Table of Contents
Parallel and Perpendicular Lines
Parallel lines are lines that run in the same direction and never cross.
Perpendicular lines are lines that intersect at right angles.
Example 1: Graph the following lines on the grid below.
a) y=2x-3
Slope = 2
y-intercept = -3
Plot they intercept =-3
Use the slope to get a point m=21or-2-1
y=2x+1
Slope = 2
y-intercept = 1
Plot y-intercept = 1
Use slope to get a point
m=2=21or-2-1
b) What do you notice about the slopes of the lines?
The slopes are the same (equal).
Conclusion: Parallel lines have the same (equal) slopes.
Example 2: Graph the following lines on the grid below
a) y=2x+3
Slope =2
y-intercept = 3
Plot the y-int =3
Use slope to get a point
m1=2
=21 or -2-1
y=-12x+1
Slope =-4/2
y-intercept =1
Plot the y-int =1
Use the slope to get a second point
m2=-12
=-12or1-2
b) Multiply the slopes. What do you notice about the product?
m1m2=2 ×-12
=-1
So, m1m2=-1
The product of the slopes of the two lines is equal to -1 if the lines are perpendicular.
m1m2=-1
m1=-1m2 or m2=-1m1
Negative Reciprocal: 2/3 its reciprocal is 3/2
2/1 its reciprocal is 1/2
2/3 its negative reciprocal is -3/2
2 its negative reciprocal is -1/2
Conclusion: The slope of one line is equal to the negative reciprocal of the slope of the other line. (m1=2 and m2=-12)
Example 3: Given: 2x 3y + 6 = 0
a) Write the equation of a line parallel to the given line.
b) Write the equation of a line perpendicular to the given line
Solution: Write 2x-3y+6=0 in the form
y=mx+b (slope y-intercept form)
2x-3y+6=0
2x+6=0+3y
3y=2x+6
3y3=23x+63
y=23x+2
m=23
a) The line whose equation we are trying to write has a slope of 2/3 since this line is parallel to y=2/3x+2 (parallel lines have equal slopes)
y=2/3x-5
b) The negative reciprocal of 2/3 is -3/2. Slope of the new line is -3/2.
y=-3/2x+b, where b is any real number.
y=-3/2x+5 is an equation of a line perpendicular to y=2/3x+2
