To find the perimeter of the parallelogram defined by the vertices \((2, 1)\), \((2, 3)\), \((4, 4)\), and \((4, 2)\), we first need to determine the lengths of the sides of the parallelogram.
Step 1: Identify the sides of the parallelogram
The vertices can be paired in order:
- \((2, 1)\) to \((2, 3)\)
- \((2, 3)\) to \((4, 4)\)
- \((4, 4)\) to \((4, 2)\)
- \((4, 2)\) to \((2, 1)\)
Step 2: Calculate the lengths of the sides
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Length from \((2, 1)\) to \((2, 3)\): \[ \text{Distance} = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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Length from \((2, 3)\) to \((4, 4)\): \[ \text{Distance} = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
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Length from \((4, 4)\) to \((4, 2)\): \[ \text{Distance} = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
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Length from \((4, 2)\) to \((2, 1)\): \[ \text{Distance} = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
Step 3: Compute the perimeter
The perimeter \(P\) of the parallelogram is the sum of all side lengths. Since opposite sides of a parallelogram are equal, we can write:
\[ P = 2 \times (\text{Length of one side} + \text{Length of the adjacent side}) = 2 \times (2 + \sqrt{5}) \]
Calculating \(P\):
\[ P = 2 \times (2 + \sqrt{5}) \approx 2 \times (2 + 2.236) = 2 \times 4.236 = 8.472 \]
Step 4: Round to the nearest tenth
Rounding \(8.472\) to the nearest tenth gives:
\[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram is approximately 8.5.