In this lesson, the slope of a line segment connecting two points will be compared to the slope of segments parallel and perpendicular. A general formula for finding the slope of a perpendicular line segment will be developed and used.

Parallel Lines:

The line segment shown below connects the points (1, 2) and (3, –2). As we look at this diagram from left to right, the segment slopes downward. We will define a negative number to describe the downward slope and how steeply this line segment slopes. To do this, we notice that the vertical distance (rise) between these points is 4 and the horizontal distance (run) is 2. The slope is a ratio of vertical to horizontal distances: .

Any line parallel to this segment will also have the same slope of –2.

Perpendicular Lines:

Shown below is another segment of the same length perpendicular at (1, 2). The other endpoint is (– 3, 0). The rise between endpoints is 2 and the run is 4, the opposite of the rise and run for the segment connecting (1, 2) and (3, – 2). Also this new segment is sloping upward. The slope will be positive. We have slope = . Contrast this with the slope of the perpendicular which is – 2. This is the reciprocal, and differs in sign.

Summary:

1. If two lines are perpendicular, their slopes are negative reciprocals.
2. Parallel lines have the same slope.

Let's Practice:

1. If a line has a slope of 5, what is the slope of a line parallel and a line perpendicular?

The parallel line will have the same slope which is 5.
The perpendicular line will have a slope of which is the negative reciprocal.

To assist you in seeing this relationship, look at 5 as the fraction . Its negative reciprocal would then be .

2. If the equation of a line is given by , what is the slope of a line perpendicular?

Since the slope of the given line is , the negative reciprocal is .


An extension of this example would be to calculate the equation of the line that is perpendicular to our original line, and passes through the same y-intercept, (0,7). Using the slope-intercept form of a line, y = mx + b, we can generate its equation as y = x + 7. As shown below, these two lines would cross on the y-axis and be oriented at 90º to each other. Note that the dimensions of the graphing calculator's screen unfortunately do not give the correct "impression of perpendicularity."

3. If the equation of a line is given by , what is an equation of a line parallel and containing the point (2, 3)?

The equation of a line is given by . In this case, m = 4, . We have an equation given by .

4. What is an equation of a line perpendicular to the line given by the equation at the point
   (-1, -1)?

We find the slope of the line by solving for y: we have . Therefore the slope is and the slope of a perpendicular is . A line with this slope and containing the point (-1, -1) will have an equation given by .

Important facts:

1. If two line segments are parallel, their slopes are the same.
2. If two line segments are perpendicular, their slopes are negative reciprocals.