To find the perimeter of the rectangle with vertices at coordinates (0,0), (4,4), (5,3), and (1,−1), we need to calculate the lengths of the sides of the rectangle.
First, we can analyze the vertices to determine the correct arrangement:
- (0,0) to (1,-1)
- (1,-1) to (4,4)
- (4,4) to (5,3)
- (5,3) to (0,0)
Next, we will calculate the distance between each pair of vertices.
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Distance between (0,0) and (1,-1): \[ d = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414 \]
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Distance between (1,-1) and (4,4): \[ d = \sqrt{(4-1)^2 + (4-(-1))^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.831 \]
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Distance between (4,4) and (5,3): \[ d = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414 \]
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Distance between (5,3) and (0,0): \[ d = \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.831 \]
Now that we have the distances, we can find the perimeter \(P\) of the rectangle by adding twice the lengths of adjacent sides.
We have two sides of approximately \(1.414\) and two sides of approximately \(5.831\): \[ P \approx 2 \times 1.414 + 2 \times 5.831 \approx 2.828 + 11.662 = 14.490 \]
Rounding this to the nearest tenth, the perimeter is approximately: \[ \boxed{14.5} \text{ units}. \]