Question

A rectangle is drawn on a coordinate plane with vertices (0,0)%0D%0A(%0D%0A0%0D%0A,%0D%0A0%0D%0A)%0D%0A, (2,2)%0D%0A(%0D%0A2%0D%0A,%0D%0A2%0D%0A)%0D%0A, (9,−4)%0D%0A(%0D%0A9%0D%0A,%0D%0A−%0D%0A4%0D%0A)%0D%0A, and (7,−6)%0D%0A(%0D%0A7%0D%0A,%0D%0A−%0D%0A6%0D%0A)%0D%0A. Using the coordinates, what is the approximate area of the rectangle?(1 point)%0D%0AResponses%0D%0A%0D%0A12.05 units2%0D%0A12.05%0D%0A %0D%0Aunits%0D%0A2%0D%0A12.05 units squared%0D%0A%0D%0A52.2 units2%0D%0A52.2%0D%0A %0D%0Aunits%0D%0A2%0D%0A52.2 units squared%0D%0A%0D%0A25.76 units2%0D%0A25.76%0D%0A %0D%0Aunits%0D%0A2%0D%0A26.07 units squared%0D%0A%0D%0A24.1 units2

Answer 1

To find the approximate area of the rectangle defined by the given vertices \((0,0)\), \((2,2)\), \((9,-4)\), and \((7,-6)\), we first need to verify that these points actually form a rectangle and then calculate the area.

1. **Identifying the Rectangle**:
For a quadrilateral to be a rectangle, opposite sides must be equal in length, and adjacent sides must be perpendicular. We can find the distances between the points.

Calculate the length of each side:
- Distance between \( (0,0) \) and \( (2,2) \):
\[
\sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
- Distance between \( (2,2) \) and \( (9,-4) \):
\[
\sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]
- Distance between \( (9,-4) \) and \( (7,-6) \):
\[
\sqrt{(7-9)^2 + (-6+4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
- Distance between \( (7,-6) \) and \( (0,0) \):
\[
\sqrt{(0-7)^2 + (0+6)^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]

From these calculations, we can see that opposite sides are equal, indicating that the shape formed by these points is a rectangle.

2. **Calculating the Area**:
The area \(A\) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
We have \( \text{length} \approx 9.22 \) and \( \text{width} \approx 2.83 \).

Therefore, the approximate area is:
\[
A \approx 9.22 \times 2.83 \approx 26.07 \text{ units}^2
\]

So, the area of the rectangle is approximately **26.07 units²**.