Question

A rectangle is drawn on a coordinate plane with verticies (0,0), (2,2), (9,-4), and (7,-6). Using the coordinates, what is the approximate area of the rectangle?

a. 24.1 units^2
b. 52.2 units^2
c. 26.08 units^2
d. 12.05 units^2

Answer 1

To find the area of the rectangle formed by the given vertices, we need to first confirm that these points actually form a rectangle and then calculate the area.

The vertices given are:
- \( A(0, 0) \)
- \( B(2, 2) \)
- \( C(9, -4) \)
- \( D(7, -6) \)

We can find the lengths of the sides of the quadrilateral by calculating the distances between the points:

1. Distance \( AB \):
\[
AB = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

2. Distance \( BC \):
\[
BC = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]

3. Distance \( CD \):
\[
CD = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

4. Distance \( DA \):
\[
DA = \sqrt{(0 - 7)^2 + (0 + 6)^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]

From the calculations, we see that:
- \( AB = CD = 2\sqrt{2} \)
- \( BC = DA = \sqrt{85} \)

This indicates that the opposite sides are equal, and since we have one right angle at each vertex, this implies that the quadrilateral is indeed a rectangle.

Now, we can calculate the area of the rectangle using the lengths of its sides:
\[
\text{Area} = \text{length} \times \text{width} = AB \times BC = (2\sqrt{2}) \times (\sqrt{85}) = 2\sqrt{170}
\]

Next, we calculate \( 2\sqrt{170} \):
\[
\sqrt{170} \approx 13.038
\]
\[
2\sqrt{170} \approx 2 \times 13.038 \approx 26.076
\]

Thus, the area of the rectangle is approximately \( 26.08 \) units².

The closest answer choice given is:
**c. 26.08 units²**.