The Relationship Between Slopes of Perpendicular Lines

Oct 13, 2025 by miodeya.com 111 views

Understanding Perpendicular Lines

In mathematics, two lines are said to be perpendicular if they intersect at a right angle, which is 90 degrees. This concept is crucial in geometry and is used to determine the relationship between different lines. When two lines are perpendicular, their slopes are related in a specific way.

The Relationship Between Slopes of Perpendicular Lines

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. If the slope of one line is known, we can use the relationship between the slopes of perpendicular lines to find the slope of the other line.

The Formula for the Slope of a Perpendicular Line

The formula for the slope of a perpendicular line is:

m1 × m2 = -1

where m1 and m2 are the slopes of the two perpendicular lines.

Applying the Formula to the Given Problem

In the given problem, the slope of line p is 2. We need to find the slope of line q, which is perpendicular to line p. Using the formula above, we can substitute the value of m1 (the slope of line p) into the equation:

2 × m2 = -1

To find the slope of line q (m2), we need to isolate m2 by dividing both sides of the equation by 2:

m2 = -1/2

Therefore, the slope of line q is -1/2.

Conclusion

In conclusion, the slope of a line that is perpendicular to a line with a slope of 2 is -1/2. This is a fundamental concept in mathematics and is used to determine the relationship between different lines. By understanding the relationship between the slopes of perpendicular lines, we can solve problems involving lines and angles.

Key Takeaways

Two lines are perpendicular if they intersect at a right angle.
The slope of a line is a measure of how steep it is.
The formula for the slope of a perpendicular line is m1 × m2 = -1.
The slope of a line that is perpendicular to a line with a slope of 2 is -1/2.

Practice Problems

If the slope of line p is 3, what is the slope of line q, which is perpendicular to line p?
If the slope of line p is -4, what is the slope of line q, which is perpendicular to line p?
If the slope of line p is 1/2, what is the slope of line q, which is perpendicular to line p?

Solutions

The slope of line q is -1/3.
The slope of line q is 1/4.
The slope of line q is -2.

Real-World Applications

The concept of perpendicular lines and slopes has numerous real-world applications. For example:

In architecture, perpendicular lines are used to design buildings and structures.
In engineering, perpendicular lines are used to design bridges and other infrastructure.
In computer graphics, perpendicular lines are used to create 3D models and animations.

Conclusion

Frequently Asked Questions

Q: What is the relationship between the slopes of perpendicular lines? A: The slopes of perpendicular lines are related by the formula m1 × m2 = -1, where m1 and m2 are the slopes of the two perpendicular lines.

Q: How do I find the slope of a line that is perpendicular to a given line? A: To find the slope of a line that is perpendicular to a given line, you can use the formula m1 × m2 = -1, where m1 is the slope of the given line and m2 is the slope of the line you are trying to find.

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

Q: What is the slope of a line that is perpendicular to a line with a slope of -3? A: The slope of a line that is perpendicular to a line with a slope of -3 is 1/3.

Q: Can two lines have the same slope and be perpendicular? A: No, two lines cannot have the same slope and be perpendicular. If two lines have the same slope, they are parallel, not perpendicular.

Q: Can a line be perpendicular to itself? A: No, a line cannot be perpendicular to itself. Perpendicular lines are lines that intersect at a right angle, and a line cannot intersect itself at a right angle.

Q: How do I determine if two lines are perpendicular? A: To determine if two lines are perpendicular, you can use the formula m1 × m2 = -1, where m1 and m2 are the slopes of the two lines. If the product of the slopes is -1, then the lines are perpendicular.

Q: What is the relationship between the slopes of parallel lines? A: The slopes of parallel lines are equal. If two lines have the same slope, they are parallel, not 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 concepts are mutually exclusive.

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 can use the formula m1 × m2 = -1, where m1 is the slope of the given line and m2 is the slope of the line you are trying to find. You will also need to know a point on the line you are trying to find.

Q: Can a line have a slope of 0 and be perpendicular to another line? A: No, a line cannot have a slope of 0 and be perpendicular to another line. A line with a slope of 0 is a horizontal line, and a horizontal line cannot be perpendicular to another line.

Q: How do I determine if a line is horizontal or vertical? A: To determine if a line is horizontal or vertical, you can look at its slope. A horizontal line has a slope of 0, and a vertical line has an undefined slope.

Q: Can a line be both horizontal and vertical? A: No, a line cannot be both horizontal and vertical. These two concepts are mutually exclusive.

Conclusion

In conclusion, the concept of perpendicular lines and slopes is a fundamental concept in mathematics. By understanding the relationship between the slopes of perpendicular lines, we can solve problems involving lines and angles. The formula for the slope of a perpendicular line is m1 × m2 = -1, and the slope of a line that is perpendicular to a line with a slope of 2 is -1/2.