To compute the area of the rectangle formed by the given coordinates A(0,0), B(4,-4), C(7,-1), and D(3,3), we should confirm if these points indeed form a rectangle and then calculate the area accordingly.
The area of a rectangle can be calculated using the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
First, we will find the lengths of the sides using the distance formula which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will calculate the lengths of opposite sides to check if they are equal:
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Calculate the distance AB: \[ AB = \sqrt{(4 - 0)^2 + (-4 - 0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Calculate the distance CD: \[ CD = \sqrt{(3 - 7)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (3 + 1)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Calculate the distance BC: \[ BC = \sqrt{(7 - 4)^2 + (-1 - (-4))^2} = \sqrt{3^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Calculate the distance AD: \[ AD = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
Since opposite sides are equal (AB = CD and BC = AD), we can conclude that these points do indeed form a rectangle.
Now we can calculate the area:
\[ \text{Area} = AB \times BC \]
Substituting in the lengths:
\[ \text{Area} = (4\sqrt{2}) \times (3\sqrt{2}) = 4 \times 3 \times (\sqrt{2} \times \sqrt{2}) = 12 \times 2 = 24 \text{ units}^2 \]
Thus, the exact area of the rectangle is 24 units².
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