To classify the quadrilateral formed by the vertices \( L(-4,4) \), \( A(3,2) \), \( K(-5,5) \), and \( E(-2,-3) \), we need to determine the properties of the figure, including the lengths of the sides and the angles.
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Calculate the lengths of the sides and determine if opposite sides are equal:
- Distance \( LA \): \[ LA = \sqrt{(3 - (-4))^2 + (2 - 4)^2} = \sqrt{(3 + 4)^2 + (2 - 4)^2} = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \]
- Distance \( AK \): \[ AK = \sqrt{(-5 - 3)^2 + (5 - 2)^2} = \sqrt{(-8)^2 + (3)^2} = \sqrt{64 + 9} = \sqrt{73} \]
- Distance \( KE \): \[ KE = \sqrt{(-2 - (-5))^2 + (-3 - 5)^2} = \sqrt{(3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73} \]
- Distance \( EL \): \[ EL = \sqrt{(-4 - (-2))^2 + (4 - (-3))^2} = \sqrt{(-2)^2 + (7)^2} = \sqrt{4 + 49} = \sqrt{53} \]
Now we have:
- \( LA = \sqrt{53} \)
- \( AK = \sqrt{73} \)
- \( KE = \sqrt{73} \)
- \( EL = \sqrt{53} \)
The opposite sides are equal: \( LA = EL \) and \( AK = KE \), confirming it is a Parallelogram.
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Check for rectangle properties (right angles or equal diagonal lengths):
- Calculate the lengths of the diagonals \( LK \) and \( AE \):
- Distance \( LK \): \[ LK = \sqrt{(-5 - (-4))^2 + (5 - 4)^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Distance \( AE \): \[ AE = \sqrt{(-2 - 3)^2 + (-3 - 2)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]
Since \( LK \) and \( AE \) are not the same length, the diagonals are not equal, so it's not a rectangle.
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Check for rhombus properties (equal sides):
- We already noted that while the opposite sides are equal, adjacent sides show different properties: \( LA \neq AK \).
Since the quadrilateral is a parallelogram with opposite sides equal but does not meet the criteria for rectangles or rhombuses, we determine that the best classification that still aligns with the properties is a Parallelogram.
Thus, the final classification is:
Parallelogram
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