To find the perimeter of the parallelogram defined by the vertices (2,1), (2,3), (4,4), and (4,2), we will first calculate the lengths of the sides.
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Identify the vertices: Let the vertices be:
- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(4, 4) \)
- \( D(4, 2) \)
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Calculate the lengths of the sides:
- Length of side AB: \[ AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
- Length of side BC: \[ BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
- Length of side CD: \[ CD = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
- Length of side DA: \[ DA = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
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Perimeter calculation: The perimeter \( P \) of the parallelogram can be calculated as: \[ P = AB + BC + CD + DA = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5} \]
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Evaluate \( 2\sqrt{5} \): The approximate value of \( \sqrt{5} \) is about 2.236. Thus, \[ 2\sqrt{5} \approx 2 \times 2.236 = 4.472 \]
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Calculating the perimeter: Adding this to 4 gives: \[ P \approx 4 + 4.472 = 8.472 \]
Rounding this to the nearest tenth: \[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram is \( \boxed{8.5} \).
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