To classify the quadrilateral formed by the vertices Q(−8,1), U(3,5), A(7,−1), and Y(−4,−4), we can follow these steps:
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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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Calculate the midpoints of the diagonals to see if they bisect each other.
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Check the slopes of the opposite sides to see if they are parallel.
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Check the lengths of diagonals and if they are equal (for rectangle and square).
Step 1: Calculate the lengths of the sides
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Length of QU: \[ d_{QU} = \sqrt{(3 - (-8))^2 + (5 - 1)^2} = \sqrt{(3 + 8)^2 + (5 - 1)^2} = \sqrt{11^2 + 4^2} = \sqrt{121 + 16} = \sqrt{137} \]
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Length of UA: \[ d_{UA} = \sqrt{(7 - 3)^2 + (-1 - 5)^2} = \sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
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Length of AY: \[ d_{AY} = \sqrt{(-4 - 7)^2 + (-4 - (-1))^2} = \sqrt{(-11)^2 + (-3)^2} = \sqrt{121 + 9} = \sqrt{130} \]
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Length of YQ: \[ d_{YQ} = \sqrt{(-8 - (-4))^2 + (1 - (-4))^2} = \sqrt{(-4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41} \]
Step 2: Check the midpoints of the diagonals
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Midpoint of QU: \[ M_{QU} = \left( \frac{-8 + 3}{2}, \frac{1 + 5}{2} \right) = \left( \frac{-5}{2}, 3 \right) \]
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Midpoint of AY: \[ M_{AY} = \left( \frac{7 + (-4)}{2}, \frac{-1 + (-4)}{2} \right) = \left( \frac{3}{2}, -\frac{5}{2} \right) \]
Step 3: Check slopes to verify parallelism
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Slope of QU: \[ \text{slope}_{QU} = \frac{5 - 1}{3 - (-8)} = \frac{4}{11} \]
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Slope of AY: \[ \text{slope}_{AY} = \frac{-4 - (-1)}{-4 - 7} = \frac{-3}{-11} = \frac{3}{11} \]
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Slope of UA: \[ \text{slope}_{UA} = \frac{-1 - 5}{7 - 3} = \frac{-6}{4} = -\frac{3}{2} \]
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Slope of YQ: \[ \text{slope}_{YQ} = \frac{1 - (-4)}{-8 - (-4)} = \frac{5}{-4} = -\frac{5}{4} \]
Parallel Sides
We have neither pairs of sides parallel nor equal in length.
Conclusion
The sides have different lengths, and the slopes indicate that no sides are parallel.
Thus, the most specific classification for the shape formed by the vertices Q(−8,1), U(3,5), A(7,−1), and Y(−4,−4) is a quadrilateral.