 0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Between \((1,-1)\) and \((5,3)\): \[ d_3 = \sqrt{(5 - 1)^2 + (3 - (-1))^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Between \((4,4)\) and \((5,3)\): \[ d_4 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Identify the lengths of the sides:
- The distances \(d_1\) and \(d_3\) are both \(4\sqrt{2}\), and the distances \(d_2\) and \(d_4\) are both \(\sqrt{2}\).
- It seems we have two pairs of equal lengths which indicates that we could have a rectangle.
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Calculating Perimeter: The perimeter \(P\) of a rectangle is given by: \[ P = 2 \times (\text{length} + \text{width}) \] Here, we can take \(\text{length} = 4\sqrt{2}\) and \(\text{width} = \sqrt{2}\): \[ P = 2 \times (4\sqrt{2} + \sqrt{2}) = 2 \times (5\sqrt{2}) = 10\sqrt{2} \]
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Approximate \(10\sqrt{2}\): Using \(\sqrt{2} \approx 1.414\): \[ 10\sqrt{2} \approx 10 \times 1.414 = 14.14 \] Rounding to the nearest tenth, we get \(14.1\).
Thus, the perimeter of the rectangle is approximately 14.1 units.
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