To determine the area of the rectangle defined by the vertices \((0,0)\), \((5,3)\), \((7,0)\), and \((2,-3)\), we need to check the arrangement of these points to confirm that they indeed form a rectangle.
First, let's visualize the points:
- \((0,0)\) - This is the origin.
- \((5,3)\) - This point is to the right and above the origin.
- \((7,0)\) - This point is to the right of the origin and on the x-axis.
- \((2,-3)\) - This point is to the left of \( (5,3) \) and below the x-axis.
Now, we can see that the points do not form a rectangle because:
- A rectangle's opposite sides must be parallel, and the given configurations do not satisfy this condition.
- The coordinates do not represent a configuration of a rectangle where opposite corners can be aligned vertically or horizontally on the Cartesian plane.
To confirm further, let's find the lengths of the sides, if possible:
- Length between \((0,0)\) and \((7,0)\): This distance is \(7 - 0 = 7\).
- Length between \((0,0)\) and \((5,3)\): We use the distance formula: \[ \sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34} \]
- Length between \((5,3)\) and \((2,-3)\): Using the distance formula: \[ \sqrt{(2 - 5)^2 + (-3 - 3)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \]
- Length between \((7,0)\) and \((2,-3)\): \[ \sqrt{(2 - 7)^2 + (-3 - 0)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \]
At this point, we can see that there are unequal lengths among the sides, and they do not fit the definition of a rectangle.
Since the points do not form a rectangle, we find that we cannot calculate its area because the vertices themselves do not create such a geometric figure.
Thus, the conclusion is:
The vertices given do not form a rectangle. Therefore, the area of this shape cannot be calculated as a rectangle.