To find the perimeter of the rectangle defined by the given vertices, we first need to ascertain which of these points form the rectangle, as the coordinates given do not form a rectangle by inspection.
Step 1: Identify the vertices
The given vertices are:
- A: (0, 0)
- B: (4, 4)
- C: (5, 3)
- D: (1, -1)
We need to check if these points can indeed represent the corners of a rectangle. The diagonals of a rectangle should be equal, and adjacent sides must be perpendicular.
Step 2: Check sides for length
We can calculate the lengths of the sides between each pair of points:
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Distance between A (0, 0) and B (4, 4): \[ \text{Distance} = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance between A (0, 0) and C (5, 3): \[ \text{Distance} = \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34} \]
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Distance between A (0, 0) and D (1, -1): \[ \text{Distance} = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance between B (4, 4) and C (5, 3): \[ \text{Distance} = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance between B (4, 4) and D (1, -1): \[ \text{Distance} = \sqrt{(1-4)^2 + (-1-4)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]
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Distance between C (5, 3) and D (1, -1): \[ \text{Distance} = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = 4\sqrt{2} \]
Step 3: Find the shape
The calculated distances:
- From A to B: \( 4\sqrt{2} \)
- From A to C: \( \sqrt{34} \)
- From A to D: \( \sqrt{2} \)
- From B to C: \( \sqrt{2} \)
- From B to D: \( \sqrt{34} \)
- From C to D: \( 4\sqrt{2} \)
Identifying pairs:
- Sides: \( \sqrt{34} \) appears between opposite pairs of points (A to C and B to D).
The computed distances do not match a rectangle's properties (opposite sides being equal).
Final step: Conclusion
Since the point arrangement does not create a rectangular structure, we conclude to verify a valid rectangle.
Correcting the conclusion:
From the initial lineup of vertices, we do not have a rectangle, thus finding the perimeter of an assumed rectangle isn't achievable with the current set; we should check which other point configurations might yield valid rectangles or reconsider the problem statement.
However, if the problem were stated correctly with unique rectangular properties, input point coordinates would need revision for perimeter calculation to hold true.
Thus, the final response regarding the rectangle perimeter is undefined, as the set provided doesn't yield a rectangle dimension or structure physically alignable within defined geometric properties.
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