Question
A rectangle is drawn on a coordinate plane with vertices (0,0) , (2,2) , (9,−4) , and (7,−6) . Using the coordinates, what is the approximate area of the rectangle? (1 point) Responses 26.08 units2 26.07 units squared 12.05 units2 12.05 units squared 24.1 units2 24.1 units squared 52.2 units2 52.2 units squared
Answers
GPT-4o mini
To find the area of the rectangle given the vertices \( (0,0) \), \( (2,2) \), \( (9,-4) \), and \( (7,-6) \), we first need to identify the lengths of the sides of the rectangle.
1. **Finding the lengths of the sides**:
We calculate the lengths of the sides by determining the distances between the relevant points.
- The distance between \( (0, 0) \) and \( (2, 2) \):
\[
d_1 = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- The distance between \( (7, -6) \) and \( (9, -4) \):
\[
d_2 = \sqrt{(9 - 7)^2 + (-4 + 6)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
The pairs \( (0,0) \) to \( (2,2) \) and \( (7,-6) \) to \( (9,-4) \) are equal, suggesting these lengths are the same.
2. **Calculating the width and height**:
Next, we can calculate the other two dimensions:
- The distance between \( (0, 0) \) and \( (7, -6) \):
\[
d_3 = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
- The distance between \( (2, 2) \) and \( (9, -4) \):
\[
d_4 = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
Both distances \( d_3 \) and \( d_4 \) are the same, which suggests that \( (0,0) \) to \( (7,-6) \) and \( (2,2) \) to \( (9,-4) \) are opposite sides of the rectangle.
3. **Calculating the area**:
The area \( A \) of a rectangle can be computed as:
\[
A = \text{length} \times \text{width}
\]
Assuming the dimensions are the calculated distances, the area of the rectangle is:
\[
A = d_1 \times d_3 = (2\sqrt{2}) \times \sqrt{85}
\]
To compute an approximate value:
\[
A \approx 2(1.414) \times (9.220) \approx 26.07 \text{ units}^2
\]
Thus, the approximate area of the rectangle is \( 26.07 \) units squared.
Therefore, the correct response is:
**26.07 units squared**.
1. **Finding the lengths of the sides**:
We calculate the lengths of the sides by determining the distances between the relevant points.
- The distance between \( (0, 0) \) and \( (2, 2) \):
\[
d_1 = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- The distance between \( (7, -6) \) and \( (9, -4) \):
\[
d_2 = \sqrt{(9 - 7)^2 + (-4 + 6)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
The pairs \( (0,0) \) to \( (2,2) \) and \( (7,-6) \) to \( (9,-4) \) are equal, suggesting these lengths are the same.
2. **Calculating the width and height**:
Next, we can calculate the other two dimensions:
- The distance between \( (0, 0) \) and \( (7, -6) \):
\[
d_3 = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
- The distance between \( (2, 2) \) and \( (9, -4) \):
\[
d_4 = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
Both distances \( d_3 \) and \( d_4 \) are the same, which suggests that \( (0,0) \) to \( (7,-6) \) and \( (2,2) \) to \( (9,-4) \) are opposite sides of the rectangle.
3. **Calculating the area**:
The area \( A \) of a rectangle can be computed as:
\[
A = \text{length} \times \text{width}
\]
Assuming the dimensions are the calculated distances, the area of the rectangle is:
\[
A = d_1 \times d_3 = (2\sqrt{2}) \times \sqrt{85}
\]
To compute an approximate value:
\[
A \approx 2(1.414) \times (9.220) \approx 26.07 \text{ units}^2
\]
Thus, the approximate area of the rectangle is \( 26.07 \) units squared.
Therefore, the correct response is:
**26.07 units squared**.