Question
a rectangle is drawn on a coordinate plane with vertices (0,0),(2,2),(9,-4),&(7,-6) using the coordinates what's the area of the rectangle?
A. 24.1 units
B. 25.76 units
C. 12.05 units
D. 52.2 units
A. 24.1 units
B. 25.76 units
C. 12.05 units
D. 52.2 units
Answers
GPT-4o mini
To determine the area of the rectangle formed by the vertices \((0,0)\), \((2,2)\), \((9,-4)\), and \((7,-6)\), we first need to confirm that these points actually form a rectangle.
To check this, we can find the lengths of the sides formed by the given points.
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Calculating the distances between the points:
1. Distance between \((0,0)\) and \((2,2)\):
\[
d_1 = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
2. Distance between \((2,2)\) and \((9,-4)\):
\[
d_2 = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
3. Distance between \((9,-4)\) and \((7,-6)\):
\[
d_3 = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. Distance between \((7,-6)\) and \((0,0)\):
\[
d_4 = \sqrt{(0 - 7)^2 + (0 + 6)^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]
From our calculations, we see that:
- Two pairs of opposite sides are equal:
- \(d_1 = d_3 = 2\sqrt{2}\)
- \(d_2 = d_4 = \sqrt{85}\)
Since opposite pairs of sides are equal, we can conclude that the points do indeed form a rectangle.
To find the area of the rectangle, we can multiply the lengths of the sides. We set one pair of opposite sides to calculate the area:
- Side lengths are \(2\sqrt{2}\) and \(\sqrt{85}\).
Thus, the area \(A\) is given by:
\[
A = \text{length} \times \text{width} = (2\sqrt{2}) \times (\sqrt{85})
\]
Calculating the area:
\[
A = 2\sqrt{2} \cdot \sqrt{85} = 2 \cdot \sqrt{170} \approx 2 \cdot 13.038 = 26.076 \text{ units}^2
\]
Comparing with the options given:
- None of the options match exactly; however, the closest value for \(A\) is approximately \(25.76\) units.
Thus, the closest answer is:
**B. 25.76 units**.
To check this, we can find the lengths of the sides formed by the given points.
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Calculating the distances between the points:
1. Distance between \((0,0)\) and \((2,2)\):
\[
d_1 = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
2. Distance between \((2,2)\) and \((9,-4)\):
\[
d_2 = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
3. Distance between \((9,-4)\) and \((7,-6)\):
\[
d_3 = \sqrt{(7 - 9)^2 + (-6 + 4)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. Distance between \((7,-6)\) and \((0,0)\):
\[
d_4 = \sqrt{(0 - 7)^2 + (0 + 6)^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]
From our calculations, we see that:
- Two pairs of opposite sides are equal:
- \(d_1 = d_3 = 2\sqrt{2}\)
- \(d_2 = d_4 = \sqrt{85}\)
Since opposite pairs of sides are equal, we can conclude that the points do indeed form a rectangle.
To find the area of the rectangle, we can multiply the lengths of the sides. We set one pair of opposite sides to calculate the area:
- Side lengths are \(2\sqrt{2}\) and \(\sqrt{85}\).
Thus, the area \(A\) is given by:
\[
A = \text{length} \times \text{width} = (2\sqrt{2}) \times (\sqrt{85})
\]
Calculating the area:
\[
A = 2\sqrt{2} \cdot \sqrt{85} = 2 \cdot \sqrt{170} \approx 2 \cdot 13.038 = 26.076 \text{ units}^2
\]
Comparing with the options given:
- None of the options match exactly; however, the closest value for \(A\) is approximately \(25.76\) units.
Thus, the closest answer is:
**B. 25.76 units**.