One of the key topics in Grade 10 Ontario Math (Chapter 1: Line Segment) is understanding the concept Slope of Perpendicular Lines. In this blog post, we will explore the mathematical principles behind perpendicular lines, focusing on reciprocal numbers, negative reciprocals, and how they relate to slopes.
Understanding Slope
The slope of a line measures its steepness and is defined as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run). It is represented by the letter m
It can be calculated using the formula
To get more knowledge about the concept of slope you can click here.
To take a quiz on the topic of slope you can click here
Reciprocal Numbers Slope of Perpendicular Lines
A reciprocal of a number is the result of flipping the numerator and denominator of a fraction. For any nonzero number a, its reciprocal is given by:
For example:
- The reciprocal of 2 is 1/2.
- The reciprocal of -3 is -1/3.
- The reciprocal of 1/4 is 4.
Reciprocal numbers play a crucial role in determining the slopes of perpendicular lines.Slope of Perpendicular Lines
Negative Reciprocals
A negative reciprocal is obtained by taking the reciprocal of a number and changing its sign. In other words, the negative reciprocal of a is:
For example:
- The negative reciprocal of 2 is -1/2.
- The negative reciprocal of -3 is 1/3.
- The negative reciprocal of 4 is -1/4.
Negative reciprocals are essential for understanding perpendicular lines in coordinate geometry.
Perpendicular Lines and their Slope
Two lines are perpendicular if they meet at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other.
Mathematical Proof
Consider two perpendicular lines with slopes m₁ and m₂. By the property of perpendicular slopes, we have:
This means that if the slope of one line is known, the slope of a perpendicular line can be found using:
Examples
Example 1 Slope of Perpendicular Lines
Given a line with slope m = 3, find the slope of a perpendicular line.
Solution:
- The negative reciprocal of 3 is -1/3.
- Therefore, the perpendicular line has a slope of -1/3.
Example 2 Slope of Perpendicular Lines
If a line has a slope of -2/5, what is the slope of a perpendicular line?
Solution:
- The negative reciprocal of -2/5 is 5/2.
- Hence, the perpendicular line has a slope of 5/2.
Graphical Representation of Slope of Perpendicular Lines
When plotted on a coordinate plane, two perpendicular lines intersect at a right angle. Slope of Perpendicular Lines visually demonstrate the negative reciprocal relationship, making it clear that the product of their slopes is always -1.
Real-World Applications of Perpendicular Slopes
Understanding perpendicular slopes is not just theoretical—it has practical applications in various fields:
1. Engineering and Architecture
Engineers and architects use perpendicular lines to design buildings, bridges, and other structures that require right angles for stability and functionality.
2. Road Intersections
Traffic intersections often include perpendicular roads to facilitate smooth vehicle movement and pedestrian safety.
3. Sports Fields and Courts
Many sports fields and courts (e.g., basketball, soccer, and tennis) are designed with perpendicular boundaries to ensure fair play and proper measurement.
4. Computer Graphics and Design
In digital design, perpendicular slopes are used to align elements perfectly at right angles, ensuring symmetry and accuracy in layouts.
Conclusion
Understanding the concept of perpendicular lines and their slopes is fundamental in coordinate geometry. By learning about reciprocal numbers, negative reciprocals, and the mathematical relationship between perpendicular slopes, students can gain a deeper appreciation for geometry’s role in the real world.
Practice Questions (Slope of Perpendicular Lines)- Click here
Online Quiz (Slope of Perpendicular Lines) – Click here
Our other links
Principles of Mathematics (MPM2D) notes
Chapter 1 Line Segment
1.1 Distance Between Two Points – class notes click here
1.2 Midpoint – class notes click here
1.3 Slope of a Line – class notes click here
1.4 Slopes of Parallel Lines – class notes click here
1.5 Slopes of Perpendicular Lines –class notes click here
Principles of Mathematics (MPM2D) Quiz
1.1 Distance Between Two Points – quiz click here
1.2 Midpoint – quiz click here
1.3 Slope of a Line – quiz click here
1.4 Slopes of Parallel Lines – quiz click here
1.5 Slopes of Perpendicular Lines -quiz click here