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.
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
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:
Reciprocal numbers play a crucial role in determining the slopes of perpendicular lines.Slope of Perpendicular Lines
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:
Negative reciprocals are essential for understanding perpendicular lines in coordinate geometry.
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.
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:
Given a line with slope m = 3, find the slope of a perpendicular line.
Solution:
If a line has a slope of -2/5, what is the slope of a perpendicular line?
Solution:
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.
Understanding perpendicular slopes is not just theoretical—it has practical applications in various fields:
Engineers and architects use perpendicular lines to design buildings, bridges, and other structures that require right angles for stability and functionality.
Traffic intersections often include perpendicular roads to facilitate smooth vehicle movement and pedestrian safety.
Many sports fields and courts (e.g., basketball, soccer, and tennis) are designed with perpendicular boundaries to ensure fair play and proper measurement.
In digital design, perpendicular slopes are used to align elements perfectly at right angles, ensuring symmetry and accuracy in layouts.
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.
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
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
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