To compute the perimeter of a polygon defined by the vertices \((2,1)\), \((2,3)\), \((4,4)\), and \((4,2)\), we first need to find the lengths of the sides between each pair of consecutive vertices.
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Distance between \((2,1)\) and \((2,3)\): \[ d_1 = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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Distance between \((2,3)\) and \((4,4)\): \[ d_2 = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
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Distance between \((4,4)\) and \((4,2)\): \[ d_3 = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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Distance between \((4,2)\) and \((2,1)\): \[ d_4 = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
Now, we calculate the total perimeter \(P\) by adding the lengths of all sides:
\[ P = d_1 + d_2 + d_3 + d_4 = 2 + \sqrt{5} + 2 + \sqrt{5} \] \[ P = 4 + 2\sqrt{5} \approx 4 + 2 \times 2.236 \approx 4 + 4.472 \approx 8.472 \]
Rounding this to the nearest tenth:
\[ P \approx 8.5 \]
Thus, the perimeter of the shape formed by the given vertices is approximately 8.5.
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