To determine the area of a rectangle defined by specific vertices, let's first verify whether the given coordinates truly form a rectangle by checking the distances between these points.
The vertices given are:
- \( A(0, 0) \)
- \( B(2, 2) \)
- \( C(9, -4) \)
- \( D(7, -6) \)
Next, we can calculate the distance between each pair of vertices.
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Distance between A and B: \[ AB = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between B and C: \[ BC = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
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Distance between C and D: \[ CD = \sqrt{(7-9)^2 + (-6-(-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between D and A: \[ DA = \sqrt{(0-7)^2 + (0 - (-6))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
Now, we can see that:
- \( AB = CD = 2\sqrt{2} \)
- \( BC = DA = \sqrt{85} \)
Since opposite sides are equal, we must also ensure that adjacent sides are perpendicular.
Next, we can check the slopes:
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Slope of AB: \[ \text{slope of } AB = \frac{2-0}{2-0} = 1 \]
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Slope of BC: \[ \text{slope of } BC = \frac{-4-2}{9-2} = \frac{-6}{7} \]
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Slope of CD: \[ \text{slope of } CD = \frac{-6 - (-4)}{7 - 9} = \frac{-2}{-2} = 1 \]
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Slope of DA: \[ \text{slope of } DA = \frac{0-(-6)}{0-7} = \frac{6}{-7} = -\frac{6}{7} \]
Checking perpendicularity:
- The slopes \( 1 \) (AB and CD) are parallel.
- The slopes \( -\frac{6}{7} \) (DA) and \( \frac{7}{6} \) are negative reciprocals confirming they are indeed perpendicular.
Thus, we have established that \( AB \) is parallel to \( CD \) and \( BC \) is parallel to \( DA \). This confirms that we indeed have a rectangle.
Now, finally, we calculate the area of the rectangle using the lengths of the sides:
- The length of side \( AB \) (or \( CD \)) is \( 2\sqrt{2} \).
- The length of side \( BC \) (or \( DA \)) is \( \sqrt{85} \).
The area \( A \) of the rectangle is given by: \[ A = \text{length} \times \text{width} = (2\sqrt{2}) \times (\sqrt{85}) = 2\sqrt{170} \approx 2 \times 13.0384 = 26.0768 \]
Thus, the approximate area of the rectangle is: \[ \boxed{26.08} \]