To find the perimeter of the parallelogram formed by the given vertices, we first need to identify the sides of the parallelogram using the given points: \( A(0,0) \), \( B(2,3) \), \( C(5,2) \), and \( D(3,-1) \).
The perimeter \( P \) of a parallelogram can be computed as \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the sides.
First, we need to find the lengths of the sides \( AB \) and \( BC \).
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Calculate the length of \( AB \): \[ AB = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Calculate the length of \( BC \): \[ BC = \sqrt{(5 - 2)^2 + (2 - 3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
Both pairs of opposite sides are equal in length; so, \( CD \) will also be \( \sqrt{13} \), and \( AD \) will also be \( \sqrt{10} \).
Now, we can use the lengths of the two distinct sides to calculate the perimeter: \[ P = 2(AB + BC) = 2(\sqrt{13} + \sqrt{10}) \]
Using approximate values: \[ \sqrt{13} \approx 3.60555 \quad \text{and} \quad \sqrt{10} \approx 3.16228 \] Thus, \[ P \approx 2(3.60555 + 3.16228) = 2(6.76783) \approx 13.53566 \]
Rounding to the nearest tenth, the perimeter of the parallelogram is approximately: \[ \boxed{13.5} \text{ units.} \]
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