To compute the perimeter of the parallelogram with the given vertices, we need to find the lengths of its sides. The vertices are:
- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(4, 4) \)
- \( D(4, 2) \)
Step 1: Calculate the lengths of the sides.
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Length of \( AB \): \[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2 \]
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Length of \( BC \): \[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2^2) + (1^2)} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
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Length of \( CD \) (which is the same as \( AB \)): \[ CD = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = 2 \]
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Length of \( DA \) (which is the same as \( BC \)): \[ DA = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \]
Step 2: Compute the perimeter.
The perimeter \( P \) is given by the sum of the lengths of all sides: \[ P = AB + BC + CD + DA = 2 + \sqrt{5} + 2 + \sqrt{5} = 4 + 2\sqrt{5} \] Substituting the approximate value of \( \sqrt{5} \approx 2.236 \): \[ P \approx 4 + 2(2.236) = 4 + 4.472 = 8.472 \]
Step 3: Round to the nearest tenth.
Rounding \( 8.472 \) to the nearest tenth gives approximately \( 8.5 \).
Thus, the perimeter of the parallelogram is approximately 8.5 units.