To find the area of the rectangle defined by the vertices (0, 0), (2, 2), (9,-4), and (7, -6), we need to check whether these points actually represent the vertices of a rectangle.
First, let's verify the coordinates by plotting or calculating the lengths of the sides between points.
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Finding Distances Between Points:
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The distance between (0, 0) and (2, 2): \[ d_1 = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \]
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The distance between (2, 2) and (9, -4): \[ d_2 = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22 \]
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The distance between (9, -4) and (7, -6): \[ d_3 = \sqrt{(7-9)^2 + (-6-(-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \]
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The distance between (7, -6) and (0, 0): \[ d_4 = \sqrt{(0-7)^2 + (0-(-6))^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22 \]
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Checking if Opposite Sides are Equal:
- We find that opposite sides have the same lengths:
- \( d_1 \approx d_3 \) (both \( \sqrt{8} \approx 2.83 \))
- \( d_2 \approx d_4 \) (both \( \sqrt{85} \approx 9.22 \))
- We find that opposite sides have the same lengths:
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Calculating Area: Since it appears that the given vertices form a rectangle, we can use the lengths of its sides to find the area. The lengths of the sides are approximately:
- \( \text{Length} = \sqrt{85} \approx 9.22 \)
- \( \text{Width} = \sqrt{8} \approx 2.83 \)
Therefore, the area \( A \) can be calculated as: \[ A \approx \text{Length} \times \text{Width} \approx 9.22 \times 2.83 \approx 26.12 \text{ units}^2 \]
However, since our calculated area does not match any option precisely, let’s check the area directly derived from the coordinates with the shoelace formula.
Shoelace Formula:
Using the vertices:
- \( (0, 0) \)
- \( (2, 2) \)
- \( (9, -4) \)
- \( (7, -6) \)
Calculating using the shoelace formula: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \] Substituting the values: \[ \text{Area} = \frac{1}{2} \left| 0 \cdot 2 + 2 \cdot (-4) + 9 \cdot (-6) + 7 \cdot 0 - (0 \cdot 2 + 2 \cdot 9 + (-4) \cdot 7 + (-6) \cdot 0) \right| \]
Calculating: \[ = \frac{1}{2} \left| 0 - 8 - 54 + 0 - (0 + 18 - 28 + 0) \right| \] \[ = \frac{1}{2} \left| -62 - (-10) \right| = \frac{1}{2} \left| -62 + 10 \right| = \frac{1}{2} \times 52 = 26 \]
Thus, finding the closest match within the given options: None of the provided options closely represent 26 units\(^2\).
Unfortunately, it seems the choices provided may not align with the correct computations for the area from the provided coordinates. If strictly looking for the closest estimation within the choices, 25.76 would be the nearest approximation.
Thus, the best choice is:
- 25.76 units\(^2\).