4-7B Perpendicular Lines
Algebra 1 Glencoe McGraw-Hill Linda Stamper
What is the reciprocal of 8?What is the opposite reciprocal of 8?
In this lesson you will need to find and use opposite reciprocals.
81
81
Two lines in a coordinate plane are perpendicular if they intersect at right angles (90 degree angles).
Horizontal and vertical lines are perpendicular to each other.
x
y
In a coordinate plane, two non vertical lines are perpendicular if and only if the product of their slopes is –1.
x
y
Do you think these lines are perpendicular?
How to determine whether lines are perpendicular.
1. Write each line in slope-intercept form.
2. Multiply the slopes together.
3. If the product is –1, then the lines are perpendicular.
Write the equation of each line in the graph. Determine whether the lines are perpendicular.
x
y
Green line
y-intercept is 3
bmxy
3x23
y
Blue liney-intercept is –
1
32
slope
bmxy
1x32
y
1x32
y 1
32
23
The lines are perpendicular!
23
slope
•
• ••
These are
opposite reciproca
ls!
Example 1 Write the equation of each line in the graph. Determine whether the lines are perpendicular.
x
y
Green liney-intercept is
3
bmxy 3xy
Blue liney-intercept is –
443
slope
bmxy
4x43
y
4x43
y
143
1
The lines are not perpendicular!
1 slope••
•
•
Example 2 Write the equation of each line in the graph. Determine whether the lines are perpendicular.
x
y
Green liney-intercept is
1
bmxy
1x23
y
Blue liney-intercept is –
432
slope
bmxy
4x32
y
4x32
y 1
32
23
The lines are perpendicular!
23
slope •
•
••
Write the slope-intercept form of an equation for the line that passes through (-4,-3) and is perpendicular to the graph of 4x + y = 2.
x
y
(-4,-3)
•
••
bmxy
b441
3
b441
3
Find the slope of the given line. 4m
Find the opposite reciprocal.
41
m
Use slope-intercept or point-slope to write the equation.
b13 11
b2
2x41
y
•
2yx4
Write the slope-intercept form of an equation for the line that passes through (-4,-3) and is perpendicular to the graph of 4x + y = 2.
Find the slope of the given line. 4m
Find the opposite reciprocal. 4
1
Use slope-intercept or point-slope to write the equation.
bmxy
b441
3
b13 11
b2
2x41
y
11 xxmyy
4x41
3y
3 3
4x41
3y
1x41
3y
2x41
y
2yx4 x4 x4
2x4y
(when no graph is given)
Example 3 Write the slope-intercept form of an equation for the line that passes through (-3,-2) and is perpendicular to the graph of x + 4y = 12.
Example 4 Write the slope-intercept form of an equation for the line that passes through (4,7) and is perpendicular to the graph of -2x + 3y = -3.
Example 5 Write the slope-intercept form of an equation for the line that passes through (4,-1) and is perpendicular to the graph of 7x - 2y = 3.
Find slope of given line.
Find the opposite reciprocal.
Use slope-intercept or point-slope form to write the equation.
Example 3 Write the slope-intercept form of an equation for the line that passes through (-3,-2) and is perpendicular to the graph of x + 4y = 12.
Find slope of given line. 4
1m
Find the opposite reciprocal. 4
Use slope-intercept or point-slope form to write the equation.bmxy
b342
b122 1212
b1010x4y
11 xxmyy
3x42y
2 2
3x42y
12x42y
10x4y
12y4x x x
12xy4 4 4 4
3x41
y
Example 4 Write the slope-intercept form of an equation for the line that passes through (4,7) and is perpendicular to the graph of -2x + 3y = -3.
Find slope of given line. 3
2m
Find the opposite reciprocal.
2
3
Use slope-intercept or point-slope form to write the equation.bmxy
b423
7
11 xxmyy
4x23
7y
7 7
4x23
7y
6x23
7y
13x23
y
3y3x2 x2 x2
3x2y3 3 3 3
1x32
y b67
66b13
13x23
y
Example 5 Write the slope-intercept form of an equation for the line that passes through (4,-1) and is perpendicular to the graph of 7x - 2y = 3.
Find slope of given line. 2
7m
Find the opposite reciprocal. 7
2
Use slope-intercept or point-slope form to write the equation.bmxy
b472
1
11 xxmyy
4x72
1y
1 77
4x72
1y
78
x72
1y
71
x72
y
3y2x7 x7 x7
3x7y2 2 2 2
23
x27
y b
78
78
78
b71
71
x72
y
177
Write an equation in slope-intercept form for a line perpendicular to the graph of -3x + 2y = 6 that passes through the x-intercept of the line.
What is the problem
asking you to do?x
y
•
-3x+2y=6
Write equation for
this line.
(but no graph is given)
Write an equation in slope-intercept form for a line perpendicular to the graph of -3x + 2y = 6 that passes through the x-intercept of the line.
Find slope of given line. 2
3m
Find the opposite reciprocal.
32
Find the x-intercept of the given line. 6y2x3
602x3 6y2x3
x3 x36x3y2
2 2 2
3x23
y 336x3
2x
Use slope-intercept or point-slope form to write the equation.
Coordinate of x-intercept is (-2,0).
bmxy
b232
0
b34
0
34
34
b34
34
x32
y
Write an equation in slope-intercept form for a line perpendicular to the graph of -3x + 2y = 6 that passes through the x-intercept of the line.
x
y
•
34
x32
y
Example 6 Write an equation in slope-intercept form for a line perpendicular to the graph of x + 3y = 6 that passes through the x-intercept of the line.
Example 7 Write an equation in slope-intercept form for a line perpendicular to the graph of 3x + 2y = 8 that passes through the y-intercept of the line.
Find slope of given line.
Find the opposite reciprocal.
Use slope-intercept or point-slope form to write the equation.
Find the intercept of given line.
Example 6 Write an equation in slope-intercept form for a line perpendicular to the graph of x + 3y = 6 that passes through the x-intercept of the line.
Find slope of given line. 3
1m
Find the opposite reciprocal.
3
Find the x-intercept of the given line. 6y3x
603x 6y3x
x x6xy3
3 3 3
2x31
y
6x
Use slope-intercept or point-slope form to write the equation.
Coordinate of x-intercept is (6,0).
bmxy
b630
b180 1818b18
18x3y
Example 7 Write an equation in slope-intercept form for a line perpendicular to the graph of 3x + 2y = 8 that passes through the y-intercept of the line.
Find slope of given line. 2
3m
Find the opposite reciprocal. 3
2
Find the x-intercept of the given line. 8y2x3
8y203 8y2x3
x3 x38x3y2
2 2 2
4x23
y 228y2
4y
Use slope-intercept or point-slope form to write the equation.
Coordinate of y-intercept is (0,4).
bmxy
b032
4
b4
4x32
y
4-A15 Pages 240-241 #18–25,28-30,36 and
Skills Practice Wkbk. Page 30 #13-24.Review nightly for trimester 1 exam.