To determine whether the given vertices form a rectangle and to find its area, we need to analyze the coordinates provided: (0,0), (2,2), (9,−4), and (7,−6).
First, let's organize the coordinates:
- \( A(0, 0) \)
- \( B(2, 2) \)
- \( C(9, -4) \)
- \( D(7, -6) \)
Next, we can calculate the distances between the points to check for the sides of a rectangle:
-
Distance \( AB \): \[ d_{AB} = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
-
Distance \( BC \): \[ d_{BC} = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
-
Distance \( CD \): \[ d_{CD} = \sqrt{(7-9)^2 + (-6 - (-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
-
Distance \( DA \): \[ d_{DA} = \sqrt{(0-7)^2 + (0-(-6))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85} \]
Now, we can group the distances:
- \( AB \) and \( CD \) are both \( 2\sqrt{2} \).
- \( BC \) and \( DA \) are both \( \sqrt{85} \).
Since opposite sides are equal and all angles would be right angles (which we can check further), the four points (0,0), (2,2), (9,-4), and (7,-6) represent vertices of a rectangle.
Next, we will compute the area of the rectangle, which is given by the formula: \[ \text{Area} = \text{length} \times \text{width} \] Here, we can take:
- Length = \( \sqrt{85} \)
- Width = \( 2\sqrt{2} \)
Now we calculate the area: \[ \text{Area} = \sqrt{85} \times 2\sqrt{2} = 2\sqrt{170} \]
To approximate the numerical value of the area, let's compute:
- Approximating \( \sqrt{170} \): \[ \sqrt{170} \approx 13.0384 \quad (\text{using a calculator}) \] Therefore, \[ \text{Area} \approx 2 \times 13.0384 \approx 26.0768 \]
Thus, the approximate area of the rectangle is: \[ \boxed{26.08} \]
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