To find the area of the rectangle given its vertices A(0, 0), B(2, 2), C(9, −4), and D(7, −6), we first need to verify which pairs of points represent the sides of the rectangle by calculating the distances between the vertices.
We will use the distance formula, which is defined as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Now let's calculate the distances for the pairs of vertices:
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Distance AB: \[ AB = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance BC: \[ BC = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
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Distance CD: \[ CD = \sqrt{(7 - 9)^2 + (-6 - (-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance DA: \[ DA = \sqrt{(0 - 7)^2 + (0 - (-6))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
Now, we have the lengths of the sides:
- Length AB = 2√2 (which corresponds to height)
- Length BC = √85 (which corresponds to the base)
- Length CD = 2√2 (same as AB)
- Length DA = √85 (same as BC)
Height (h) = 2√2 and Base (b) = √85
Now let's find the area (A) of the rectangle: \[ A = \text{base} \times \text{height} \] \[ A = \sqrt{85} \times 2\sqrt{2} \] \[ A = 2\sqrt{170} \] Calculating \(2\sqrt{170}\): \[ \sqrt{170} \approx 13.0384 \implies 2\sqrt{170} \approx 26.0768 \]
Thus, the area of the rectangle is approximately \(26.08\) units².
The closest answer is 26.08 units².