Question
A rectangle is drawn on a coordinate plane with vertices (0,0), (5,3), (7,0), and (2,−3)
. Using the coordinates, what is the exact area of the rectangle?
. Using the coordinates, what is the exact area of the rectangle?
Answers
GPT-4o mini
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:
1. \((0,0)\) - This is the origin.
2. \((5,3)\) - This point is to the right and above the origin.
3. \((7,0)\) - This point is to the right of the origin and on the x-axis.
4. \((2,-3)\) - 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:
1. Length between \((0,0)\) and \((7,0)\): This distance is \(7 - 0 = 7\).
2. Length between \((0,0)\) and \((5,3)\): We use the distance formula:
\[
\sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34}
\]
3. Length between \((5,3)\) and \((2,-3)\): Using the distance formula:
\[
\sqrt{(2 - 5)^2 + (-3 - 3)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45}
\]
4. 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.
First, let's visualize the points:
1. \((0,0)\) - This is the origin.
2. \((5,3)\) - This point is to the right and above the origin.
3. \((7,0)\) - This point is to the right of the origin and on the x-axis.
4. \((2,-3)\) - 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:
1. Length between \((0,0)\) and \((7,0)\): This distance is \(7 - 0 = 7\).
2. Length between \((0,0)\) and \((5,3)\): We use the distance formula:
\[
\sqrt{(5-0)^2 + (3-0)^2} = \sqrt{25 + 9} = \sqrt{34}
\]
3. Length between \((5,3)\) and \((2,-3)\): Using the distance formula:
\[
\sqrt{(2 - 5)^2 + (-3 - 3)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45}
\]
4. 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.