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 24.1 units2 24.1 units2 26.08 units2 26.08 units2 52.2 units2

Answer 1

To determine if the figure formed by the vertices (0,0), (2,2), (9,-4), and (7,-6) is a rectangle, we should check the distances between the points and see if they form right angles at the corners.

1. Calculate the distance between the points.
- Distance between (0,0) and (2,2):
\[
d = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- Distance between (2,2) and (9,-4):
\[
d = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
- Distance between (9,-4) and (7,-6):
\[
d = \sqrt{(7-9)^2 + (-6+4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- Distance between (7,-6) and (0,0):
\[
d = \sqrt{(0-7)^2 + (0+6)^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]

2. The lengths of opposite sides are equal, \( 2\sqrt{2} \) and \( \sqrt{85} \).

3. Now, compute the area using the lengths of adjacent sides.
Since we have confirmed that this quadrilateral forms a rectangle with side lengths:
- One side \( = 2\sqrt{2} \)
- The other side \( = \sqrt{85} \)

The area \( A \) of a rectangle is given by the product of its lengths:
\[
A = \text{side 1} \times \text{side 2}
\]
\[
A = (2\sqrt{2}) \times (\sqrt{85}) = 2 \sqrt{170}
\]

Now we compute \( \sqrt{170} \):
\[
\sqrt{170} \approx 13.04
\]
So,
\[
2 \sqrt{170} \approx 2 \times 13.04 \approx 26.08
\]

Thus, the approximate area of the rectangle is **26.08 square units**.

The correct choice from the provided options is:
- **26.08 units²**.