To classify the quadrilateral formed by the vertices P(-5, 4), L(4, 10), U(10, 1), and G(1, -5), we can follow these steps:
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Calculate the lengths of the sides:
- Distance between P and L: \[ PL = \sqrt{(-5 - 4)^2 + (4 - 10)^2} = \sqrt{(-9)^2 + (-6)^2} = \sqrt{81 + 36} = \sqrt{117} \]
- Distance between L and U: \[ LU = \sqrt{(4 - 10)^2 + (10 - 1)^2} = \sqrt{(-6)^2 + (9)^2} = \sqrt{36 + 81} = \sqrt{117} \]
- Distance between U and G: \[ UG = \sqrt{(10 - 1)^2 + (1 - (-5))^2} = \sqrt{(9)^2 + (6)^2} = \sqrt{81 + 36} = \sqrt{117} \]
- Distance between G and P: \[ GP = \sqrt{(1 - (-5))^2 + (-5 - 4)^2} = \sqrt{(6)^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117} \]
All four sides have the same length (\( \sqrt{117} \)), so we have a rhombus.
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Check the angles: To check if it is a rectangle or a square, we need to determine if the angles are right angles. We can calculate the slopes of adjacent sides to check for perpendicularity.
- Slope of PL: \[ \text{slope}_{PL} = \frac{10 - 4}{4 - (-5)} = \frac{6}{9} = \frac{2}{3} \]
- Slope of LU: \[ \text{slope}_{LU} = \frac{1 - 10}{10 - 4} = \frac{-9}{6} = -\frac{3}{2} \]
- Slope of UG: \[ \text{slope}_{UG} = \frac{-5 - 1}{1 - 10} = \frac{-6}{-9} = \frac{2}{3} \]
- Slope of GP: \[ \text{slope}_{GP} = \frac{4 - (-5)}{-5 - 1} = \frac{9}{-6} = -\frac{3}{2} \]
The product of the slopes of each pair of adjacent sides:
- \( \frac{2}{3} \times -\frac{3}{2} = -1 \) This indicates that PL and LU are perpendicular.
The same applies for UG and GP.
Since the quadrilateral has all sides equal and angles that are right angles, it meets both criteria for being a square.
Therefore, the most specific classification for the quadrilateral formed by the points P, L, U, and G is a Square.