F&P Math 10 - Lesson 6.2

Introduction

This tutorial focuses on understanding the slopes of parallel and perpendicular lines, fundamental concepts in coordinate geometry. By mastering these concepts, you'll enhance your ability to analyze and create linear equations, which are essential in various applications, including engineering and economics.

Step 1: Understanding Slope

• Definition of Slope: The slope of a line measures its steepness and direction, calculated as the rise over run.

Slope Formula

[ m = \frac{y_2 - y_1}{x_2 - x_1} ] where (m) is the slope, and ((x_1, y_1)) and ((x_2, y_2)) are two points on the line.
• Practical Tip: To visualize slope, plot two points on a graph and draw a line between them. The vertical change divided by the horizontal change gives you the slope.

Step 2: Identifying Parallel Lines

• Definition: Parallel lines have the same slope but different y-intercepts.

Key Point: Since their slopes are equal, if two lines have slopes (m_1) and (m_2), then

[ m_1 = m_2 ]
• Example: If line A has a slope of 3, any line parallel to it will also have a slope of 3, regardless of its y-intercept.

Step 3: Identifying Perpendicular Lines

• Definition: Perpendicular lines have slopes that are negative reciprocals of each other.

Key Point: If one line has a slope (m_1), then the slope of the line perpendicular to it, (m_2), will satisfy

[ m_1 \times m_2 = -1 ]
• Example: If a line has a slope of 2, the slope of a line perpendicular to it will be (-\frac{1}{2}).

Step 4: Graphing Parallel and Perpendicular Lines

• Graphing Parallel Lines:

  1. Choose a slope for your line (e.g., 3).
  2. Use various y-intercepts to create parallel lines (e.g., (y = 3x + 1) and (y = 3x - 2)).
  3. Plot these lines on a graph to see they never intersect.

• Graphing Perpendicular Lines:

  1. Start with a line, e.g., (y = 2x + 1) (slope of 2).
  2. Calculate the negative reciprocal slope: (-\frac{1}{2}).
  3. Create a perpendicular line, such as (y = -\frac{1}{2}x + 3).
  4. Plot both lines to observe their intersection at a right angle.

Conclusion

Understanding the slopes of parallel and perpendicular lines is crucial in geometry. Remember that parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals. Practice graphing these lines to solidify your understanding. As a next step, explore real-world applications of these concepts, such as in architecture or design, where the relationships between lines are vital.