To calculate the perimeter of the parallelogram with given vertices, we can follow these steps:
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Identify the vertices:
- A = (0, 0)
- B = (1, 2)
- C = (2, 1)
- D = (1, -1)
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Calculate the lengths of the sides: We will calculate the lengths of AB, BC, CD, and DA.
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Length of AB: \[ AB = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
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Length of BC: \[ BC = \sqrt{(2-1)^2 + (1-2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Length of CD: \[ CD = \sqrt{(1-2)^2 + (-1-1)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
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Length of DA: \[ DA = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Sum the lengths to find the perimeter: Since opposite sides of a parallelogram are equal, we can calculate the perimeter as follows: \[ \text{Perimeter} = 2(AB + BC) \] Plugging in the values we calculated: \[ \text{Perimeter} = 2(\sqrt{5} + \sqrt{2}) \]
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Numerical approximation: We can also calculate the approximate values of \(\sqrt{5}\) and \(\sqrt{2}\): \[ \sqrt{5} \approx 2.236, \quad \sqrt{2} \approx 1.414 \] Thus, \[ \text{Perimeter} \approx 2(2.236 + 1.414) \approx 2(3.650) \approx 7.3 \]
Since the values don't exactly match the options provided but are close to 7.2, the best estimate would be:
A. 7.2 units