To find the perimeter of the rectangle defined by the vertices \((0,0)\), \((4,4)\), \((5,3)\), and \((1,-1)\), we first need to determine the order of the vertices and the lengths of the sides.
Let's visualize the points:
- \((0,0)\)
- \((4,4)\)
- \((5,3)\)
- \((1,-1)\)
To confirm these points form a rectangle, we should check the distances between adjacent points and the angles between the sides.
Calculating the distances between the pairs of consecutive points:
-
Distance between \((0,0)\) and \((1,-1)\): \[ \sqrt{(1 - 0)^2 + (-1 - 0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
-
Distance between \((1,-1)\) and \((4,4)\): \[ \sqrt{(4 - 1)^2 + (4 - (-1))^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \]
-
Distance between \((4,4)\) and \((5,3)\): \[ \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
-
Distance between \((5,3)\) and \((0,0)\): \[ \sqrt{(0 - 5)^2 + (0 - 3)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]
Now we can summarize the sides:
- Two sides of length \(\sqrt{2}\)
- Two sides of length \(\sqrt{34}\)
The perimeter \(P\) of the rectangle can be calculated as: \[ P = 2 \times (\text{length 1} + \text{length 2}) = 2 \times (\sqrt{2} + \sqrt{34}) \]
Calculating this:
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Approximating the values:
- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{34} \approx 5.831\)
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Now substituting these values: \[ P \approx 2 \times (1.414 + 5.831) = 2 \times 7.245 = 14.49 \]
Rounding to the nearest tenth: \[ P \approx 14.5 \]
Thus, the perimeter of the rectangle is \(\boxed{14.5}\).
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