To compute the area of the rectangle formed by the given points A (0,0), B (4,-4), C (7,-1), and D (3,3), we first need to identify which pairs of vertices form the sides of the rectangle.
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Let's find the lengths of the sides formed by the pairs:
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Length between A (0,0) and B (4,-4): \[ \text{Length} = \sqrt{(4-0)^2 + (-4-0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Length between A (0,0) and D (3,3): \[ \text{Length} = \sqrt{(3-0)^2 + (3-0)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Length between B (4,-4) and C (7,-1): \[ \text{Length} = \sqrt{(7-4)^2 + (-1-(-4))^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Length between C (7,-1) and D (3,3): \[ \text{Length} = \sqrt{(3-7)^2 + (3-(-1))^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Since we have found two pairs of parallel sides with lengths \(4\sqrt{2}\) and \(3\sqrt{2}\), we can identify the dimensions of the rectangle:
- Width (short side) = \(3\sqrt{2}\)
- Height (long side) = \(4\sqrt{2}\)
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The area \(A\) of a rectangle is calculated as: \[ A = \text{Width} \times \text{Height} = (3\sqrt{2}) \times (4\sqrt{2}) = 12 \times 2 = 24 \text{ units}^2 \]
Therefore, the exact area of the rectangle is 24 units².
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