To find the area of the rectangle defined by the vertices (0,0), (2,2), (9,-4), and (7,-6), we can begin by checking if the vertices indeed form a rectangle.
Firstly, to determine the lengths of the sides of the rectangle, we will calculate the distances between the vertices.
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Distance between (0,0) and (2,2): \[ d = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between (2,2) and (9,-4): \[ d = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
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Distance between (9,-4) and (7,-6): \[ d = \sqrt{(7-9)^2 + (-6+4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between (7,-6) and (0,0): \[ d = \sqrt{(0-7)^2 + (0+6)^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} \]
From these calculations, we observe that:
- The opposite sides (0,0)-(2,2) and (7,-6)-(9,-4) have equal lengths of \(2\sqrt{2}\).
- The other opposite sides (2,2)-(9,-4) and (0,0)-(7,-6) have equal lengths of \(\sqrt{85}\).
Now that we have confirmed it is a rectangle, we can find the area. The area \(A\) of a rectangle is given by the product of its width (\(w\)) and height (\(h\)):
Assuming the lengths of the sides are:
- \(w = 2\sqrt{2}\)
- \(h = \sqrt{85}\)
The area can be calculated as: \[ A = w \times h = (2\sqrt{2}) \times (\sqrt{85}) = 2\sqrt{170} \]
To get an approximate area, we can calculate: \[ \sqrt{170} \approx 13.038 \] So, \[ 2\sqrt{170} \approx 2 \times 13.038 \approx 26.076 \]
Thus, the approximate area of the rectangle is: \[ \boxed{26.08} \]
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