Parallel and Perpendicular Lines
Parallel Lines
Consider the two lines shown in the figure below.
Each of these lines has a slope of
, and
these lines are parallel. In general, we have the following fact.
Parallel Lines
Nonvertical parallel lines have equal slopes.
Of course, any two vertical lines are parallel, but we cannot say that they have equal
slopes because slope is not defined for vertical lines.
Example 1
Parallel lines
Line l goes through the origin and is parallel to the line through (-2, 3) and (4,
-5). Find the slope of line l.
Solution
The line through (-2, 3) and (4, -5) has slope
Because line l is parallel to a line with slope
, the slope of line
l is
also.
Perpendicular Lines
The lines shown in the figure below have slopes 2 and
.
These two lines appear to be
perpendicular to each other. It can be shown that a line is perpendicular to another
line if its slope is the negative of the reciprocal of the slope of the other.
Perpendicular Lines
Two lines with slopes m1 and m2 are perpendicular if and only if
Of course, any vertical line and any horizontal line are perpendicular, but we cannot
give a relationship between their slopes because slope is undefined for vertical lines.
Example 2
Perpendicular lines
Line l contains the point (1, 6) and is perpendicular to the line through
(-4, 1) and (3, -2). Find the slope of line l.
Solution
The line through (-4, 1) and (3, -2) has slope
Because line l is perpendicular to a line with slope
, the slope of line l is
.
Applications of Slope
When a geometric figure is located in a coordinate system, we can use slope to
determine whether it has any parallel or perpendicular sides.
Example 3
Using slope with geometric figures
Determine whether (-3, 2), (-2, -1), (4, 1), and (3, 4) are the vertices of a
rectangle.
Solution
The figure above shows the quadrilateral determined by these points. If a parallelogram
has at least one right angle, then it is a rectangle. Calculate the slope of each side.
Because the opposite sides have the same slope, they are parallel, and the figure is
a parallelogram. Because
is the opposite of the reciprocal of
-3, the intersecting
sides are perpendicular. Therefore the figure is a rectangle.
