To classify the quadrilateral with the given vertices A(7, 0), B(3, 1), C(4, 7), and D(8, 6), we can calculate the lengths of its sides and the slopes of its sides. This will help us determine if it falls into any of the specified categories (rectangle, parallelogram, trapezoid, or square).
Step 1: Calculate the lengths of the sides
Using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\):
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Length of AB: \[ AB = \sqrt{(3 - 7)^2 + (1 - 0)^2} = \sqrt{(-4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
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Length of BC: \[ BC = \sqrt{(4 - 3)^2 + (7 - 1)^2} = \sqrt{(1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37} \]
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Length of CD: \[ CD = \sqrt{(8 - 4)^2 + (6 - 7)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
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Length of DA: \[ DA = \sqrt{(8 - 7)^2 + (6 - 0)^2} = \sqrt{(1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37} \]
Step 2: Check the parallelism of opposite sides
To classify the quadrilateral, we should check if opposite sides are equal and parallel:
- \(AB = CD = \sqrt{17}\)
- \(BC = DA = \sqrt{37}\)
This indicates that opposite sides are equal, which is a condition for a parallelogram.
Step 3: Calculate the slopes
We'll calculate the slopes between pairs of points:
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Slope of AB: \[ m_{AB} = \frac{1 - 0}{3 - 7} = \frac{1}{-4} = -\frac{1}{4} \]
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Slope of BC: \[ m_{BC} = \frac{7 - 1}{4 - 3} = \frac{6}{1} = 6 \]
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Slope of CD: \[ m_{CD} = \frac{6 - 7}{8 - 4} = \frac{-1}{4} = -\frac{1}{4} \]
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Slope of DA: \[ m_{DA} = \frac{6 - 0}{8 - 7} = \frac{6}{1} = 6 \]
From the slope calculations, we can see:
- \(m_{AB} = m_{CD} = -\frac{1}{4}\) (meaning sides AB and CD are parallel)
- \(m_{BC} = m_{DA} = 6\) (meaning sides BC and DA are parallel)
Conclusion
Since both pairs of opposite sides are equal and parallel, the quadrilateral ABCD is classified as a parallelogram.
Therefore, the answer is Parallelogram.