Recalling the Properties of Perpendicular Lines

Oct 13, 2025 by miodeya.com 30 views

Understanding the Problem

When dealing with lines in mathematics, it's essential to understand the properties of lines, including their slopes and y-intercepts. The slope of a line is a measure of how steep it is, and the y-intercept is the point where the line crosses the y-axis. In this problem, we're given a line and asked to find the equation of a line that is perpendicular to it and has the same y-intercept.

Recalling the Properties of Perpendicular Lines

Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that if we have a line with a slope of m, then the slope of a line perpendicular to it will be -1/m. Additionally, the y-intercept of the perpendicular line will be the same as the y-intercept of the original line.

Analyzing the Given Options

We're given four options for the equation of the line that is perpendicular to and has the same y-intercept as the given line. Let's analyze each option and determine which one meets the criteria.

Option A: y = (1/5)x + 1

The slope of this line is 1/5, which means that the slope of a line perpendicular to it will be -5. However, the y-intercept of this line is 1, which is not the same as the y-intercept of the original line.

Option B: y = (1/5)x + 5

The slope of this line is 1/5, which means that the slope of a line perpendicular to it will be -5. However, the y-intercept of this line is 5, which is not the same as the y-intercept of the original line.

Option C: y = 5x + 1

The slope of this line is 5, which means that the slope of a line perpendicular to it will be -1/5. This meets the criteria for the slope of a perpendicular line. Additionally, the y-intercept of this line is 1, which is the same as the y-intercept of the original line.

Option D: y = 5x + 5

The slope of this line is 5, which means that the slope of a line perpendicular to it will be -1/5. However, the y-intercept of this line is 5, which is not the same as the y-intercept of the original line.

Conclusion

Based on our analysis, the correct answer is Option C: y = 5x + 1. This line has a slope of -1/5, which is the negative reciprocal of the slope of the original line, and it has the same y-intercept as the original line.

Understanding the Concept

The concept of perpendicular lines is a fundamental idea in mathematics, and it has many real-world applications. For example, in architecture, perpendicular lines are used to design buildings and bridges. In engineering, perpendicular lines are used to design machines and mechanisms. In art, perpendicular lines are used to create perspective and depth.

Real-World Applications

Perpendicular lines have many real-world applications, including:

Architecture: Perpendicular lines are used to design buildings and bridges.
Engineering: Perpendicular lines are used to design machines and mechanisms.
Art: Perpendicular lines are used to create perspective and depth.
Navigation: Perpendicular lines are used to navigate through space and time.
Science: Perpendicular lines are used to describe the motion of objects and the behavior of physical systems.

Conclusion

In conclusion, the equation of the line that is perpendicular to and has the same y-intercept as the given line is y = 5x + 1. This line has a slope of -1/5, which is the negative reciprocal of the slope of the original line, and it has the same y-intercept as the original line. The concept of perpendicular lines is a fundamental idea in mathematics, and it has many real-world applications.

Understanding Perpendicular Lines

Perpendicular lines are a fundamental concept in mathematics, and they have many real-world applications. In this article, we'll answer some frequently asked questions about perpendicular lines.

Q: What is the definition of perpendicular lines?

A: Perpendicular lines are two lines that intersect at a 90-degree angle. This means that the lines are at right angles to each other.

Q: How do you find the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line, you need to find the negative reciprocal of the slope of the original line. This means that if the slope of the original line is m, then the slope of the perpendicular line will be -1/m.

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

A: The slopes of perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is m, then the slope of the perpendicular line will be -1/m.

Q: How do you find the y-intercept of a line that is perpendicular to another line?

A: The y-intercept of a line that is perpendicular to another line is the same as the y-intercept of the original line.

Q: What is the difference between perpendicular lines and parallel lines?

A: Perpendicular lines intersect at a 90-degree angle, while parallel lines never intersect. This means that perpendicular lines have a point of intersection, while parallel lines do not.

Q: Can two lines be both perpendicular and parallel?

A: No, two lines cannot be both perpendicular and parallel. If two lines are perpendicular, they intersect at a 90-degree angle, while if two lines are parallel, they never intersect.

Q: How do you use perpendicular lines in real-world applications?

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

Architecture: Perpendicular lines are used to design buildings and bridges.
Engineering: Perpendicular lines are used to design machines and mechanisms.
Art: Perpendicular lines are used to create perspective and depth.
Navigation: Perpendicular lines are used to navigate through space and time.
Science: Perpendicular lines are used to describe the motion of objects and the behavior of physical systems.

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:

Confusing perpendicular lines with parallel lines: Make sure to understand the difference between perpendicular and parallel lines.
Not finding the negative reciprocal of the slope: Make sure to find the negative reciprocal of the slope of the original line to find the equation of the perpendicular line.
Not using the same y-intercept: Make sure to use the same y-intercept as the original line when finding the equation of the perpendicular line.

Conclusion

In conclusion, perpendicular lines are a fundamental concept in mathematics, and they have many real-world applications. By understanding the definition, relationship, and applications of perpendicular lines, you can use them to solve problems and make informed decisions in various fields.

Additional Resources

If you're interested in learning more about perpendicular lines, here are some additional resources:

Math textbooks: Check out math textbooks for a comprehensive overview of perpendicular lines.
Online resources: Visit online resources such as Khan Academy, Mathway, and Wolfram Alpha for interactive lessons and practice problems.
Math communities: Join online math communities such as Reddit's r/learnmath and r/math to ask questions and get help from other math enthusiasts.

Final Thoughts

Perpendicular lines are a powerful tool in mathematics, and they have many real-world applications. By understanding the definition, relationship, and applications of perpendicular lines, you can use them to solve problems and make informed decisions in various fields. Remember to avoid common mistakes and use the negative reciprocal of the slope to find the equation of the perpendicular line.