1. Module 5 Parallel Lines and Perpendicular Lines

2. Key Concepts of a Line The graph of a linear function of the form y = mx + b is a straight line with slope = m and the y- intercept = b. slope The equation of the line 2x + 3y = 6 can be changed to the “Slope-Intercept” form: the y- intercept

3. Facts about Parallel Lines If the two lines are parallel, they have the same slope. If two lines have the same slope then they are parallel. Let’s observe the graphs of the following functions: a) L 1 : y = 2x – 4 has slope = 2 and the y-intercept = -4 b) L 2 : y = 2x + 4 has slope = 2 and the y-intercept = 4 c) L 3 : y = 2x – 8 has slope = 2 and the y-intercept = -8 d) L 4 : y = 2x + 8 has slope = 2 and the y-intercept = 8 The graphs of these 4 lines will be shown in the next page.

4. Graphs of the Parallel lines L2L2 L1L1 L3L3 L4L4 All the lines have the same slope of 2 and are parallel to each other.

5. If two lines have the same slope, then they are parallel.

6. (0,0) (1,1) (0,2) (-2,0) Slope of red line = Slope of green line = 2 - 0 0 + 2 = 1 1 - 0 = 1 1 - 0 Since the slopes are equal, the two lines are parallel. x y

7. Facts about Perpendicular Lines If two lines are perpendicular, the product of their slopes is –1. That is, if then = – 1. Where is the slope of the line and is the slope of the line or =,

8. Example: The two lines: 2x – 3y = 4 and 3x + 2y = 6 are perpendicular to each other because the line 2x – 3y = 4 has slope and the line 3x + 2y = 6 has slope the product of the slopes =

9. If the slopes of two lines are negative reciprocals, then the lines are perpendicular.

12. If a line L passing through the point (-4, 2) and perpendicular to the line 4x – y = 5, can you Find the equation of the line L? Some guidelines to find the equation of the line L: 1) Find the slope of the line 4x – y = 5; 2)Then find the slope of the line L; ( Remember, the line L and the line 4x – y = 5 are perpendicular) 3)Since the line L passing through the point (-4, 2), then the coordinates (-4,2) and the slope of the line L will satisfy the equation y = mx + b, where b-the y-intercept can be found.

13. Self Check For each of the following pairs of equations, tell whether the lines are parallel or perpendicular or neither. 1) 2x – y = 3; y = 2x + 5 2) 3x + 5y = 8; 5x – 3y = 9 3) 2(x – 1) = 3y; y = 2x –1 4) 4x = 5(y – 2) ; 4y = 3 – 5x

14. Problem Set (Con’d) 5)Find the equation of the line passing through the point (-2, 3) and parallel to the line 5x – y = 12. 6) Find the equation of the line passing through the point (-3, -2) and perpendicular to the line 3x – 4y = 12.