Finding Perpendicular Lines

Oct 13, 2025 by miodeya.com 104 views

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. Perpendicular lines have a special property: the product of their slopes is -1.

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 a line has a slope of 2, it means that for every 1 unit of horizontal change, the line rises 2 units.

Finding Perpendicular Lines

To find a line that is perpendicular to a given line, we need to find a line with a slope that is the negative reciprocal of the given line's slope. The negative reciprocal of a number is obtained by multiplying the number by -1 and then taking its reciprocal.

Example: Finding Perpendicular Lines

Let's say we have a line with a slope of $-\frac{1}{3}$ . To find a line that is perpendicular to this line, we need to find a line with a slope that is the negative reciprocal of $-\frac{1}{3}$ .

The negative reciprocal of $-\frac{1}{3}$ is 3. Therefore, a line with a slope of 3 is perpendicular to a line with a slope of $-\frac{1}{3}$ .

Analyzing the Options

Now, let's analyze the options given in the problem:

A. line MN B. line AB C. line EF D. line JK

We need to determine which of these lines has a slope of 3.

Option A: Line MN

Let's assume that line MN has a slope of 3. To verify this, we need to calculate the slope of line MN using the coordinates of points M and N.

Option B: Line AB

Let's assume that line AB has a slope of 3. To verify this, we need to calculate the slope of line AB using the coordinates of points A and B.

Option C: Line EF

Let's assume that line EF has a slope of 3. To verify this, we need to calculate the slope of line EF using the coordinates of points E and F.

Option D: Line JK

Let's assume that line JK has a slope of 3. To verify this, we need to calculate the slope of line JK using the coordinates of points J and K.

Calculating Slopes

To calculate the slope of each line, we need to use the coordinates of the points that define each line. Let's assume that the coordinates of the points are as follows:

Line MN: M(1, 2), N(3, 4)
Line AB: A(2, 3), B(4, 5)
Line EF: E(3, 4), F(5, 6)
Line JK: J(4, 5), K(6, 7)

Using the coordinates of the points, we can calculate the slope of each line as follows:

Line MN: m = (4 - 2) / (3 - 1) = 2 / 2 = 1
Line AB: m = (5 - 3) / (4 - 2) = 2 / 2 = 1
Line EF: m = (6 - 4) / (5 - 3) = 2 / 2 = 1
Line JK: m = (7 - 5) / (6 - 4) = 2 / 2 = 1

Conclusion

Based on the calculations, we can see that none of the lines (MN, AB, EF, or JK) have a slope of 3. Therefore, none of the options A, B, C, or D are correct.

However, we can still determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$ . Since the negative reciprocal of $-\frac{1}{3}$ is 3, a line with a slope of 3 is perpendicular to a line with a slope of $-\frac{1}{3}$ .

Unfortunately, none of the lines (MN, AB, EF, or JK) have a slope of 3. Therefore, we cannot determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$ based on the options given.

Final Answer

$Since none of the options A, B, C, or D are correct, we cannot determine which line is perpendicular to a line with a slope of $-\frac{1}{3}$ based on the information provided. However, we can conclude that a line with a slope of 3 is perpendicular to a line with a slope of $-\frac{1}{3}$.<br/> **Frequently Asked Questions (FAQs)**$

Q: What is the slope of a line?

A: 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.

Q: How do I calculate the slope of a line?

A: To calculate the slope of a line, you need to use the coordinates of two points on the line. The formula for calculating the slope is:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What is the negative reciprocal of a number?

A: The negative reciprocal of a number is obtained by multiplying the number by -1 and then taking its reciprocal. For example, the negative reciprocal of 2 is -1/2.

Q: How do I find a line that is perpendicular to a given line?

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

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

A: The product of the slopes of two perpendicular lines is -1. This means that if two lines are perpendicular, their slopes are negative reciprocals of each other.

Q: Can a line have a slope of 0?

A: Yes, a line can have a slope of 0. This means that the line is horizontal and does not change in the vertical direction.

Q: Can a line have an undefined slope?

A: Yes, a line can have an undefined slope. This means that the line is vertical and does not change in the horizontal direction.

Q: How do I determine if two lines are parallel or perpendicular?

A: To determine if two lines are parallel or perpendicular, you need to compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: Can a line be both parallel and perpendicular to another line?

A: No, a line cannot be both parallel and perpendicular to another line. These two properties are mutually exclusive.

Q: What is the difference between a line and a plane?

A: A line is a one-dimensional object that extends infinitely in two directions. A plane is a two-dimensional object that extends infinitely in all directions.

Q: Can a line be perpendicular to a plane?

A: Yes, a line can be perpendicular to a plane. This means that the line intersects the plane at a 90-degree angle.

Q: Can a plane be perpendicular to a line?

A: No, a plane cannot be perpendicular to a line. A plane is a two-dimensional object, and a line is a one-dimensional object. These two objects cannot be perpendicular to each other.

Q: What is the relationship between the slope of a line and its equation?

A: The slope of a line is a measure of how steep it is, and it is related to the equation of the line. The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: Can a line have a slope that is not a rational number?

A: Yes, a line can have a slope that is not a rational number. This means that the slope is an irrational number, such as the square root of 2.

Q: Can a line have a slope that is not a real number?

A: No, a line cannot have a slope that is not a real number. The slope of a line is always a real number, which means it can be positive, negative, or zero.