To find the perimeter of the rectangle given the vertices at coordinates (0,0), (4,4), (5,3), and (1,-1), we first need to confirm which pairs of vertices actually form the sides of the rectangle.
A rectangle has opposite sides that are equal and all angles are right angles. To check the pairs, we will calculate the lengths of each side between these points.
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Distance between (0,0) and (4,4): \[ d = \sqrt{(4-0)^2 + (4-0)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.7 \]
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Distance between (4,4) and (5,3): \[ d = \sqrt{(5-4)^2 + (3-4)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
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Distance between (5,3) and (1,-1): \[ d = \sqrt{(1-5)^2 + (-1-3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.7 \]
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Distance between (1,-1) and (0,0): \[ d = \sqrt{(0-1)^2 + (0-(-1))^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.4 \]
We have found two pairs of opposite sides with the same lengths:
- The sides between (0,0) and (4,4) and between (5,3) and (1,-1) both are approximately 5.7 units long.
- The sides between (4,4) and (5,3) and between (1,-1) and (0,0) both are approximately 1.4 units long.
Now we sum the lengths of all four sides to find the perimeter of the rectangle:
\[ \text{Perimeter} = 2 \times (5.7 + 1.4) \approx 2 \times 7.1 \approx 14.2 \]
Thus, the perimeter of the rectangle is approximately 14.2 units when rounded to the nearest tenth.