To calculate the perimeter of the parallelogram defined by the vertices \((0,0)\), \((2,3)\), \((5,2)\), and \((3,-1)\), we need to compute the lengths of the sides.
Step 1: Calculate the lengths of the sides.
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Length between (0,0) and (2,3): \[ d_1 = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Length between (2,3) and (5,2): \[ d_2 = \sqrt{(5-2)^2 + (2-3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
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Length between (5,2) and (3,-1): \[ d_3 = \sqrt{(3-5)^2 + (-1-2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Length between (3,-1) and (0,0): \[ d_4 = \sqrt{(0-3)^2 + (0-(-1))^2} = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]
Step 2: Compute the perimeter.
The perimeter \(P\) of the parallelogram is given by the sum of the lengths of all sides: \[ P = d_1 + d_2 + d_3 + d_4 = \sqrt{13} + \sqrt{10} + \sqrt{13} + \sqrt{10} = 2\sqrt{13} + 2\sqrt{10} \]
To find the numerical values, we calculate:
- \(\sqrt{13} \approx 3.6056\)
- \(\sqrt{10} \approx 3.1623\)
Now substituting these values: \[ P \approx 2(3.6056) + 2(3.1623) = 7.2112 + 6.3246 \approx 13.5358 \]
Step 3: Round the perimeter to the nearest tenth.
Rounding \(13.5358\) to the nearest tenth gives: \[ \boxed{13.5} \]
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