Finding the Slope of a Perpendicular Line

Oct 13, 2025 by miodeya.com 7 views

Introduction

In mathematics, the concept of perpendicular lines is crucial in understanding the relationship between different lines. Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). In this article, we will explore how to find the equation of a line perpendicular to another line, given the equation of the original line and a point on the new line.

The Equation of Line CD

The equation of line CD is given as y = 3x - 3. This is in slope-intercept form, where the slope (m) is 3 and the y-intercept (b) is -3.

Finding the Slope of a Perpendicular Line

To find the equation of a line perpendicular to line CD, we need to find the slope of the new line. The slope of a perpendicular line is the negative reciprocal of the slope of the original line. In this case, the slope of line CD is 3, so the slope of a perpendicular line is -1/3.

The Equation of a Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find its equation. The point-slope form is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

We are given the point (3, 1) on the new line, so we can substitute these values into the equation:

y - 1 = (-1/3)(x - 3)

Simplifying the Equation

To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:

3(y - 1) = -1(x - 3)

Expanding the left-hand side, we get:

3y - 3 = -x + 3

Adding x to both sides and adding 3 to both sides, we get:

3y = -x + 6

Dividing both sides by 3, we get:

y = (-1/3)x + 2

Conclusion

In this article, we found the equation of a line perpendicular to line CD, given the equation of the original line and a point on the new line. We used the concept of negative reciprocals to find the slope of the perpendicular line and then used the point-slope form to find its equation. The final equation of the perpendicular line is y = (-1/3)x + 2.

Comparison with the Options

Now that we have found the equation of the perpendicular line, we can compare it with the options given:

A. y = 3x + 0 B. y = (-1/3)x + 2

Clearly, option B is the correct answer.

Discussion

The concept of perpendicular lines is crucial in mathematics, particularly in geometry and trigonometry. Understanding how to find the equation of a perpendicular line is essential in solving problems involving right triangles and circles. In this article, we used the concept of negative reciprocals to find the slope of the perpendicular line and then used the point-slope form to find its equation. This method can be applied to find the equation of a perpendicular line in any situation where the equation of the original line and a point on the new line are given.

Real-World Applications

The concept of perpendicular lines has numerous real-world applications, including:

Architecture: Perpendicular lines are used in the design of buildings and bridges to ensure stability and structural integrity.
Engineering: Perpendicular lines are used in the design of machines and mechanisms to ensure proper function and efficiency.
Art: Perpendicular lines are used in the creation of geometric shapes and patterns to add visual interest and depth.

Introduction

In our previous article, we explored how to find the equation of a line perpendicular to another line, given the equation of the original line and a point on the new line. In this article, we will answer some frequently asked questions (FAQs) related to the concept of perpendicular lines and their equations.

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

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

Q: How do I find the equation of a line perpendicular to another line?

A: To find the equation of a line perpendicular to another line, you need to follow these steps:

Find the slope of the original line.
Find the negative reciprocal of the slope (this will be the slope of the perpendicular line).
Use the point-slope form of a line to find the equation of the perpendicular line, given a point on the new line.

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

A: The point-slope form of a line is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

Q: How do I use the point-slope form to find the equation of a perpendicular line?

A: To use the point-slope form to find the equation of a perpendicular line, you need to follow these steps:

Substitute the values of the point (x1, y1) and the slope m into the equation.
Simplify the equation to get the final equation of the perpendicular line.

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 given by:

y = mx + b

where m is the slope and b is the y-intercept.

The point-slope form of a line is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

The main difference between the two forms is that the slope-intercept form gives the equation of a line in terms of the slope and y-intercept, while the point-slope form gives the equation of a line in terms of a point on the line and the slope.

Q: Can I use the slope-intercept form to find the equation of a perpendicular line?

A: Yes, you can use the slope-intercept form to find the equation of a perpendicular line. However, you need to find the negative reciprocal of the slope of the original line and then substitute it into the equation.

Q: What are some real-world applications of perpendicular lines?

A: Perpendicular lines have numerous real-world applications, including:

Architecture: Perpendicular lines are used in the design of buildings and bridges to ensure stability and structural integrity.
Engineering: Perpendicular lines are used in the design of machines and mechanisms to ensure proper function and efficiency.
Art: Perpendicular lines are used in the creation of geometric shapes and patterns to add visual interest and depth.

Conclusion

In this article, we answered some frequently asked questions related to the concept of perpendicular lines and their equations. We hope that this article has provided you with a better understanding of how to find the equation of a line perpendicular to another line and its real-world applications.