Which Line Is Perpendicular To A Line With A Slope Of $-\frac{1}{3}$?A. Line MN B. Line AB C. Line EF D. Line JK

Introduction

In mathematics, particularly in geometry and trigonometry, understanding the concept of perpendicular lines is crucial for solving various problems. A perpendicular line is a line that intersects another line at a 90-degree angle. In this article, we will explore the concept of perpendicular lines and determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$.

What are Perpendicular Lines?

Perpendicular lines are lines that intersect each other at a 90-degree angle. This means that if two lines are perpendicular, they form a right angle (90 degrees) at the point of intersection. In other words, if you draw a line perpendicular to another line, it will always form a right angle with the original line.

Slope of a Line

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be represented by the letter 'm'. For example, if the slope of a line is $m = \frac{1}{3}$, it means that for every 3 units of horizontal change, the line rises by 1 unit.

Finding the Perpendicular Line

To find the perpendicular line to a line with a slope of $-\frac{1}{3}$, we need to find a line that has a slope that is the negative reciprocal of $-\frac{1}{3}$. The negative reciprocal of a number is obtained by changing the sign of the number and then taking its reciprocal.

Negative Reciprocal

The negative reciprocal of $-\frac{1}{3}$ is $\frac{3}{1} = 3$. This means that the slope of the perpendicular line is 3.

Determining the Perpendicular Line

Now that we have found the slope of the perpendicular line, we need to determine which line is perpendicular to the line with a slope of $-\frac{1}{3}$. To do this, we need to examine the slopes of the lines given in the options.

Option A: Line MN

The slope of line MN is not given. Therefore, we cannot determine if it is perpendicular to the line with a slope of $-\frac{1}{3}$.

Option B: Line AB

The slope of line AB is not given. Therefore, we cannot determine if it is perpendicular to the line with a slope of $-\frac{1}{3}$.

Option C: Line EF

The slope of line EF is not given. Therefore, we cannot determine if it is perpendicular to the line with a slope of $-\frac{1}{3}$.

Option D: Line JK

The slope of line JK is not given. Therefore, we cannot determine if it is perpendicular to the line with a slope of $-\frac{1}{3}$.

Conclusion

In conclusion, to determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$, we need to find a line that has a slope that is the negative reciprocal of $-\frac{1}{3}$. The negative reciprocal of $-\frac{1}{3}$ is 3. However, since the slopes of the lines given in the options are not provided, we cannot determine which line is perpendicular to the line with a slope of $-\frac{1}{3}$.

Recommendation

To determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$, we need to have the slopes of the lines given in the options. Once we have the slopes, we can determine which line is perpendicular to the line with a slope of $-\frac{1}{3}$ by finding the negative reciprocal of the slope of the line with a slope of $-\frac{1}{3}$ and comparing it with the slopes of the lines given in the options.

Final Answer

Since the slopes of the lines given in the options are not provided, we cannot determine which line is perpendicular to the line with a slope of $-\frac{1}{3}$. However, if we assume that the slopes of the lines given in the options are provided, we can determine which line is perpendicular to the line with a slope of $-\frac{1}{3}$ by finding the negative reciprocal of the slope of the line with a slope of $-\frac{1}{3}$ and comparing it with the slopes of the lines given in the options.

Example

Let's assume that the slopes of the lines given in the options are provided. For example, let's assume that the slope of line MN is 3. In this case, we can determine that line MN is perpendicular to the line with a slope of $-\frac{1}{3}$ because the slope of line MN is the negative reciprocal of the slope of the line with a slope of $-\frac{1}{3}$.

Conclusion

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Q: What is the concept of perpendicular lines in mathematics?

A: Perpendicular lines are lines that intersect each other at a 90-degree angle. This means that if two lines are perpendicular, they form a right angle (90 degrees) at the point of intersection.

Q: How do you determine if two lines are perpendicular?

A: To determine if two lines are perpendicular, you need to check if the product of their slopes is equal to -1. If the product of their slopes is equal to -1, then the lines are perpendicular.

Q: What is the negative reciprocal of a number?

A: The negative reciprocal of a number is obtained by changing the sign of the number and then taking its reciprocal. For example, the negative reciprocal of 3 is -1/3.

Q: How do you find the perpendicular line to a line with a given slope?

A: To find the perpendicular line to a line with a given slope, you need to find a line that has a slope that is the negative reciprocal of the given slope.

Q: What is the slope of the perpendicular line to a line with a slope of $-\frac{1}{3}$?

A: The slope of the perpendicular line to a line with a slope of $-\frac{1}{3}$ is 3.

Q: How do you determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$?

A: To determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$, you need to find a line that has a slope that is the negative reciprocal of $-\frac{1}{3}$. The negative reciprocal of $-\frac{1}{3}$ is 3.

Q: What is the relationship between the slopes of perpendicular lines?

A: The slopes of perpendicular lines are negative reciprocals of each other.

Q: Can two lines have the same slope and be perpendicular?

A: No, two lines cannot have the same slope and be perpendicular. If two lines have the same slope, they are parallel, not perpendicular.

Q: Can a line be perpendicular to itself?

A: No, a line cannot be perpendicular to itself. Perpendicular lines are lines that intersect each other at a 90-degree angle, and a line cannot intersect itself at a 90-degree angle.

Q: What is the significance of perpendicular lines in real-life applications?

A: Perpendicular lines have many real-life applications, such as in architecture, engineering, and physics. They are used to design buildings, bridges, and other structures, and to calculate distances and angles.

Q: How do you use perpendicular lines in problem-solving?

A: To use perpendicular lines in problem-solving, you need to identify the slopes of the lines and determine if they are perpendicular. If the lines are perpendicular, you can use the negative reciprocal of one slope to find the other slope.

Q: What are some common mistakes to avoid when working with perpendicular lines?

A: Some common mistakes to avoid when working with perpendicular lines include:

• Assuming that two lines are perpendicular just because they intersect at a point.
• Failing to check if the product of the slopes of two lines is equal to -1.
• Not using the negative reciprocal of one slope to find the other slope.

Q: How do you check if two lines are perpendicular using a graphing calculator?

A: To check if two lines are perpendicular using a graphing calculator, you need to enter the equations of the lines and use the calculator to graph the lines. If the lines intersect at a 90-degree angle, then they are perpendicular.

Q: What are some real-life examples of perpendicular lines?

A: Some real-life examples of perpendicular lines include:

• The lines that form the corners of a building.
• The lines that form the edges of a piece of paper.
• The lines that form the sides of a rectangular box.

Q: How do you use perpendicular lines in geometry and trigonometry?

A: Perpendicular lines are used extensively in geometry and trigonometry to solve problems involving right triangles, angles, and distances. They are used to calculate the lengths of sides, the measures of angles, and the areas of triangles.