To compute the perimeter of the parallelogram with the given vertices (2, 1), (2, 3), (4, 4), and (4, 2), we first need to find the lengths of the sides.
We can pair up the vertices to determine the lengths of the sides of the parallelogram:
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The distance between points \( (2, 1) \) and \( (2, 3) \): \[ d_1 = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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The distance between points \( (2, 3) \) and \( (4, 4) \): \[ d_2 = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
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The distance between points \( (4, 4) \) and \( (4, 2) \): \[ d_3 = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = 2 \]
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The distance between points \( (4, 2) \) and \( (2, 1) \): \[ d_4 = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
The sides of the parallelogram consist of two lengths equal to 2, and two lengths equal to \(\sqrt{5}\).
Now, we find the perimeter, \( P \): \[ P = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5} \]
Calculating \(\sqrt{5}\) numerically: \[ \sqrt{5} \approx 2.236 \] So, \[ 2\sqrt{5} \approx 2 \times 2.236 = 4.472 \] Thus: \[ P \approx 4 + 4.472 = 8.472 \]
Rounding this to the nearest tenth, we get: \[ P \approx 8.5 \]
Therefore, the perimeter of the parallelogram is approximately 8.5 units.
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