Parallel and Perpendicular Lines

Now that we have a better understanding of
lines and angles
, we are ready to begin applying some of these concepts
onto the
Caresian coordinate plane
. We will use our previous knowledge of slopes
and
algebraic equations
to learn about parallel and perpendicular lines in the
coordinate plane.

Although the coordinate plane is used extensively in the study of algebra, it is
very useful in geometry as well. In algebra, when you study slope, essentially what
you are dealing with is angles. More specifically, the slope of a line is the measure
of an angle of a line from a perfectly horizontal line (or the x-axis). This concept
is illustrated below.This goes to show that different areas of mathematics are connected
and consistent with each other.

You can take an angle formed by two lines and place one of the lines on the x-axis
to see a relationship between angles and slopes.

Parallel Lines

Recall that two lines in a plane that never intersect are called parallel
lines. Working with parallel lines in the coordinate plane is fairly straightforward.
The reason for this is because the slope of a line is essentially the measure of
an angle of a line from a perfectly horizontal line (or the x-axis). Thus, in the
coordinate plane, if we want two different lines to never intersect, we simply apply
the same slopes to them.

Let’s take a look at the following equations:

How do we determine if these lines are parallel or if they intersect at some point?

First, it will help to put both equations in
slope-intercept form
. The first equation is already of this form so we do
not need to change it. The second equation, however, needs to be manipulated. Let’s
work it out:

Now, we add y to both sides of the equation to get

Subtracting 4x from both sides of the equation gives

Now, if we look at both equations, we notice that they both have slopes of 2. Since
both lines “rise” two units for every one unit they “run,” they will never intersect.
Thus, they are parallel lines. The graph of these equations is shown below.

We now see that the two lines are parallel. But how many more lines can we find
that are parallel to them? The answer is infinitely many. As long as the lines have
slopes of 2, they will never intersect.

Now let’s try a type of problem that requires a bit more work.

Example 1

Find the equation of a line that passes through the point (3, 1) and
is parallel to the line

In order to solve this kind of problem, we will need to keep in mind that parallel
lines have the same slope. We will also have to utilize what we know about equations
in slope-intercept form.

In slope-intercept form, x and y are variables that
will change, so we do not determine an exact value for them. All that is left to
solve for are m and b, where m is our
slope and b is the y-intercept of our line. Recall that parallel lines
have the same slope, so m = 2/3 in this example. We have:

We only need to solve for b now. We do this by plugging in the given
point, (3, 1), that lies on our line. This method is shown below.

Since we determined that m = 2/3 and that b = -1, we
can plug these values straight into our slope-intercept formula. This yields

We can take a look at the graph of these lines to see that this line is indeed parallel
to the given line and that it passes through (3, 1).

Perpendicular Lines

Pairs of lines that intersect each other at right angles are called perpendicular
lines. The symbol that represents perpendicularity between two lines is ?. Thus,
if line AB meets line CD at a 90° angle, we express it mathematically as
. Perpendicular lines are shown below.

The intersection of line AB with line CD forms a 90° angle

There is also a way of determining if two lines are perpendicular to each other
in the coordinate plane. While parallel lines have the same slope, lines that are
perpendicular to each other have opposite reciprocal slopes. We can determine
perpendicularity just by looking at the equations of lines just as we did with parallel
lines. For instance, consider the line

If we want to find the equation of a line that is perpendicular to the given line
we just need to follow two simple steps.

(1) Take the reciprocal (or flip the fraction) of the slope:

(2) Make it the opposite sign:

Any line with a slope of 2 will be perpendicular to the given line. Since there
are infinitely many lines with this slope, there are infinitely many lines perpendicular
to the given line.

Note: It is a common mistake to only take the reciprocal of a line’s slope
and forget about taking the opposite of the slope. Why doesn’t this work? If we
did not take the opposite sign of the slope, we would have two lines with either
positive or negative slopes. This would make it impossible for the lines to ever
meet at a 90°. In short, remember that perpendicular lines have opposite reciprocal
slopes.

Let’s try another example.

Example 2

Find the equation of the line that passes through the point (8, 1)
and is perpendicular to the line

Similar to the Example 1, we first identify what the slope of our equation should
be. The slope of the line we are given is -4, so we perform the following steps
to find the slope:

(1) Take the reciprocal of the slope:

(2) Make it the opposite sign:

Now we have

So, we plug in the the x and y values of the point we
were given to get

We now plug in the m and b values we have found, so
the equation of our line is

We see that there does indeed exist a right angle at the intersection of the two
lines in the figure shown below.

The lines are perpendicular to each other

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