To compute the area of the rectangle given the coordinates of its vertices, let's first verify the coordinates and determine the lengths of the sides.
The vertices provided are:
- A: (0, 0)
- B: (4, -4)
- C: (7, -1)
- D: (3, 3)
However, the shape described needs to be assessed better since the given points don't seem to define a rectangle.
Let's calculate the distances between consecutive points to see if sides are perpendicular and could form a rectangle. The key here is not necessarily to find lengths of edges but their relationships to understand if the provided points can form a rectangle.
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Distance AB: \[ d_{AB} = \sqrt{(4 - 0)^2 + (-4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance BC: \[ d_{BC} = \sqrt{(7 - 4)^2 + (-1 + 4)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
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Distance CD: \[ d_{CD} = \sqrt{(3 - 7)^2 + (3 + 1)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance DA: \[ d_{DA} = \sqrt{(3 - 0)^2 + (3 - 0)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]
From the distances, we see that \( AB = CD \) and \( BC = DA \), suggesting these edges could be parallel. However, we still need to prove that the angles are right angles (to confirm it's a rectangle) or identify if it is a specific type of quadrilateral.
Now, we will find the area using the coordinates with the Shoelace Theorem.
According to the Shoelace Theorem, if a polygon's vertices are given as \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\), the area \( A \) can be calculated as: \[ A = \frac{1}{2} \left| \sum_{i=1}^{n}(x_i y_{i+1} - y_i x_{i+1}) \right| \]
We will apply this theorem, ensuring to loop back to the first point:
- A(0, 0)
- B(4, -4)
- C(7, -1)
- D(3, 3)
- (Back to A(0, 0))
Calculating:
- \((0)(-4) + (4)(-1) + (7)(3) + (3)(0) = 0 - 4 + 21 + 0 = 17\)
- \((0)(4) + (-4)(7) + (-1)(3) + (3)(0) = 0 - 28 - 3 + 0 = -31\)
Therefore, \[ A = \frac{1}{2} \left| 17 - (-31) \right| = \frac{1}{2} \left| 17 + 31 \right| = \frac{1}{2} \times 48 = 24 \text{ units}^2 \]
Thus, the exact area of the shape is 24 units squared.
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