To determine the area of the rectangle with the given vertices (0, 0), (2, 2), (9, -4), and (7, -6), we first need to confirm that these points actually form a rectangle.
The properties of a rectangle include that opposite sides must be equal in length, and the angles must be 90 degrees.
Let's start by checking the distances between the points:
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Distance between (0, 0) and (2, 2): \[ d_1 = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between (2, 2) and (9, -4): \[ d_2 = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
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Distance between (9, -4) and (7, -6): \[ d_3 = \sqrt{(7-9)^2 + (-6+4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
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Distance between (7, -6) and (0, 0): \[ d_4 = \sqrt{(0-7)^2 + (0-(-6))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
From our calculations, we see that:
- The lengths of opposite sides are equal (d1 = d3 = 2√2 and d2 = d4 = √85).
- The pairs of lengths show that the points can be viewed as two separate lengths of rectangle sides.
Next, the area of a rectangle can be calculated using the formula: \[ \text{Area} = \text{length} \times \text{width} \]
Where:
- Length (d2 or d4) = √85
- Width (d1 or d3) = 2√2
Thus, the area can be calculated as follows: \[ \text{Area} = (2\sqrt{2}) \times (\sqrt{85}) \]
At this point: \[ \text{Area} = 2\sqrt{170} \]
Now, we can compute the numerical value: \[ \sqrt{170} \approx 13.038 \] So, \[ \text{Area} \approx 2 \times 13.038 \approx 26.076 \]
Hence, the approximate area of the rectangle is: \[ \text{Area} \approx 26.08 \]
Thus, the final answer is approximately 26.08 square units.
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