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? (1 point) Responses 12.05 units2 12.05 units2 52.2 units2 52.2 units2 24.1 units2 24.1 units2 26.08 units2

Answer 1

To find the area of the rectangle defined by the vertices (0,0), (2,2), (9,-4), and (7,-6), we first confirm that these points correctly form a rectangle. We do this by checking the slopes and lengths of the sides.

1. **Calculate the lengths of the sides**:
- The distance between (0, 0) and (2, 2):
\[
\text{Distance} = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}
\]

- The distance between (2, 2) and (9, -4):
\[
\text{Distance} = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]

- The distance between (9, -4) and (7, -6):
\[
\text{Distance} = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

- The distance between (7, -6) and (0, 0):
\[
\text{Distance} = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]

Since opposite sides are equal in length (\(2\sqrt{2}\) and \(\sqrt{85}\)), these points can define a rectangle.

2. **Calculate the area of the rectangle**:
The area of a rectangle can be calculated as:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Therefore, we can plug in the distances:
\[
\text{Area} = (2\sqrt{2}) \times \sqrt{85}
\]

Now, we compute the numerical value:
\[
\sqrt{2} \approx 1.414
\]
\[
\sqrt{85} \approx 9.219
\]
So:
\[
2\sqrt{2} \approx 2 \times 1.414 \approx 2.828
\]
Now multiplying:
\[
\text{Area} \approx 2.828 \times 9.219 \approx 26.083
\]

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

Therefore, the closest response is \(26.08 \text{ units}^2\).