In Exercises 19–22, find the area of the parallelogram whose vertices are listed.
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Transcribed Image Text:### Question 22
Given the points: \((0, -2)\), \((5, -2)\), \((-3, 1)\), \((2, 1)\)
This question presents four coordinates or points on a Cartesian plane. Each point is defined by an x-value (horizontal position) and a y-value (vertical position). These coordinates can be plotted on a graph to analyze their relationships, such as determining if they form a specific geometric shape or to use them in further calculations, such as finding the distance between points or determining the slope of the line connecting two points.
#### Explanation of Points:
1. \((0, -2)\): The x-coordinate is 0, and the y-coordinate is -2.
2. \((5, -2)\): The x-coordinate is 5, and the y-coordinate is -2.
3. \((-3, 1)\): The x-coordinate is -3, and the y-coordinate is 1.
4. \((2, 1)\): The x-coordinate is 2, and the y-coordinate is 1.
These points may be used in various mathematical exercises such as plotting on a graph, determining the shape they form, calculating distances, or identifying lines and their slopes.
![### Finding the Area of a Parallelogram from Vertex Coordinates
In Exercises 19–22, you are tasked with finding the area of a parallelogram whose vertices are provided. To complete this exercise, follow these steps:
1. **List the Coordinates**: Identify the vertices of the parallelogram. Typically, these will be given as points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and \((x_4, y_4)\).
2. **Use the Area Formula for a Parallelogram**: Recall that the area \(A\) of a parallelogram can be calculated if you have the coordinates of the vertices. The formula is:
\[
A = \left| \frac{1}{2} \left( x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - y_1 x_2 - y_2 x_3 - y_3 x_4 - y_4 x_1 \right) \right|
\]
This formula derives from the Shoelace Theorem, which provides a method to find the area of any polygon when the vertices are known.
3. **Calculate Each Term**: Plug the coordinates of the vertices into the formula, perform the multiplication and addition inside the absolute value, and then multiply by \(\frac{1}{2}\).
4. **Absolute Value**: Make sure to take the absolute value of the final result to ensure the area is a positive number.
5. **Interpret the Result**: The result from the formula will give you the area of the parallelogram in square units.
These steps allow you to determine the area of a parallelogram using only its vertex coordinates.
Here is an example to illustrate the process:
**Example**:
Given vertices \(A(0, 0)\), \(B(3, 0)\), \(C(4, 2)\), and \(D(1, 2)\), calculate the area.
- Identify coordinates:
\[
(x_1, y_1) = (0, 0), \, (x_2, y_2) = (3, 0), \, (x_3, y_3) = (4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb1a8ba3-371b-48d7-95b6-c118f740fa83%2Fd8773f14-1cf9-4741-9e4b-1bfb86ae1131%2F2hxujw_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Area of a Parallelogram from Vertex Coordinates
In Exercises 19–22, you are tasked with finding the area of a parallelogram whose vertices are provided. To complete this exercise, follow these steps:
1. **List the Coordinates**: Identify the vertices of the parallelogram. Typically, these will be given as points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and \((x_4, y_4)\).
2. **Use the Area Formula for a Parallelogram**: Recall that the area \(A\) of a parallelogram can be calculated if you have the coordinates of the vertices. The formula is:
\[
A = \left| \frac{1}{2} \left( x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 - y_1 x_2 - y_2 x_3 - y_3 x_4 - y_4 x_1 \right) \right|
\]
This formula derives from the Shoelace Theorem, which provides a method to find the area of any polygon when the vertices are known.
3. **Calculate Each Term**: Plug the coordinates of the vertices into the formula, perform the multiplication and addition inside the absolute value, and then multiply by \(\frac{1}{2}\).
4. **Absolute Value**: Make sure to take the absolute value of the final result to ensure the area is a positive number.
5. **Interpret the Result**: The result from the formula will give you the area of the parallelogram in square units.
These steps allow you to determine the area of a parallelogram using only its vertex coordinates.
Here is an example to illustrate the process:
**Example**:
Given vertices \(A(0, 0)\), \(B(3, 0)\), \(C(4, 2)\), and \(D(1, 2)\), calculate the area.
- Identify coordinates:
\[
(x_1, y_1) = (0, 0), \, (x_2, y_2) = (3, 0), \, (x_3, y_3) = (4
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