Math Tutorial: Graphs of Parallel and Perpendicular Lines
In this example, the line perpendicular to a line with negative slope is found.
Review of parallel and perpendicular lines:
From geometry you know that (1) parallel lines do not intersect and (2) perpendicular lines do intersect at a 90-degree angle. This is true of all lines in Euclidean two-dimensional space.
These concepts are also true on the Cartesian coordinate plane, which consists of an underlying grid of parallel and perpendicular lines. The difference is that on the Cartesian plane, a line can be defined based on its slope and y-interecept, using the slope-intercept form, shown here.
y = mx + b
When two lines intersect, there are two coordinates (x, y) that the two lines share in common. When two lines are parallel, they have no points in common. When they are perpendicular, they have one point in common.
Parallel lines. Two lines on the Cartesian plane are parallel if they have the same slope. Let's look at various scenarios involving parallel lines (in each case assume that two lines have the same value for slope, m):
- Math Tutorial: Different values for the y-intercept. This is the common example of two distinct lines on the Cartesian plane that are parallel. The equation of each parallel line has the same value for m, but a different value for b.
- Math Tutorial: Same value for the y-intercept. You may come across a situation where two linear equations may look different, but may result in the same graph. For example, one equation may be written in standard form and another in slope-intercept, but their graph is identical. In such a situation, you have a line parallel to itself.
- Math Tutorial: Two horizontal lines parallel to each other. Recall that the slope of a horizontal line is zero. So, of you see two equations of the form y = c, where c is a constant, then the lines are parallel, because the slope is zero. In such cases you won't see an x-term, since m = 0. But the lines are parallel.
- Math Tutorial: Two vertical lines parallel to each other. Recall that the slope of a vertical line is undefined. So, of you see two equations of the form x = c, where c is a constant, then the lines are parallel, because the slope is undefined for each. In such cases you won't see a y-term, since m is undefined. But the lines are parallel.
Perpendicular lines. Two lines on the Cartesian plane are perpendicular if their slopes are negative reciprocals of each other. Let's look at various scenarios involving perpendicular lines (in each case assume that two lines are perpendicular):
- Math Tutorial: Positive and negative slopes. If one line has a positive slope, then the perpendicular line has a negative slope. And vice-versa.
- Math Tutorial: One perpendicular line is horizontal. If one of the lines is horizontal, then its slope is zero. This means that the perpendicular line has an undefined slope, since this would result in having a zero in the denominator of the slope. This means that the perpendicular line is vertical.
- Math Tutorial: One perpendicular line is vertical. If one of the lines is vertical, then its slope is undefined. This means that the line perpendicular to it is horizontal.
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