Lines are parallel if their slopes are equal. Solve the following problems. Click “Submit” when finished.

You have seen that parallel lines do not intersect.You know that lines with unequal slopes are not parallel, and they do intersect.

When lines intersect, they form an angle. A right angle has a measure of <EQUATION>. If the angle formed by the two lines is a right angle, then the lines are called |B| perpendicular lines |/B| .

Let’s look at several numbers that have a product of <EQUATION>. <EQUATION>, <EQUATION>, and <EQUATION>. Do you notice anything special about these pairs of numbers? In order for two numbers to have a product of <EQUATION>, the numbers must be |B| negative reciprocals |/B| . That is, if the first number is <EQUATION>, the second number is <EQUATION>. To find the negative reciprocal of a number, you change the sign, and invert the fraction. For example <EQUATION> is equal to <EQUATION>, so its negative reciprocal is <EQUATION>.

The red and blue lines are perpendicular. Change the slope of the blue line, and see how the two slopes change. What do you notice when the blue line is horizontal? Can the slopes of perpendicular lines have the same sign? Can the slopes both be greater than <EQUATION> in absolute value? What happens when you make the slope of one line close to zero? What do you get when you multiply the slopes of perpendicular lines? Click “Done Exploring” when finished.

The line perpendicular to a horizontal line is vertical. A horizontal line has slope <EQUATION>, and a vertical line has undefined slope. For non-vertical perpendicular lines, the slopes have opposite signs—one is negative, and the other is positive. The smaller the slope of one line in absolute value, the larger the slope of the other line in absolute value. And, the slopes of perpendicular lines multiply to <EQUATION>.

If the slopes of two lines multiply to <EQUATION>, then the lines are perpendicular. So, if we know the equations of two lines, we can determine if the lines are perpendicular. Is the line <EQUATION> perpendicular to the line <EQUATION>? The graphs of these two lines look perpendicular, so let's compute their slopes to check. You know several methods for finding the slope of a line. You can find the ratio of the change in y to the change in <EQUATION> for two points on the line. Or you can convert to slope-intercept form. Or, if your line is in standard form, the slope is <EQUATION>. For the line <EQUATION>, let’s compute the slope between the two points <EQUATION> and <EQUATION>. The slope is <EQUATION>.

We will find the slope of the line <EQUATION> by solving for <EQUATION> to write the equation in slope-intercept form. The coefficient of <EQUATION>, <EQUATION>, is the slope. The <EQUATION>intercept agrees with the graph of the line.

Now, to determine if the lines are perpendicular, check whether the product of their slopes is <EQUATION>. The product is <EQUATION>, so the two lines are not perpendicular.

Any one line has many perpendicular lines. But there is only one line that is perpendicular to a given line and that passes through a given point.

What is the equation of the line perpendicular to <EQUATION> that passes through the point <EQUATION>? To find the equation of the perpendicular line, we need to know the slope of the given line. The slope of <EQUATION> is <EQUATION>.

The product of the slope of the given line and the slope of the perpendicular line must be <EQUATION>. So the slope of the perpendicular line is <EQUATION>, because <EQUATION>. So we want the equation of a line with slope <EQUATION> that passes through the point <EQUATION>. We will use the slope-intercept form to find the equation of the line. Substitute <EQUATION> for <EQUATION>, and substitute <EQUATION> for <EQUATION> and <EQUATION>. Then solve for <EQUATION>. We find that <EQUATION>, the <EQUATION>intercept, is <EQUATION>.

So, the slope-intercept form of the equation of the perpendicular line is <EQUATION>. We can graph this line, and verify that it appears to be perpendicular.

Select any of the lines that are perpendicular to <EQUATION>. Click “Submit” when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education