To find the area of the rectangle defined by the vertices (0,0), (5,3), (7,0), and (2,-3), we can first confirm that these points actually form a rectangle.
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Step 1: Verify the vertices form a rectangle
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Identify the lengths of sides:
- The distance between (0, 0) and (7, 0) is simply the x-coordinate difference: \[ \text{Length} = 7 - 0 = 7 \]
- The distance between (0, 0) and (5, 3) is calculated using the distance formula: \[ \text{Length} = \sqrt{(5 - 0)^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]
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Calculate other side lengths:
- The distance between (2, -3) and (5, 3): \[ \text{Length} = \sqrt{(5 - 2)^2 + (3 - (-3))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} \]
Step 2: Determine whether the rectangle's vertices define valid dimensions
Simply calculating lengths isn't sufficient; we need the appropriate opposite sides.
Looking at these coordinates, note that they do not successfully define a rectangle, as vertices do not match opposite pairs properly.
Step 3: Not all vertices form a closed shape successfully
Let's find the area using the shoelace formula to ensure whether they outline a valid polygon:
The shoelace formula for the area of a polygon given vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is:
\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \] where \( (x_{n+1}, y_{n+1}) = (x_1, y_1) \).
So for our points: (0,0), (5,3), (7,0), (2,-3).
Plugging in:
\[ \text{Area} = \frac{1}{2} \left| 03 + 50 + 7*(-3) + 20 - (05 + 37 + 02 + (-3)*0)\right| \] \[ = \frac{1}{2} \left| 0 + 0 - 21 + 0 - (0 + 21 + 0 + 0) \right| \] \[ = \frac{1}{2} \left| -21 - 21 \right| = \frac{1}{2} \cdot 42 = 21 \]
Conclusion
\[ \text{Area} = 21 \text{ square units} \]
Therefore, the exact area of the rectangle based on the vertices is:
\[ \boxed{21.02 \text{ units squared}} \]