To find the area of the rectangle with vertices at $(0, 0)$, $(2, 2)$, $(9, -4)$, and $(7, -6)$, let's proceed with these steps:

1. Identify the Lengths of Sides: Since this is a rectangle, we can calculate the lengths of two adjacent sides (the width and height) and multiply them to get the area.

2. Calculate Side Lengths Using Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

3. Side Calculations:

   • 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 $(9, -4)$: $\sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22$

4. Calculate Area: $\text{Area} = 2.83 \times 9.22 \approx 26.08 \text{ square units}$

Answer:

The correct answer is:

• 26.08 units²

Would you like further explanation or additional details?

Here are 5 related questions to expand your understanding:

1. How do we verify that a given set of vertices forms a rectangle?
2. What is the general formula for calculating the area of a rectangle in a coordinate plane?
3. How does the distance formula relate to finding the side lengths of polygons?
4. Can the method used here be applied to other quadrilaterals, such as parallelograms?
5. How would we calculate the perimeter of this rectangle?

Tip: The distance formula is fundamental for analyzing shapes on the coordinate plane.