Parallel and
Perpendicular Lines
Using parallelism and
perpendicularity to solve problems
In the graph below, the two lines are parallel.
Parallel lines - are lines in the same plane that never
intersect. The equation of line 1 is y = 2x + 1. the
equation of line 2 is y = 2x -2
Slopes of Parallel Lines
Nonvertical lines are
parallel if they have the
same slope and
different y-intercepts.
Any two vertical lines are
parallel.
Any two horizontal lines
are parallel
You can use slope-intercept form of an equation
to determine whether lines are parallel.
Are the graphs of y = -1/3x + 5 and 2x + 6y = 12
parallel? Explain.
Write 2x + 6y = 12 in
slope-intercept form,
then compare with
y = -1/3x + 5
2x + 6y = 12
6y = -2x + 12
6y = - 2x + 12
6
6
y = - 1/3x + 2
Compare to
y = -1/3x + 5
The lines are parallel. The
equations have the
same slope, -1/3, and
different y-intercepts.
Are the graphs of -6x + 8y = -24 and y = 3/4x – 7
parallel? Explain.
You can use the fact that the slopes of parallel
lines are the same to write the equation of a line
parallel to a given line. To write the equation, you
use the slope of the given line and the point-slope
form of a linear equation.
Step 1 Identify the slope of the given line.
y = 3/5x – 4
Step 2 Write the equation of the line through (5, 1) using point-slope
form.
y – y1 = m(x – x1)
point-slope form.
y – 1 = 3/5(x – 5)
Substitute (5, 1) for (x1,Y1) and 3/5 for m.
y – 1 = 3/5x – 3/5(5)
Use the distributive property.
y – 1 = 3/5x – 3
Simplify.
y = 3/5x – 2
Add 1 to each side.
TRY ONE
Write an equation for the line that contains
(2, -6) and is parallel to y = 3x + 9
Step 1 Identify the slope of the given line.
Step 2 Write the equation of the line through (2, -6)
using point-slope form of a linear equation.
y – y1 = m(x – x1)
Write an equation for the line that is parallel
to the given line and that passes through the
given point.
1) Y = 6x - 2; (0, 0)
2) Y = -3x; (3, 0)
3) Y =-2x + 3; (-3, 5)
4) Y = -7/2x + 6; (-4, -6)
The two lines in the graph below are perpendicular.
Perpendicular lines – are lines that intersect to form
right angles. The line y = 2x + 1 is perpendicular to
the line y = -1/2x + 1.
Slopes of perpendicular
lines
Two lines are
perpendicular if the
product of their slopes
is -1. A vertical and a
horizontal line are also
perpendicular.
The product of two numbers is -1 if one number
is the negative reciprocal of the other. Here is
how to find the negative reciprocal of a number.
Start with a fraction:
-1/2
Find its reciprocal:
-2/1
Write the negative reciprocal:
2/1 or 2
Since -1/2 • 2/1 = -1, 2/1 is the negative
reciprocal of -1/2
TRY THESE
Find the negative reciprocal of each:
1) 4
2) 3/4
3) -1/2
4) -2
5) -4/3
You can use the negative reciprocal of the slope of a
given line to write an equation of a line perpendicular
to that line. To write the equation, you use the
negative reciprocal of the slope of the given line and
the point-slope form of a linear equation.
Step 1 Identify the slope of the given line.
y = 5x + 3
Step 2 Find the negative reciprocal of the slope.
5 • -1/5 = -1
Step 3 Use the point-slope form to write an equation that contains
(0, -2) and is perpendicular to y = 5x + 3
y – y1 = m(x – x1)
Point-slope form.
y – (-2) = -1/5(x – 0) Substitute (0, -2) for (x1,y1) and -1/5 for m.
y + 2 = -1/5x – 0
y = -1/5x – 2
Use the distributive property.
Subtract 2 from each side. Simplify.
TRY ONE
Write an equation of the line that contains (6, 2)
and is perpendicular to y = -2x + 7
Step 1 Identify the slope of the given line.
Step 2 Find the negative reciprocal of the slope.
Step 3 Use the point-slope form of an equation that
contains (6, 2) and is perpendicular to y = -2x + 7
Write an equation for the line that is
perpendicular to the given line and that passes
through the given point.
1) Y = 2x + 7; (0, 0)
2) Y = -1/3x + 2; (4, 2)
3) Y = x – 3; (4, 6)
4) 4x – 2y = 9; (8, 2)
Write the equation of each line. Determine if
the lines are parallel or perpendicular. Explain
why or why not.
Problem Solving
Problem Solving
Entrance
A city’s civil engineer
is planning a new
parking garage and a
new street. The new
street will go from
the entrance of the
parking garage to
Handel St. It will be
perpendicular to
Handel St. What is
the equation of the
line representing the
new street?