Introduction

Two lines are called perpendicular if they intersect in a 90° angle.  When perpendicular lines are found on the coordinate plane, there are several ways to analyze them.  The slopes, equations, and graphs of the lines can all be used to determine whether two lines are perpendicular or not. 

Lesson

Observe the two lines to the right.  They meet in a 90° angle, so they are perpendicular.  In this lesson, we will focus on the characteristics of these lines. The first characteristic that you should observe is the slopes of the lines.

• Line #1 has a steep positive slope
• Line #2 is not as steep and has a negative slope

The most obvious observation of the two perpendicular lines is that one has a positive slope while the other has a negative slope.  If the lines were rotated around the point of intersection, one slope would always be positive, while the other remained negative (except in the case of a vertical line and a horizontal line, where the slopes are 0 and ∞).  As the lines are rotated, one line becomes steeper as the other line becomes less steep.

The actual relationship between the slopes of perpendicular lines is as follows:

The slopes can be described as opposite reciprocals.  Their signs are opposite and the numerical parts are reciprocals.   

Determining whether two existing lines are perpendicular (or not)

Two lines can appear perpendicular on a graph, but to make sure they are actually perpendicular you have to do a little more work.  One option is to use a protractor and determine whether the angle of intersection is exactly 90°.  When lines are on the coordinate plane, they can be analyzed using their slopes.

Example 1:  Determine whether the lines are perpendicular in the graph on the left.

Equations:and

The slopes of the lines areand 3.  They are opposite reciprocals, so therefore the lines are perpendicular.

In example #2, find the slopes to determine which of the lines are perpendicular.

Example 2:  Determine which of the two blue lines is perpendicular to the red line.

Line #1’s slope is

Line #2’s slope is

Line #3’s slope is

 Line #2 and line #3 have opposite reciprocal slopes, so these two  lines are perpendicular.  Line #1 appears to be perpendicular to line #3 on the diagram, but the slopes indicate that the lines are not quite perpendicular.

 Construct a line that is perpendicular to a given line

In the diagram to the right, there are an infinite number of lines that can go through point A.  However, there is only one line that goes through point A that is perpendicular to line B. 

Perpendicular lines have opposite reciprocal slopes.  This information can be used to find the slope of any line that is perpendicular to line B, including the perpendicular line that goes through point A.

Example 3:  Find the equation of the line that goes through the point (4, 6) and is perpendicular to the line

The most common error in finding the equation of a perpendicular line is in figuring out the slope.  Remember that parallel lines have the same slope, but perpendicular lines have opposite reciprocal slopes.  The slope of the original line (in example 3) is -­½ , while the slope of the perpendicular line is 2.  One final example is done in much the same way…

Example 4:  Find the equation of the line that goes through the point (4, 6) and is perpendicular to the line  

Review

Find the equation of the perpendicular line.

1)  Find the equation of the line that goes through the point (2, 5) and is perpendicular to the line y = 2x + 1.

2)  Find the equation of the line that goes through the point (3, -2) and is perpendicular to the line 6y + 2x = 12.

Determine whether the lines are perpendicular

3)  y = x + 2  and  y = -x + 4

4)  y + 5 = -2x  and  y = 2 – 2x

5)  2x + 4y = 10  and  y = 2x

6)  5y = 10x + 15  and  2y + x = 6

Scroll down for answers…

Answers:

1)  The perpendicular line has an equation of y = -½x + 6.  

2)  The perpendicular line has an equation of y = 3x - 11.

Determine whether the lines are parallel

3)  Yes.  The slopes are 1 and -1 (opposite reciprocals)

4)  No.  The slopes are -2 and -2

5)  Yes.  The slopes are -½ and 2 (opposite reciprocals)   

6)  Yes.  The slopes are 2 and -½.