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Algebra 1

1.14.4 Writing an Equation of a Line Perpendicular to a Given Line

Algebra 11.14.4 Writing an Equation of a Line Perpendicular to a Given Line
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Activity

1.

Find the equation of the line using point-slope form: m = 2 m = 2 , ( 3 , 1 ) ( 3 , 1 )

2.

Rewrite your equation from Question 1 in slope-intercept form.

3.

Find the equation of the line using point-slope form: m = − 1 2 m = − 1 2 , ( − 1 , 4 ) ( − 1 , 4 )

4.

Rewrite your equation from Question 3 in slope-intercept form.

5.

Find the equation of the line using point-slope form: m = − 4 m = − 4 , ( 2 , 1 ) ( 2 , 1 )

6.

Rewrite your equation from Question 5 in slope-intercept form.

7.

Find the equation of the line using point-slope form: m = 1 4 m = 1 4 , ( 3 , 2 ) ( 3 , 2 )

8.

Rewrite your equation from Question 7 in slope-intercept form.

Use the graphing tool or technology outside the course.

Access multimedia content
9.

Graph your equations from questions 2 and 4 on the same graph using the Desmos tool.

Use the graphing tool or technology outside the course.

Access multimedia content
10.

Graph your equations from questions 6 and 8 using the Desmos tool.

11.

What is the slope of the line perpendicular to a line with a slope of  1 4 1 4 ?

12.

What is the slope of the line perpendicular to a line with a slope of -8?

13.

What is the slope of the line perpendicular to a line with a slope of − 6 11 − 6 11 ?

Self Check

Which equation represents an equation of a line perpendicular to the line  y = 1 2 x − 3 that contains the point ( 6 , 4 ) ?
  1. y = − 2 x − 2
  2. y = 2 x − 8
  3. y = − 2 x + 14
  4. y = − 2 x + 16

Additional Resources

Finding a Perpendicular Line Given an Equation and Point

In this section, you will use what you know about writing the equation of a line given the slope and a point to write the equation of perpendicular lines. The graph shows the equation of the line y = 2 x − 3 y = 2 x − 3 . The point P ( − 2 , 1 ) P ( − 2 , 1 ) is also plotted. You can use the fact that perpendicular lines have slopes that are opposite reciprocals to find a line that is perpendicular to line l l and goes through point P ( − 2 , 1 ) P ( − 2 , 1 ) .

A graph with a blue line with negative slope. A light blue dot is at the point (-2, 1). The x- and y-axes range from -6 to 6.

From the equation, you know the slope of the line is 2. The second line will pass through P ( − 2 , 1 ) P ( − 2 , 1 ) and have slope − 1 2 − 1 2 . To graph the line, start at ( − 2 , 1 ) ( − 2 , 1 ) and count out the rise and run. The slope is r i s e r u n = − 1 2 r i s e r u n = − 1 2 . Count out the rise, − 1 − 1 , and run, 2, and plot the point. You can graph the line as shown. Line n n is perpendicular to line l l and goes through point ( − 2 , 1 ) ( − 2 , 1 ) .

A graph with x- and y-axes showing two intersecting lines. A red right triangle highlights the vertical and horizontal distances between the two points where the lines intersect the grid.

To find the equation of a line perpendicular to a line through a given point algebraically, you can use what you know about finding the equation of a line given the slope and a point.

Example

Write an equation of a line perpendicular to y = 2 x − 3 y = 2 x − 3 that contains the point ( − 2 , 1 ) ( − 2 , 1 ) . Write the equation in slope-intercept form.

Step 1 - Find the slope of the given line.

The line is in slope-intercept form y = 2 x − 3 y = 2 x − 3

m = 2 m = 2

Step 2 - Find the slope of the perpendicular line perpendicular lines have the inverse slope

m 1 = − 1 / 2 m 1 = − 1 / 2

Step 3 - Identify the point

The given point is ( − 2 , 1 ) ( − 2 , 1 )

( x 1 , y 1 ) = ( − 2 , 1 ) ( x 1 , y 1 ) = ( − 2 , 1 )

Step 4 - Substitute values into the point-slope form y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 )

Simplify

y − y 1 = m ( x − x 1 ) y − 1 = − 1 2 ( x − ( − 2 ) ) y − 1 = − 1 2 ( x + 2 ) y − 1 = − 1 2 x − 1 y − y 1 = m ( x − x 1 ) y − 1 = − 1 2 ( x − ( − 2 ) ) y − 1 = − 1 2 ( x + 2 ) y − 1 = − 1 2 x − 1

Step 5 - Write the equation in slope-intercept form

y = − 1 2 x y = − 1 2 x

Use this table for reference to write an equation of a line perpendicular to a given line:

Step 1 - Find the slope of the given line.

Step 2 - Find the slope of the perpendicular line.

Step 3 - Identify the point.

Step 4 - Substitute the values into the point-slope form:

y − y 1 = m ( x − x 1 ) y − y 1 = m ( x − x 1 ) .

Step 5 - Simplify.

Step 6 - Write the equation in slope-intercept form ( y = m x + b ) ( y = m x + b ) .

Try it

Try It: Finding a Perpendicular Line Given an Equation and Point

Find an equation of a line perpendicular to the line y = 3 x + 1 y = 3 x + 1 that contains the point ( 4 , 2 ) ( 4 , 2 ) . Write the equation in slope-intercept form.

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