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
52.2 units252.2 units2
12.05 units212.05 units2
26.08 units226.08 units2
24.1 units2
Responses
52.2 units252.2 units2
12.05 units212.05 units2
26.08 units226.08 units2
24.1 units2
Answers
GPT-4o mini
To find the area of a rectangle given its vertices, we need to first verify that the given points form a rectangle. The vertices provided are (0,0), (2,2), (9,-4), and (7,-6).
Let's check the distances between the points to ensure they form a rectangle:
1. **Distance between (0,0) and (2,2)**:
\[
d = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
2. **Distance between (0,0) and (7,-6)**:
\[
d = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]
3. **Distance between (2,2) and (9,-4)**:
\[
d = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]
4. **Distance between (2,2) and (7,-6)**:
\[
d = \sqrt{(7 - 2)^2 + (-6 - 2)^2} = \sqrt{(5)^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43
\]
5. **Distance between (9,-4) and (7,-6)**:
\[
d = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
6. **Distance between (0,0) and (9,-4)**:
\[
d = \sqrt{(9 - 0)^2 + (-4 - 0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85
\]
These calculations reveal that the sides are not perpendicular, which means the figure formed by the vertices is not a rectangle.
To find the area directly, we can use the formula for the area of a convex shape defined by vertices:
\[
\text{Area} = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |
\]
Using coordinates (0,0), (2,2), (9,-4), (7,-6):
\[
= \frac{1}{2} | 0*2 + 2*(-4) + 9*(-6) + 7*0 - (0*2 + 2*9 + (-4)*7 + (-6)*0) |
\]
\[
= \frac{1}{2} | 0 - 8 - 54 + 0 - (0 + 18 - 28 + 0) |
\]
\[
= \frac{1}{2} | -62 - (-10) |
\]
\[
= \frac{1}{2} | -62 + 10 | = \frac{1}{2} * 52 = 26
\]
The area of the figure formed by these points is approximately 26 square units.
Thus, the closest response from the options provided is:
**26.08 units²**.
Let's check the distances between the points to ensure they form a rectangle:
1. **Distance between (0,0) and (2,2)**:
\[
d = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
2. **Distance between (0,0) and (7,-6)**:
\[
d = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]
3. **Distance between (2,2) and (9,-4)**:
\[
d = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85} \approx 9.22
\]
4. **Distance between (2,2) and (7,-6)**:
\[
d = \sqrt{(7 - 2)^2 + (-6 - 2)^2} = \sqrt{(5)^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43
\]
5. **Distance between (9,-4) and (7,-6)**:
\[
d = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83
\]
6. **Distance between (0,0) and (9,-4)**:
\[
d = \sqrt{(9 - 0)^2 + (-4 - 0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85
\]
These calculations reveal that the sides are not perpendicular, which means the figure formed by the vertices is not a rectangle.
To find the area directly, we can use the formula for the area of a convex shape defined by vertices:
\[
\text{Area} = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |
\]
Using coordinates (0,0), (2,2), (9,-4), (7,-6):
\[
= \frac{1}{2} | 0*2 + 2*(-4) + 9*(-6) + 7*0 - (0*2 + 2*9 + (-4)*7 + (-6)*0) |
\]
\[
= \frac{1}{2} | 0 - 8 - 54 + 0 - (0 + 18 - 28 + 0) |
\]
\[
= \frac{1}{2} | -62 - (-10) |
\]
\[
= \frac{1}{2} | -62 + 10 | = \frac{1}{2} * 52 = 26
\]
The area of the figure formed by these points is approximately 26 square units.
Thus, the closest response from the options provided is:
**26.08 units²**.