To compute the perimeter of the parallelogram with vertices at \((2, 1)\), \((2, 3)\), \((4, 4)\), and \((4, 2)\), we first need to calculate the lengths of the sides.
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Identify the vertices:
- A = (2, 1)
- B = (2, 3)
- C = (4, 4)
- D = (4, 2)
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Calculate the lengths of the sides: We check two sides: AB and AD.
Length of AB: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2 \]
Length of BC: \[ BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
Length of CD (which is equal to AB since it's opposite): \[ CD = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + (-2)^2} = \sqrt{4} = 2 \]
Length of DA (which is equal to BC since it's opposite): \[ DA = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
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Compute the perimeter: The perimeter \(P\) of the parallelogram can be calculated using: \[ P = 2(AB + BC) = 2(2 + \sqrt{5}) \] Approximating \(\sqrt{5} \approx 2.236\), we have: \[ P \approx 2(2 + 2.236) = 2(4.236) \approx 8.472 \]
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Round to the nearest tenth: \[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram is approximately 8.5 units.
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