Dear grade 10 students we are providing detailed class notes for chapter 1 line segment topic – Slope of Parallel Lines based on Ontario curriculum. A key property of parallel lines is that they have the same slope. This chapter will cover how to determine the slope of parallel lines, apply it to equations of lines, and explore its real-world applications.
Slope of Parallel Lines – intro
In coordinate geometry, the concept of parallel lines is fundamental in understanding the relationship between different lines on a Cartesian plane. A key property of parallel lines is that they have the same slope. These class notes will cover how to determine the slope of parallel lines, apply it to equations of lines, and explore its real-world applications.
Slope of Parallel Lines Definition
Two lines are said to be parallel if they never intersect, no matter how far they are extended. In coordinate geometry, parallel lines have identical slopes but different y-intercepts.
For two lines with equations:
- y=m1x+b1
- y=m2x+b2
If the lines are parallel,
then: m1=m2
where:
- m1 and m2 are the slopes of the two lines.
- b1 and b2are the y-intercepts (which can be different).
Example Slope of Parallel Lines
The equations of the two lines below are parallel because they have the same slope:
- Line 1: y=3x+5
- Line 2: y=3x−2
Both lines have a slope of 3, so they will never intersect.
Finding the Slope of Parallel Lines
Example 1: Identifying Parallel Slopes
Given the equation of a line y = -2x + 4, find the slope of a parallel line.
Solution:
- The given line has a slope of -2.
- A parallel line must have the same slope, so the required slope is -2.
Example 2: Writing the Equation of a Parallel Line
Find the equation of a line that is parallel to y=12x−3y = \frac{1}{2}x – 3 and passes through the point (4, 2).
Solution:
- The given line has a slope of 1/2.
- A parallel line must have the same slope, so m=1/2m = 1/2.
- Using the point-slope form: y−y1=m(x−x1)y – y_1 = m(x – x_1) y−2=12(x−4)y – 2 = \frac{1}{2}(x – 4) y−2=12x−2y – 2 = \frac{1}{2}x – 2 y=12xy = \frac{1}{2}x
So, the equation of the parallel line is y=12xy = \frac{1}{2}x.
Parallel Lines and Their Graphs
When two parallel lines are graphed on the Cartesian plane, they appear as distinct lines that never meet. Because they have the same slope, they rise and run at the same rate.
Graphing Parallel Lines Slope of Parallel Lines
To graph parallel lines:
- Identify the slope mm from the given equation.
- Plot the y-intercept bb for each line.
- Use the slope to plot additional points.
- Draw the lines, ensuring they never intersect.
Parallel Lines in Different Forms of Linear Equations
1. Slope-Intercept Form (y = mx + b)
Parallel lines have the same m value but different b values. Slope of Parallel Lines
Example:
- y = 4x + 1 (Line 1)
- y = 4x – 5 (Line 2)
Both have m = 4, so they are parallel.
2. Standard Form (Ax + By + C = 0)
To check for parallel lines in standard form, convert the equation into slope-intercept form.
Example:
- 2x + 3y – 6 = 0
- 4x + 6y + 9 = 0
Rewriting both in slope-intercept form: y=−23x+2y = -\frac{2}{3}x + 2 y=−23x−32y = -\frac{2}{3}x – \frac{3}{2}
Since both have a slope of -2/3, they are parallel.
3. Point-Slope Form (y – y₁ = m(x – x₁))
If given a line in point-slope form, the equation of a parallel line can be found using the same slope but a different point.
Example:
- Given y – 3 = 5(x – 1), a parallel line passing through (2, -4) is: y+4=5(x−2)y + 4 = 5(x – 2) y=5x−10−4y = 5x – 10 – 4 y=5x−14y = 5x – 14
Real-World Applications of Parallel Lines
1. Roads and Railways
Highways and railway tracks are designed using parallel lines to ensure safe and efficient travel.
2. Architecture and Construction
Buildings, bridges, and frameworks are designed with parallel beams and supports for stability and aesthetics.
3. Sports Fields and Courts
Parallel lines are used in field markings for sports like basketball, soccer, and tennis. Slope of Parallel Lines
4. Graphic Design and Engineering
Designers and engineers use parallel lines to maintain symmetry and accuracy in technical drawings.
Practice Questions (Slope of a Line)- Click here
Online Quiz (Slope of a Line) – Click here
Conclusion
Understanding parallel lines and their slopes is an essential part of coordinate geometry. Parallel lines share the same slope, making them useful in various mathematical applications and real-life scenarios.
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
