To find the perimeter of the parallelogram formed by the vertices (0,0), (2,3), (5,2), and (3,-1), we first need to calculate the lengths of the sides.
The perimeter of a parallelogram is given by the formula:
\[ P = 2 \times (a + b) \]
where \(a\) and \(b\) are the lengths of the adjacent sides.
Step 1: Calculate the distances between adjacent vertices.
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Distance between (0, 0) and (2, 3): \[ d_1 = \sqrt{(2-0)^2 + (3-0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \]
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Distance between (2, 3) and (5, 2): \[ d_2 = \sqrt{(5-2)^2 + (2-3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.2 \]
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Distance between (5, 2) and (3, -1): \[ d_3 = \sqrt{(3-5)^2 + (-1-2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6 \]
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Distance between (3, -1) and (0, 0): \[ d_4 = \sqrt{(0-3)^2 + (0-(-1))^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \approx 3.2 \]
Step 2: Add the lengths of two adjacent sides.
From the calculations above:
- One side length is \( \sqrt{13} \approx 3.6 \).
- The other side length is \( \sqrt{10} \approx 3.2 \).
Step 3: Calculate the perimeter.
Using the lengths: \[ P \approx 2 \times (3.6 + 3.2) \approx 2 \times 6.8 = 13.6 \]
Therefore, the perimeter of the parallelogram is approximately 13.6 units.
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