### Video Transcript

In this video, we will learn how to
write the equation of a line parallel or perpendicular to another line. Weโll consider cases where we
already know the slope. And weโll also look at cases where
weโre given two points and we need to find the slope of that line before finding the
slope of a parallel or a perpendicular line.

But first, letโs review a few
things about lines. ๐ฆ equals ๐๐ฅ plus ๐ is a general
form for a straight-line equation. And in particular, we call this the
slopeโintercept form. In this form, the coefficient of ๐ฅ
โ the ๐ variable โ is the slope. And ๐ equals the ๐ฆ-intercept, the
place where this function crosses the ๐ฆ-axis. As we said on the first slide, we
want to consider parallel lines and perpendicular lines.

We know parallel lines never
intersect one another; they never cross. We usually represent that with a
small triangle on either line. And occasionally, when thereโs more
than one pair of parallel lines, you might see them noted with a multiple of the
little triangles. In that case, the parallel lines
will be the pair with the matching symbols. What weโre showing are two sets of
parallel lines.

Itโs also worth noting that you
might see this parallel symbol used to represent parallel lines. Here, line segment ๐ด๐ต is parallel
to line segment ๐ถ๐ท. But the most important thing to
remember about parallel lines is that they have the same slope. If we look at straight-line
equations given in slopeโintercept form and they have the same value for the
variable ๐ for the coefficient of ๐ฅ, they will be parallel.

Now, letโs consider what
perpendicular lines are. Perpendicular lines intersect at a
90-degree angle. And this is often noted with the
right angle symbol. Now, the set of perpendicular lines
that Iโve drawn here are horizontal and vertical. But this is not always the
case. Perpendicular lines can occur at
any orientation. And when it comes to the slope of
perpendicular lines, they are negative reciprocals of one another. So, parallel lines have the same
slope; their ๐ values are the same. And in perpendicular lines, if one
of the lines has a slope of ๐, the other line will have the negative reciprocal,
which is negative one over ๐. Itโs also worth noting here that
there are lines that intersect that are neither parallel or perpendicular. Lines that intersect forming any
angle besides a right angle do not fit in the category parallel or
perpendicular.

Letโs use this information to start
working with the equations of parallel and perpendicular lines.

Determine whether the lines ๐ฆ
equals negative one-seventh ๐ฅ minus five and ๐ฆ equals negative one-seventh ๐ฅ
minus one are parallel, perpendicular, or neither.

The categories parallel,
perpendicular, or neither are always, we categorize, intersections of lines. Parallel lines do not
intersect. Perpendicular lines intersect at a
90-degree angle. The category neither here
represents all the lines that do intersect but do not form a 90-degree angle. Parallel, perpendicular, or
neither.

But weโre not given a graph for
these two lines. Of course, we could try and draw a
graph for both of these lines. But we can determine parallel,
perpendicular, or neither without graphing these two equations. Both of these straight lines are
given in the form ๐ฆ equals ๐๐ฅ plus ๐. In both cases, the coefficient of
๐ฅ โ the ๐ variable โ is negative one-seventh. The ๐ variable represents the
slope. And so, we can say that the slope
of line one is negative one-seventh and the slope of line two is negative one
seventh, which reminds us, โparallel lines have the same slope.โ That is why they do not
intersect. Since both of these lines have a
slope of negative one-seventh, we can classify them as parallel lines without
graphing.

In this example, weโll again be
classifying lines. But this time weโre not given the
equations of the lines. Weโre only given two points that
lie on each of the lines.

Given that the coordinates of the
points ๐ด, ๐ต, ๐ถ, and ๐ท are negative 15, eight; negative six, 10; negative eight,
negative seven; and negative six, negative 16, respectively, determine whether line
๐ด๐ต and line ๐ถ๐ท are parallel, perpendicular, or neither.

Points ๐ด and ๐ต fall on the line
๐ด๐ต and points ๐ถ and ๐ท fall on the line ๐ถ๐ท. To classify the lines, we have to
remember: parallel lines have the same slope and they do not intersect. Perpendicular lines have negative
reciprocal slopes and intersect at a 90-degree angle. And neither are lines that are not
parallel or perpendicular, lines that do intersect but do not form a right
angle. This means to consider whether or
not these lines are parallel or perpendicular, we need to know the slopes of these
lines.

In the general form, ๐ฆ equals ๐๐ฅ
plus ๐, the ๐ represents the slope. And we can find the slope ๐ if we
have two points by saying ๐ equals ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ
one. In order to categorize these lines,
we need to find the slopes of line ๐ด๐ต and line ๐ถ๐ท. We can start with line ๐ด๐ต. Let point ๐ด be ๐ฅ one, ๐ฆ one and
point ๐ต be ๐ฅ two, ๐ฆ two. Then, the slope will be 10 minus
eight over negative six minus negative 15. 10 minus eight is two. Negative six minus negative 15 is
negative six plus 15, which is positive nine. So, we can say that the slope of
line ๐ด๐ต is two-ninths.

We repeat this process for line
๐ถ๐ท. Let ๐ถ be ๐ฅ one, ๐ฆ one and ๐ท be
๐ฅ two, ๐ฆ two. And weโll get ๐ equals negative 16
minus negative seven over negative six minus negative eight. Negative 16 minus negative seven is
negative 16 plus seven, which is negative nine. Negative six minus negative eight
is negative six plus eight which is two. The slope of line ๐ถ๐ท is then
negative nine over two.

If we compared these two slopes,
negative nine over two is the negative reciprocal of two over nine. And if you werenโt sure, you can
multiply them together. Reciprocals multiply together to
equal one and negative reciprocals multiply together to equal negative one. These two slopes are the negative
reciprocals of one another, making these lines perpendicular.

Hereโs another example.

Which axis is the straight line ๐ฆ
equals three parallel to?

First, we know that parallel lines
do not intersect and they have the same slope. We have the equation ๐ฆ equals
three. If we think about the general form
for a straight line ๐ฆ equals ๐๐ฅ plus ๐ and we have ๐ฆ equals three, which means
that we have a slope of zero. From there, if we sketch a ๐ฆ- and
๐ฅ-axis, graphing the line ๐ฆ equals three would look like this. We should notice here that the line
๐ฆ equals three crosses the ๐ฆ-axis. And we know thatโs true because it
has a ๐-value of three. And in the general form, the ๐
represents the ๐ฆ-intercept.

We can, therefore, say that the
straight line ๐ฆ equals three is not parallel to the ๐ฆ-axis because it intersects
the ๐ฆ-axis. But we can say that ๐ฆ equals three
is parallel to the ๐ฅ-axis. The ๐ฅ-axis is the line where ๐ฆ
equals zero. The straight line ๐ฆ equals three
is parallel to the ๐ฅ-axis.

In the next example, we need to
find the equation of a line if weโre given a point on that line and then two points
on a line perpendicular to that line.

Determine, in slopeโintercept form,
the equation of the line passing through ๐ด: 13, negative seven perpendicular to the
line passing through ๐ต: eight, negative nine and ๐ถ: negative eight, 10.

So, hereโs what weโre thinking. We have points ๐ต and ๐ถ, which
form a line. Point ๐ด does not fall on this
line. But point ๐ด falls on a line
perpendicular to the line ๐ต๐ถ. And weโre trying to find in
slopeโintercept form the equation of the line that passes through point ๐ด. Slopeโintercept form is the form ๐ฆ
equals ๐๐ฅ plus ๐. That means we need the slope of
this line and the ๐ฆ-intercept. But since we donโt know two points
along this line, weโll have to find the slope a different way.

We remember perpendicular lines
have negative reciprocal slopes. And since we do know two points
along the line ๐ต๐ถ, we can find the slope of line ๐ต๐ถ. And the slope along the line that
includes point ๐ด will be equal to negative one over the slope of the line from
๐ต๐ถ. This is just a mathematical way to
say that these two values will be negative reciprocals of one another. This means our first job is to find
the slope of line ๐ต๐ถ. If we know two points along the
line, we can find their slope by taking ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ
one.

Weโll let ๐ต be ๐ฅ one, ๐ฆ one and
๐ถ be ๐ฅ two, ๐ฆ two. And the slope of line ๐ต๐ถ will be
equal to 10 minus negative nine over negative eight minus eight. 10 minus negative nine is 19. And negative eight minus eight
equals negative 16. We can say that the slope of line
๐ต๐ถ is 19 over negative 16. But more commonly, we would include
the negative sign in the numerator and say the slope of line ๐ต๐ถ equals negative 19
over 16. The slope of the line containing
point ๐ด is the negative reciprocal of this value.

To find the reciprocal of a
fraction, we flip it. The reciprocal of negative 19 over
16 is 16 over negative 19. But we have to be careful here
because we need the negative reciprocal. And that means negative 16 over
negative 19 simplifies to 16 over 19. The slope of the line passing
through point ๐ด is then 16 over 19. At this point, we have the slope of
the line passing through point ๐ด. And we have one point that falls
along that line.

To find the ๐ฆ-intercept form of
this equation, we could then use the pointโslope formula, which says ๐ฆ minus ๐ฆ one
equals ๐ times ๐ฅ minus ๐ฅ one, where ๐ฅ one, ๐ฆ one is a point along the line. Point ๐ด is ๐ฅ one, ๐ฆ one. And so, we have ๐ฆ minus negative
seven equals 16 over 19 times ๐ฅ minus 13. Minus negative seven is plus
seven. We distribute that 16 over 19 times
๐ฅ. And 16 over 19 times negative 13
equals negative 208 over 19.

Since we want the equation in
slopeโintercept form, we need to get ๐ฆ by itself by subtracting seven from both
sides. Negative 208 over 19 minus seven is
negative 341 over 19. This line ๐ฆ equals 16 over 19๐ฅ
minus 341 over 19 is perpendicular to the line ๐ต๐ถ and passes through point ๐ด.

Hereโs another example involving
parallel lines.

The straight lines eight ๐ฅ plus
five ๐ฆ equals eight and eight ๐ฅ plus ๐๐ฆ equals negative eight are parallel. What is the value of ๐?

We know that parallel lines have
the same slope. And in slopeโintercept form, ๐ฆ
equals ๐๐ฅ plus ๐, the coefficient of the ๐ฅ-variable ๐ represents the slope. Weโve been told that these two
lines are parallel. And that means they will have the
same slope. To find the slope of these lines,
weโll convert them to slopeโintercept form. To do that, we get ๐ฆ by
itself. Since both equations have eight ๐ฅ
on the left, weโll subtract eight ๐ฅ from both sides of both equations. On the left, we would have five ๐ฆ
equals negative eight ๐ฅ plus eight. And on the right, ๐๐ฆ equals
negative eight ๐ฅ minus eight. We need to get ๐ฆ by itself for
slopeโintercept form. So, we can divide through by
five. And the equation on the left in
slopeโintercept form will be ๐ฆ equals negative eight-fifths ๐ฅ plus
eight-fifths.

On the right, we need to do
something similar. To get ๐ฆ by itself, weโll divide
through by ๐. And our second equation will be ๐ฆ
equals negative eight over ๐๐ฅ minus eight over ๐. These slopes have to be equal to
each other if these lines are parallel. Since one of the slopes is negative
eight over five, the other slope will need to be negative eight over five. And that tells us that ๐ must be
five for these two lines to be parallel. If we go back in and plug in five
for ๐, we see that the ratio of coefficients between these two equations are equal
to each other, which makes them parallel.

In our final example, weโre given
three points that form a right-angled triangle. And weโll use what we know about
parallel or perpendicular lines to solve for a missing value in one of the
points.

Suppose that the points ๐ด:
negative three, negative one; ๐ต: one, two; and ๐ถ: seven, ๐ฆ form a right-angled
triangle at ๐ต. What is the value of ๐ฆ?

We can go ahead and make a sketch
of these points. ๐ด is negative three, negative
one. ๐ต is one, two. We know the ๐ฅ-coordinate of point
๐ถ is seven. And that means ๐ถ will be located
somewhere along this line. We know the line ๐ด๐ต. And weโve been told that the right
angle of this triangle is at point ๐ต. We can get a general idea of where
we think point ๐ถ would be. But this is not a good way to find
an accurate answer. But because we know this is a right
triangle, we could say that line ๐ด๐ต is perpendicular to line ๐ต๐ถ. That means the slope of line
segment ๐ต๐ถ is the negative reciprocal of the slope of line segment ๐ด๐ต.

To solve this problem, weโll need
to do three things. First, find the slope of line
segment ๐ด๐ต. Use that slope to find the negative
reciprocal, which is the slope of line segment ๐ต๐ถ. Then, take the slope of line ๐ต๐ถ
and use that to find the ๐ฆ-value in point ๐ถ. If we have two points, we find the
slope by using ๐ equals ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one. For the points ๐ด and ๐ต, that
would be two minus negative one over one minus negative three, which equals
three-fourths. The slope of line ๐ด๐ต is then
three-fourths. And weโve completed step one.

For step two, we need to take the
negative reciprocal of the slope we found in step one. The negative reciprocal of
three-fourths is negative four-thirds. And that is step two. Now, for step three, weโll take
point ๐ต: one, two and point ๐ถ: seven, ๐ฆ. Weโll let ๐ต be ๐ฅ one, ๐ฆ one and
๐ถ be ๐ฅ two, ๐ฆ two. The slope negative four-thirds is
equal to ๐ฆ minus two over seven minus one. Seven minus one is six. To solve this, we cross
multiply. Negative four times six equals
three times ๐ฆ minus two. Negative 24 equals three ๐ฆ minus
six.

To give us a bit more room to solve
for ๐ฆ, we add six to both sides and we get negative 18 equals three ๐ฆ. Divide both sides of the equation
by three and we get negative six equals ๐ฆ. And we found from step three that
๐ฆ must be equal to negative six. This means, for this to be a right
triangle, point ๐ถ needs to be located at seven, negative six. And so, we found that missing value
to be negative six.

To wrap up, weโll review our key
points. Parallel lines do not intersect and
have the same slope. Perpendicular lines intersect at a
90-degree angle and have negative reciprocal slopes of one another.