To compute the perimeter of the parallelogram given the vertices \((2,3)\), \((6,3)\), \((7,7)\), and \((3,7)\), we'll first find the lengths of the sides of the parallelogram.
Step 1: Calculate the lengths of the sides.
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Between points \( (2,3) \) and \( (6,3) \): \[ \text{Length} = |x_2 - x_1| = |6 - 2| = 4 \]
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Between points \( (6,3) \) and \( (7,7) \): \[ \text{Length} = \sqrt{(7 - 6)^2 + (7 - 3)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \]
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Between points \( (7,7) \) and \( (3,7) \): \[ \text{Length} = |x_2 - x_1| = |7 - 3| = 4 \]
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Between points \( (3,7) \) and \( (2,3) \): \[ \text{Length} = \sqrt{(2 - 3)^2 + (3 - 7)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \]
Step 2: Compute the perimeter.
The perimeter \(P\) of a parallelogram is given by the formula: \[ P = 2(a + b) \] where \(a\) and \(b\) are the lengths of adjacent sides. Here, both lengths \(a\) and \(b\) are \(4\) and \(\sqrt{17}\) respectively.
Calculating the perimeter: \[ P = 2(4 + \sqrt{17}) \]
Step 3: Calculate \(\sqrt{17}\).
Using a calculator, \[ \sqrt{17} \approx 4.123 \]
So now, \[ P \approx 2(4 + 4.123) = 2 \times 8.123 \approx 16.246 \]
Final Step: Round the answer.
Rounding to the nearest tenth: \[ \text{Perimeter} \approx 16.2 \text{ units.} \]
Thus, the perimeter is approximately 16.2 units.