To find the area of a rectangle using its vertices, we can use the coordinates of the vertices given: A(0,0), B(4,-4), C(7,-1), and D(3,3).
However, looking closely at the coordinates, it appears these points do not form a rectangle, which typically has right angles at the corners.
To confirm if the points indeed form a rectangle, we should check the lengths of the sides and the diagonals. We can calculate distances using the distance formula:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
We need to check the sides AB, BC, CD, and DA, as well as the diagonals AC and BD.
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AB (from A(0,0) to B(4,-4)): \[ d_{AB} = \sqrt{(4-0)^2 + (-4-0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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BC (from B(4,-4) to C(7,-1)): \[ d_{BC} = \sqrt{(7-4)^2 + (-1+4)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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CD (from C(7,-1) to D(3,3)): \[ d_{CD} = \sqrt{(3-7)^2 + (3+1)^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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DA (from D(3,3) to A(0,0)): \[ d_{DA} = \sqrt{(0-3)^2 + (0-3)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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AC (from A(0,0) to C(7,-1)): \[ d_{AC} = \sqrt{(7-0)^2 + (-1-0)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]
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BD (from B(4,-4) to D(3,3)): \[ d_{BD} = \sqrt{(3-4)^2 + (3+4)^2} = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]
From the distances calculated:
- The lengths of each side are \(4\sqrt{2}\) and \(3\sqrt{2}\), which indicates that opposite sides are equal but it does not confirm a rectangle because we need to check if they are both right angles.
Since the sides are equal but do not form a proper angle at each corner, it appears these do not align properly to form a rectangle.
In conclusion, since the area can not be calculated properly due to the irregularity of points, the calculations suggest that the shape defined by these coordinates is not a rectangle, so we can not compute an area of a rectangle from the given points.
Thus, none of the options (24 units squared, 20 units squared, 9 units squared, 48 units squared) are applicable as the coordinates do not describe a rectangle.
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