To find the area of the rectangle formed by the points A(0,0), B(4,-4), C(7,-1), and D(3,3), we will first check whether these points indeed form a rectangle.
For four points to form a rectangle, the following conditions must hold:
- The opposite sides must be equal in length.
- The diagonals must also be equal in length.
Let's compute the distances between these points:
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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: \[ DA = \sqrt{(3-0)^2 + (3-0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
Now we check the diagonals:
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Distance AC: \[ AC = \sqrt{(7-0)^2 + (-1-0)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2} \]
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Distance BD: \[ BD = \sqrt{(3-4)^2 + (3-(-4))^2} = \sqrt{(-1)^2 + (7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]
The distances reveal:
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Lengths of opposite sides:
- AB = CD = \( 4\sqrt{2} \)
- BC = DA = \( 3\sqrt{2} \)
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Lengths of diagonals:
- AC = BD = \( 5\sqrt{2} \)
Since we have verified that opposite sides are equal and diagonals are equal, points A, B, C, and D indeed form a rectangle.
Now we can calculate the area: The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{width} \]
From the computed distances, the length is \( 4\sqrt{2} \) and the width is \( 3\sqrt{2} \).
\[ A = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24 \]
Thus, the exact area of the rectangle is \( \boxed{24} \).
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