To find the exact area of the rectangle defined by the given points, we first need to determine if these points represent a rectangle and then calculate its area.
The coordinates of the points are:
- Point A: (0, 0)
- Point B: (4, -4)
- Point C: (7, -1)
- Point D: (3, 3)
We will check if these points can form a rectangle. For a rectangle, opposite sides must be equal in length and the angles must be right angles.
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Calculating the lengths of the sides:
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Side AB (between points A and B): \[ \text{Length} = \sqrt{(4 - 0)^2 + (-4 - 0)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Side BC (between points B and C): \[ \text{Length} = \sqrt{(7 - 4)^2 + (-1 - (-4))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Side CD (between points C and D): \[ \text{Length} = \sqrt{(3 - 7)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Side DA (between points D and A): \[ \text{Length} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Checking the lengths for opposite sides:
- Side AB and CD have the same length: \( 4\sqrt{2} \)
- Side BC and DA have the same length: \( 3\sqrt{2} \)
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Determining the area of the rectangle: The area of a rectangle is calculated as the product of the lengths of two adjacent sides: \[ \text{Area} = \text{Length}(AB) \times \text{Length}(BC) = (4\sqrt{2}) \times (3\sqrt{2}) = 12 \times 2 = 24 \]
Thus, the exact area of the rectangle is 24 square units.
The correct answer is D. 24 units^2.
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