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
25.76 units²
25.76 units²
52.2 units²
52.2 units²
12.05 units²
12.05 units²
24.1 units²
Responses
25.76 units²
25.76 units²
52.2 units²
52.2 units²
12.05 units²
12.05 units²
24.1 units²
Answers
GPT-4o mini
To determine the area of the rectangle formed by the given vertices \((0,0)\), \((2,2)\), \((9,-4)\), and \((7,-6)\), we first need to ensure that these points indeed form a rectangle.
1. **Verify the vertices form a rectangle:**
- By examining the coordinates, we can check the slopes of the lines formed by the vertices.
- The diagonal pairs are \((0,0)\) with \((9,-4)\) and \((2,2)\) with \((7,-6)\).
- Calculate the distance between opposite points:
- Distance between \((0,0)\) and \((9,-4)\):
\[
d_1 = \sqrt{(9-0)^2 + (-4-0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}
\]
- Distance between \((2,2)\) and \((7,-6)\):
\[
d_2 = \sqrt{(7-2)^2 + (-6-2)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}
\]
- The areas are not equal, so this does not confirm a rectangle.
2. **Calculate the area using the shoelace formula:**
The shoelace formula (or surveyor's formula) for the area of a polygon given vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\), \((x_4,y_4)\) is given by:
\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\]
Using the vertices, we order them as:
\[
(0,0), (2,2), (7,-6), (9,-4)
\]
Substituting into the shoelace formula:
- \(x_1=0, y_1=0\)
- \(x_2=2, y_2=2\)
- \(x_3=9, y_3=-4\)
- \(x_4=7, y_4=-6\)
Now calculate:
\[
\text{Area} = \frac{1}{2} \left| 0\cdot2 + 2\cdot(-4) + 9\cdot(-6) + 7\cdot0 - (0\cdot2 + 2\cdot9 + (-4)\cdot7 + (-6)\cdot0) \right|
\]
Calculate each term:
- \(0 + (-8) + (-54) + 0 = -62\)
- \(0 + 18 + (-28) + 0 = -10\)
Put it together:
\[
\text{Area} = \frac{1}{2} \left| -62 + 10 \right| = \frac{1}{2} \left| -52 \right| = \frac{52}{2} = 26\text{ units}^2
\]
Given the options provided, the closest response is:
- **25.76 units²**
Hence, the approximate area of the rectangle is **25.76 units²**.
1. **Verify the vertices form a rectangle:**
- By examining the coordinates, we can check the slopes of the lines formed by the vertices.
- The diagonal pairs are \((0,0)\) with \((9,-4)\) and \((2,2)\) with \((7,-6)\).
- Calculate the distance between opposite points:
- Distance between \((0,0)\) and \((9,-4)\):
\[
d_1 = \sqrt{(9-0)^2 + (-4-0)^2} = \sqrt{9^2 + (-4)^2} = \sqrt{81 + 16} = \sqrt{97}
\]
- Distance between \((2,2)\) and \((7,-6)\):
\[
d_2 = \sqrt{(7-2)^2 + (-6-2)^2} = \sqrt{5^2 + (-8)^2} = \sqrt{25 + 64} = \sqrt{89}
\]
- The areas are not equal, so this does not confirm a rectangle.
2. **Calculate the area using the shoelace formula:**
The shoelace formula (or surveyor's formula) for the area of a polygon given vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\), \((x_4,y_4)\) is given by:
\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\]
Using the vertices, we order them as:
\[
(0,0), (2,2), (7,-6), (9,-4)
\]
Substituting into the shoelace formula:
- \(x_1=0, y_1=0\)
- \(x_2=2, y_2=2\)
- \(x_3=9, y_3=-4\)
- \(x_4=7, y_4=-6\)
Now calculate:
\[
\text{Area} = \frac{1}{2} \left| 0\cdot2 + 2\cdot(-4) + 9\cdot(-6) + 7\cdot0 - (0\cdot2 + 2\cdot9 + (-4)\cdot7 + (-6)\cdot0) \right|
\]
Calculate each term:
- \(0 + (-8) + (-54) + 0 = -62\)
- \(0 + 18 + (-28) + 0 = -10\)
Put it together:
\[
\text{Area} = \frac{1}{2} \left| -62 + 10 \right| = \frac{1}{2} \left| -52 \right| = \frac{52}{2} = 26\text{ units}^2
\]
Given the options provided, the closest response is:
- **25.76 units²**
Hence, the approximate area of the rectangle is **25.76 units²**.