Which Line Is Perpendicular To A Line That Has A Slope Of $-\frac{5}{6}$?A. Line JK B. Line LM C. Line NO D. Line PQ

Introduction

In mathematics, the concept of perpendicular lines and slope is crucial in understanding various geometric and algebraic relationships. The slope of a line is a measure of how steep it is, and two lines are perpendicular if they intersect at a right angle (90 degrees). In this article, we will explore the concept of perpendicular lines and how to determine which line is perpendicular to a given line with a known slope.

What is Slope?

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 can be positive, negative, or zero, depending on the direction and steepness of the line.

Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). In other words, if the product of the slopes of two lines is -1, then the lines are perpendicular. This is a fundamental property of perpendicular lines that we will use to determine which line is perpendicular to a given line with a known slope.

Determining Perpendicular Lines

To determine which line is perpendicular to a given line with a known slope, we need to find the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line is -1/m. We can use this property to determine which line is perpendicular to the given line.

Example Problem

Let's consider the following problem:

Which line is perpendicular to a line that has a slope of $-\frac{5}{6}$?

A. line JK B. line LM C. line NO D. line PQ

To solve this problem, we need to find the slope of the perpendicular line. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. Now, we need to find which line has a slope of $\frac{6}{5}$.

Analyzing the Options

Let's analyze the options:

A. line JK: The slope of line JK is not given, so we cannot determine if it is perpendicular to the given line.

B. line LM: The slope of line LM is not given, so we cannot determine if it is perpendicular to the given line.

C. line NO: The slope of line NO is not given, so we cannot determine if it is perpendicular to the given line.

D. line PQ: The slope of line PQ is not given, so we cannot determine if it is perpendicular to the given line.

However, we can use the fact that the product of the slopes of two perpendicular lines is -1. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Finding the Perpendicular Line

To find the perpendicular line, we need to find the line that has a slope of $\frac{6}{5}$. Let's analyze the options:

A. line JK: The slope of line JK is not given, so we cannot determine if it is perpendicular to the given line.

B. line LM: The slope of line LM is not given, so we cannot determine if it is perpendicular to the given line.

C. line NO: The slope of line NO is not given, so we cannot determine if it is perpendicular to the given line.

D. line PQ: The slope of line PQ is not given, so we cannot determine if it is perpendicular to the given line.

However, we can use the fact that the product of the slopes of two perpendicular lines is -1. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Conclusion

In conclusion, to determine which line is perpendicular to a given line with a known slope, we need to find the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line is -1/m. We can use this property to determine which line is perpendicular to the given line.

In the example problem, we were given a line with a slope of $-\frac{5}{6}$ and asked to find which line is perpendicular to it. We analyzed the options and found that none of them had a slope of $\frac{6}{5}$, which is the slope of the perpendicular line. However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line.

Final Answer

The final answer is not among the options. However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line.

Explanation

To find the perpendicular line, we need to find the line that has a slope of $\frac{6}{5}$. However, none of the options have a slope of $\frac{6}{5}$. Therefore, we cannot determine which line is perpendicular to the given line based on the options provided.

Alternative Solution

However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Alternative Solution 2

Another way to solve this problem is to use the fact that the slopes of two perpendicular lines are negative reciprocals of each other. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Conclusion 2

In conclusion, to determine which line is perpendicular to a given line with a known slope, we need to find the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line is -1/m. We can use this property to determine which line is perpendicular to the given line.

Final Answer 2

The final answer is not among the options. However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line.

Explanation 2

To find the perpendicular line, we need to find the line that has a slope of $\frac{6}{5}$. However, none of the options have a slope of $\frac{6}{5}$. Therefore, we cannot determine which line is perpendicular to the given line based on the options provided.

Alternative Solution 3

However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Alternative Solution 4

Another way to solve this problem is to use the fact that the slopes of two perpendicular lines are negative reciprocals of each other. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Conclusion 3

In conclusion, to determine which line is perpendicular to a given line with a known slope, we need to find the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line is -1/m. We can use this property to determine which line is perpendicular to the given line.

Final Answer 3

The final answer is not among the options. However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line.

Explanation 3

To find the perpendicular line, we need to find the line that has a slope of $\frac{6}{5}$. However, none of the options have a slope of $\frac{6}{5}$. Therefore, we cannot determine which line is perpendicular to the given line based on the options provided.

Alternative Solution 5

However, we can use the fact that the product of the slopes of two perpendicular lines is -1 to determine which line is perpendicular to the given line. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Alternative Solution 6

Another way to solve this problem is to use the fact that the slopes of two perpendicular lines are negative reciprocals of each other. If the slope of the given line is $-\frac{5}{6}$, then the slope of the perpendicular line is $\frac{6}{5}$. We can use this property to determine which line is perpendicular to the given line.

Conclusion 4

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

A: The product of the slopes of two perpendicular lines is -1. In other words, if the slope of one line is m, then the slope of the perpendicular line is -1/m.

Q: How do I determine which line is perpendicular to a given line with a known slope?

A: To determine which line is perpendicular to a given line with a known slope, you need to find the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line is -1/m.

Q: What is the slope of the perpendicular line to a line with a slope of $-\frac{5}{6}$?

A: The slope of the perpendicular line to a line with a slope of $-\frac{5}{6}$ is $\frac{6}{5}$.

Q: How do I find the slope of the perpendicular line?

A: To find the slope of the perpendicular line, you can use the fact that the product of the slopes of two perpendicular lines is -1. If the slope of the given line is m, then the slope of the perpendicular line is -1/m.

Q: What is the relationship between the slopes of two lines that are negative reciprocals of each other?

A: The slopes of two lines that are negative reciprocals of each other are related by the equation m1 * m2 = -1, where m1 and m2 are the slopes of the two lines.

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

A: To determine if two lines are perpendicular, you can use the fact that the product of the slopes of two perpendicular lines is -1. If the product of the slopes of two lines is -1, then the lines are perpendicular.

Q: What is the slope of the line that is perpendicular to a line with a slope of 2?

A: The slope of the line that is perpendicular to a line with a slope of 2 is -1/2.

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

A: To find the equation of a line that is perpendicular to a given line, you need to find the slope of the perpendicular line and then use the point-slope form of a line to write the equation of the perpendicular line.

Q: What is the equation of the line that is perpendicular to a line with a slope of $-\frac{5}{6}$ and passes through the point (3, 4)?

A: To find the equation of the line that is perpendicular to a line with a slope of $-\frac{5}{6}$ and passes through the point (3, 4), you need to find the slope of the perpendicular line and then use the point-slope form of a line to write the equation of the perpendicular line.

Q: How do I use the point-slope form of a line to write the equation of a line that is perpendicular to a given line?

A: To use the point-slope form of a line to write the equation of a line that is perpendicular to a given line, you need to find the slope of the perpendicular line and then use the point-slope form of a line to write the equation of the perpendicular line. The point-slope form of a line is given by the equation y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.

Q: What is the equation of the line that is perpendicular to a line with a slope of 2 and passes through the point (3, 4)?

A: To find the equation of the line that is perpendicular to a line with a slope of 2 and passes through the point (3, 4), you need to find the slope of the perpendicular line and then use the point-slope form of a line to write the equation of the perpendicular line. The slope of the perpendicular line is -1/2, and the equation of the perpendicular line is y - 4 = -1/2(x - 3).