Given the vertices of the parallelogram: \( A(2, 1) \), \( B(2, 3) \), \( C(4, 4) \), and \( D(4, 2) \), we will first find the lengths of the sides \( AB \), \( BC \), \( CD \), and \( DA \).
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Calculate distance \( AB \): \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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Calculate distance \( BC \): \[ BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
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Calculate distance \( CD \): \[ CD = \sqrt{(4 - 4)^2 + (4 - 2)^2} = \sqrt{0 + (2)^2} = \sqrt{4} = 2 \]
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Calculate distance \( DA \): \[ DA = \sqrt{(4 - 2)^2 + (2 - 1)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
Now that we have the side lengths:
- \( AB = 2 \)
- \( BC = \sqrt{5} \)
- \( CD = 2 \)
- \( DA = \sqrt{5} \)
Since opposite sides of a parallelogram are equal, we can conclude:
- Length of one pair of opposite sides \( = 2 \)
- Length of the other pair of opposite sides \( = \sqrt{5} \)
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Calculate the perimeter \( P \) of the parallelogram: \[ P = AB + BC + CD + DA = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5} \]
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Now calculate \( 2\sqrt{5} \): \[ \sqrt{5} \approx 2.236 \quad \text{so } \quad 2\sqrt{5} \approx 2 \times 2.236 \approx 4.472 \]
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Final perimeter calculation: \[ P \approx 4 + 4.472 \approx 8.472 \]
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Rounding to the nearest tenth: \[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram, rounded to the nearest tenth, is \( \boxed{8.5} \).
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