To compute the area of the rectangle defined by the points A(0,0), B(4,-4), C(7,-1), and D(3,3), we first need to verify if these points indeed form a rectangle.
A rectangle has opposite sides that are equal in length and all angles are right angles. We can use the distance formula to check the lengths of the sides and the slopes to check the angles.
The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
First, we compute the lengths of the sides:
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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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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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Distance CD: \[ CD = \sqrt{(3 - 7)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance DA: \[ DA = \sqrt{(0 - 3)^2 + (0 - 3)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
Now we can check the opposite sides:
- AB and CD are both equal to \(4\sqrt{2}\).
- BC and DA are both equal to \(3\sqrt{2}\).
Next, we can check the slopes to confirm that the angles are right angles (i.e., the product of slopes of adjacent sides should be -1):
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Slope of AB: \[ \text{slope of AB} = \frac{-4 - 0}{4 - 0} = \frac{-4}{4} = -1 \]
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Slope of BC: \[ \text{slope of BC} = \frac{-1 - (-4)}{7 - 4} = \frac{3}{3} = 1 \]
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Slope of CD: \[ \text{slope of CD} = \frac{3 - (-1)}{3 - 7} = \frac{4}{-4} = -1 \]
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Slope of DA: \[ \text{slope of DA} = \frac{0 - 3}{0 - 3} = \frac{-3}{-3} = 1 \]
Now we see that:
- The slopes of AB and BC are -1 and 1, which gives a product of -1, confirming a right angle.
- The same happens for CD and DA.
Thus, the rectangle formed by the given points has side lengths of \(AB = 4\sqrt{2}\) and \(BC = 3\sqrt{2}\).
Finally, we calculate the area: \[ \text{Area} = \text{length} \times \text{width} = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24 \]
The exact area of the rectangle is \( \boxed{24} \).
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