Slope of the Perpendicular Line

Oct 13, 2025 by recoveryshake.com 135 views

Introduction

In mathematics, the concept of lines and their equations is a fundamental topic in geometry and algebra. When dealing with lines, it's essential to understand how to find the equation of a line that is perpendicular to another given line and passes through a specific point. In this article, we will explore how to find the equation of a line that is perpendicular to the line $$y=-\frac{2}{3}x-1$$ and passes through the point $$(-4,2)$$.

Slope of the Given Line

The given line is represented by the equation $$y=-\frac{2}{3}x-1$$. To find the slope of this line, we can rewrite the equation in the slope-intercept form, which is $$y=mx+b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. Comparing this form with the given equation, we can see that the slope of the given line is $$-\frac{2}{3}$$.

Slope of the Perpendicular Line

When two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of the given line is $$-\frac{2}{3}$$, then the slope of the perpendicular line will be $$\frac{3}{2}$$.

Point-Slope Form of a Line

To find the equation of a line that passes through a specific point, we can use the point-slope form of a line, which is $$y-y_1=m(x-x_1)$$, where $$(x_1,y_1)$$ is the given point and $$m$$ is the slope of the line. In this case, the given point is $$(-4,2)$$ and the slope of the perpendicular line is $$\frac{3}{2}$$.

Finding the Equation of the Perpendicular Line

Now that we have the slope and the point, we can substitute these values into the point-slope form of a line to find the equation of the perpendicular line. Plugging in the values, we get:

y-2=\frac{3}{2}(x-(-4))

Simplifying this equation, we get:

y-2=\frac{3}{2}(x+4)

Expanding the right-hand side, we get:

y-2=\frac{3}{2}x+6

Adding 2 to both sides, we get:

y=\frac{3}{2}x+8

Conclusion

In this article, we have found the equation of a line that is perpendicular to the line $$y=-\frac{2}{3}x-1$$ and passes through the point $$(-4,2)$$. The equation of the perpendicular line is $$y=\frac{3}{2}x+8$$. This demonstrates how to use the point-slope form of a line and the concept of negative reciprocals to find the equation of a line that is perpendicular to a given line and passes through a specific point.

Example Problems

Find the equation of a line that is perpendicular to the line $$y=2x+3$$ and passes through the point $$(0,5)$$.
Find the equation of a line that is perpendicular to the line $$y=-x+2$$ and passes through the point $$(-2,3)$$.

Step-by-Step Solution

To solve these problems, follow these steps:

Find the slope of the given line.
Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
Use the point-slope form of a line to find the equation of the perpendicular line.
Simplify the equation to find the final answer.

Tips and Tricks

When finding the equation of a line that is perpendicular to a given line, make sure to take the negative reciprocal of the slope of the given line.
When using the point-slope form of a line, make sure to plug in the correct values for the slope and the point.
When simplifying the equation, make sure to add or subtract the correct values to get the final answer.

Real-World Applications

The concept of finding the equation of a line that is perpendicular to a given line and passes through a specific point has many real-world applications. For example:

In architecture, finding the equation of a line that is perpendicular to a given line can help architects design buildings and structures that are safe and functional.
In engineering, finding the equation of a line that is perpendicular to a given line can help engineers design bridges and roads that are safe and efficient.
In computer science, finding the equation of a line that is perpendicular to a given line can help programmers design algorithms and data structures that are efficient and effective.

Conclusion

In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can find the equation of a line that is perpendicular to a given line and passes through a specific point.

Introduction

In our previous article, we discussed how to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

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

A: The slope of a line that is perpendicular to a given line is the negative reciprocal of the slope of the given line.

Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point?

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point, you need to follow these steps:

Find the slope of the given line.
Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
Use the point-slope form of a line to find the equation of the perpendicular line.
Simplify the equation to find the final answer.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is $$y-y_1=m(x-x_1)$$, where $$(x_1,y_1)$$ is the given point and $$m$$ is the slope of the line.

Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point using the point-slope form?

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point using the point-slope form, you need to plug in the values for the slope and the point into the equation $$y-y_1=m(x-x_1)$$.

Q: What is the difference between the slope-intercept form and the point-slope form of a line?

A: The slope-intercept form of a line is $$y=mx+b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. The point-slope form of a line is $$y-y_1=m(x-x_1)$$, where $$(x_1,y_1)$$ is the given point and $$m$$ is the slope of the line.

Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point using the slope-intercept form?

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point using the slope-intercept form, you need to find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line, and then plug in the values for the slope and the point into the equation $$y=mx+b$$.

Q: What are some real-world applications of finding the equation of a line that is perpendicular to a given line and passes through a specific point?

A: Some real-world applications of finding the equation of a line that is perpendicular to a given line and passes through a specific point include:

In architecture, finding the equation of a line that is perpendicular to a given line can help architects design buildings and structures that are safe and functional.
In engineering, finding the equation of a line that is perpendicular to a given line can help engineers design bridges and roads that are safe and efficient.
In computer science, finding the equation of a line that is perpendicular to a given line can help programmers design algorithms and data structures that are efficient and effective.

Q: How do I know if a line is perpendicular to a given line?

A: A line is perpendicular to a given line if the product of their slopes is -1.

Q: What is the equation of a line that is perpendicular to the line $$y=2x+3$$ and passes through the point $$(0,5)$$?

A: To find the equation of a line that is perpendicular to the line $$y=2x+3$$ and passes through the point $$(0,5)$$, you need to follow these steps:

Find the slope of the given line, which is 2.
Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line, which is -1/2.
Use the point-slope form of a line to find the equation of the perpendicular line.
Simplify the equation to find the final answer.

The equation of the perpendicular line is $$y-5=-\frac{1}{2}(x-0)$$, which simplifies to $$y=-\frac{1}{2}x+5$$.

Q: What is the equation of a line that is perpendicular to the line $$y=-x+2$$ and passes through the point $$(-2,3)$$?

A: To find the equation of a line that is perpendicular to the line $$y=-x+2$$ and passes through the point $$(-2,3)$$, you need to follow these steps:

Find the slope of the given line, which is -1.
Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line, which is 1.
Use the point-slope form of a line to find the equation of the perpendicular line.
Simplify the equation to find the final answer.

The equation of the perpendicular line is $$y-3=1(x-(-2))$$, which simplifies to $$y=x+5$$.