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 24.1 units2 24.1 units squared 52.2 units2 52.2 units squared 12.05 units2
Answers
GPT-4o mini
To find the area of the rectangle with the provided vertices (0,0), (2,2), (9,−4), and (7,−6), we should first confirm whether these points indeed form a rectangle. The slopes of the diagonals should be equal if they are opposite corners of a rectangle.
Let's label the points for simplicity:
- A = (0,0)
- B = (2,2)
- C = (9,-4)
- D = (7,-6)
Next, we can find the lengths of the sides of the rectangle by calculating the distances between the adjacent points.
1. **Distance AB** (between (0,0) and (2,2)):
\[
AB = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
2. **Distance BC** (between (2,2) and (9,-4)):
\[
BC = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
3. **Distance CD** (between (9,-4) and (7,-6)):
\[
CD = \sqrt{(7-9)^2 + (-6-(-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. **Distance DA** (between (7,-6) and (0,0)):
\[
DA = \sqrt{(0-7)^2 + (0-(-6))^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]
Thus, we see that sides \( AB \) and \( CD \) are of length \( 2\sqrt{2} \), and sides \( BC \) and \( DA \) are of length \( \sqrt{85} \). This confirms that opposite sides are equal in length, suggesting that we indeed have a rectangle (assuming they meet at right angles).
Finally, the area \( A \) of the rectangle is given by:
\[
A = \text{length} \times \text{width} = (2\sqrt{2}) \times (\sqrt{85})
\]
\[
A = 2\sqrt{170}
\]
Calculating \( \sqrt{170} \):
\[
\sqrt{170} \approx 13.038
\]
Thus,
\[
A \approx 2(13.038) \approx 26.076
\]
Now rounding this value gives \( 26.1 \) units squared.
From the available responses, the closest match is **26.07 units squared**.
Let's label the points for simplicity:
- A = (0,0)
- B = (2,2)
- C = (9,-4)
- D = (7,-6)
Next, we can find the lengths of the sides of the rectangle by calculating the distances between the adjacent points.
1. **Distance AB** (between (0,0) and (2,2)):
\[
AB = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
2. **Distance BC** (between (2,2) and (9,-4)):
\[
BC = \sqrt{(9-2)^2 + (-4-2)^2} = \sqrt{7^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}
\]
3. **Distance CD** (between (9,-4) and (7,-6)):
\[
CD = \sqrt{(7-9)^2 + (-6-(-4))^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
4. **Distance DA** (between (7,-6) and (0,0)):
\[
DA = \sqrt{(0-7)^2 + (0-(-6))^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]
Thus, we see that sides \( AB \) and \( CD \) are of length \( 2\sqrt{2} \), and sides \( BC \) and \( DA \) are of length \( \sqrt{85} \). This confirms that opposite sides are equal in length, suggesting that we indeed have a rectangle (assuming they meet at right angles).
Finally, the area \( A \) of the rectangle is given by:
\[
A = \text{length} \times \text{width} = (2\sqrt{2}) \times (\sqrt{85})
\]
\[
A = 2\sqrt{170}
\]
Calculating \( \sqrt{170} \):
\[
\sqrt{170} \approx 13.038
\]
Thus,
\[
A \approx 2(13.038) \approx 26.076
\]
Now rounding this value gives \( 26.1 \) units squared.
From the available responses, the closest match is **26.07 units squared**.