### Video Transcript

In this video, we will learn how to
use the concept of slope to determine whether two lines are parallel or
perpendicular. And then, weβll see how we can use
these geometric relationships to solve problems.

The slope of a line is a very
important feature of a line, and it describes how steep a line is. The slope of a line can be
calculated from any two distinct points on a line. In general, we can say that if
there are two points on a straight line with the coordinates π₯ sub zero, π¦ sub
zero and π₯ sub one, π¦ sub one, then we can calculate the slope, often referred to
using the letter π, as π¦ sub one minus π¦ sub zero over π₯ sub one minus π₯ sub
zero. To find the slope, weβre really
dividing the vertical displacement, thatβs the change in π¦, by the horizontal
displacement, or the change in π₯. As a recap of how this works in
practice, letβs say that we have the two coordinates four, six and 12, 10. We can define four, six to have the
π₯ sub zero, π¦ sub zero values, although it wouldnβt matter if we define these with
the π₯ sub one, π¦ sub one values.

Substituting these into the slope
formula, we would have that π is equal to 10 minus six over 12 minus four. This would simplify to four over
eight, which in turn simplifies to one-half. The slope of this line is
one-half. We often think of this calculation
in very much algebraic terms, but letβs take a closer look at the geometry
involved. When weβre finding the slope of a
line, we are creating a right triangle. The lengths of the two shorter
sides are the horizontal and vertical displacements. So when we think about slope in
terms of right triangles, then we can use results that we know from trigonometry to
understand other properties of the straight line.

One property that weβre often
interested in is the acute angle that the line makes with the horizontal axis, which
we can label as πΌ. The important thing to note here is
that the angle πΌ made between line π΄π΅ and this horizontal line will be the same
as the angle πΌ made between the line π΄π΅ and the horizontal axis because these two
horizontal lines are parallel. So we can find the angle between
the straight line and the horizontal axis by finding the angle πΌ. And as already mentioned, we can do
this by using trigonometry. In this problem, we have the angle
πΌ, we have the side opposite the angle, and we have the side adjacent to the
angle.

We can therefore use the fact that
the tangent is the ratio of the opposite side and the adjacent side in a right
triangle. And so we have that tan of πΌ is
equal to π¦ sub one minus π¦ sub zero over π₯ sub one minus π₯ sub zero. In other words, tan of the angle πΌ
is simply equal to π, where π is the slope of the straight line. So remember that we worked out the
slope of this example line as one-half. If we then wanted to work out the
angle πΌ, we know that tan of πΌ is equal to one-half. Therefore, πΌ is equal to arctan of
one-half. In degrees, this is approximately
26.57 degrees to two decimal places. We can also use this approach to
find the angle between a straight line and the horizontal axis when the angle is not
acute.

We know that it is possible for the
slope of a straight line to be negative, which happens when π¦ sub one minus π¦ sub
zero and π₯ sub one minus π₯ sub zero are of an opposite sign. In this case, the straight line
will go downwards from left to right. And then the positive angle, which
is the angle measured clockwise, between the positive direction of the π₯-axis and
the straight line is then obtuse. As we can observe, by using a
calculator, the tangent of an obtuse angle is negative. And so the relationship we have
found between the slope of a line and the tangent of the positive angle the line
makes with the positive direction of the π₯-axis also holds for obtuse angles.

We can now make a more formal note
of what we have learned. Firstly, we know that the slope π
between two coordinates π₯ sub zero, π¦ sub zero and π₯ sub one, π¦ sub one is given
as π equals π¦ sub one minus π¦ sub zero over π₯ sub one minus π₯ sub zero. Furthermore, the slope is equal to
the tangent of the positive angle made between the straight line and the positive
direction of the π₯-axis such that π is equal to tan of πΌ. The angle πΌ is measured from the
positive π₯-axis to the line in a counterclockwise direction. An acute angle has a positive
tangent, whereas an obtuse angle has a negative tangent. And a final note that because the
tan of an angle of 90 degrees is undefined, then vertical lines are said to have an
undefined slope.

Weβll now take a look at an example
where we find the slope of a line given the angle it makes with the horizontal
axis.

Find, to the nearest two
decimal places, the slope of the line that makes a positive angle of 60 degrees
with the positive direction of the π₯ axis.

We can begin this problem by
visualizing a line that makes an angle of 60 degrees with the positive direction
of the π₯-axis. In order to answer this
problem, we will also need to remember that the slope of a straight line π is
equal to the tangent of the positive angle made between the straight line and
the positive direction of the π₯-axis. In this question, that angle
would be 60 degrees. So we would have that π is
equal to tan of 60 degrees. tan of 60 is equal to root
three, but as a decimal it would be 1.732 and so on. Rounded to the nearest two
decimal places then, we can say that the slope of the line is 1.73.

We will now move on to looking at
parallel and perpendicular lines. Letβs consider the fact that two
lines meet at a point unless they are parallel or coincident. Coincident lines will lie exactly
on top of one another. We know that two lines are parallel
if they have the same slope. And from what we have just seen in
this video, we can now add that lines are parallel if they make the same angle with
the positive direction of the π₯-axis. If lines have a slope of zero, then
they are parallel to the π₯-axis and parallel to each other even if they donβt cross
the π₯-axis. Two lines are parallel but not
coincident when they have the same slope but not the same π¦-intercept, as shown in
this diagram.

We should recall that perpendicular
lines meet at a point and make an angle of 90 degrees with each other. We will now see what this means for
the slopes of two perpendicular lines. Letβs take these two lines, which
have slopes of π sub one and π sub two. They make angles of πΌ and π½,
respectively, with the positive direction of the π₯-axis.

One of the things we can say is
that because the lines are perpendicular, then π½ is equal to πΌ plus 90
degrees. A property of the tangent function
is that tan of πΌ is equal to negative one over tan πΌ plus 90 degrees. Then, combining these two
equations, we have that tan of πΌ is equal to negative one over tan of π½. And then, because we know that π
sub one is equal to tan of πΌ and π sub two is equal to tan of π½, we have that π
sub one is equal to negative one over π sub two. Alternatively, this can be written
as π sub one, π sub two is equal to negative one.

You might wonder why this is
important, but what we have really demonstrated here is that the product of the
perpendicular slopes is negative one. This is a very important property
of perpendicular lines, but note that if a line is horizontal, the slope is
zero. For example, if π sub two is zero,
then to find π sub one, we would be attempting to divide by zero. This would give us an undefined
value, but of course the slope of a vertical line is undefined. This makes sense because we know
that a vertical line is perpendicular to a horizontal line. But we canβt automatically use this
algebraic fact that π sub one times π sub two equals negative one with horizontal
and vertical lines.

We can now make a quick summary of
the conditions for parallel and perpendicular lines. We can identify parallel lines as
having the same slope and a different π¦- intercept. Then, lines which are identical
have the same slope and the same π¦-intercept. And then, when the product of the
slopes is equal to negative one, then the two lines are perpendicular. And as previously noted, if the
slope of the line is zero, then the straight line is horizontal. Any line which is perpendicular to
this would not have a defined slope.

In the next example, weβll see how
we can find the slope of a straight line given the slope of a perpendicular one.

If line π΄π΅ is perpendicular
to line πΆπ· and the slope of line π΄π΅ equals two-fifths, find the slope of
line πΆπ·.

Here, we are told that we have
two perpendicular lines π΄π΅ and πΆπ·. Knowing that two lines are
perpendicular means that we know something about the relationship between their
slopes. If we define line π΄π΅ to have
a slope of π sub one and line πΆπ· to have a slope of π sub two, then we know
that π sub two is equal to negative one over π sub one. Given that the slope π sub one
of line π΄π΅ is two-fifths, then π sub two is equal to negative one over
two-fifths. This simplifies to negative
five over two. Because these two lines are
perpendicular, their slopes will be the negative reciprocal of one another. And so the slope of line πΆπ·
is negative five over two.

In the next example, weβll identify
the relationship between two straight lines.

Let πΏ be the line through the
points negative seven, negative seven and negative nine, six and π the line
through one, one and 14, three. Which of the following is true
about the lines πΏ and π? Option (A) they are parallel,
option (B) they are perpendicular, or option (C) they are intersecting but not
perpendicular.

It might be worthwhile
beginning this question with a quick sketch of the two lines through the two
sets of points. When we do this, we can observe
that the two lines do in fact intersect. We can therefore say that these
two lines are not parallel, so we can eliminate option (A). Now, we can remember that two
lines are perpendicular if they intersect or meet at right angles. From the diagram, it does
appear that the two lines are at right angles. But it could be the case that
the two lines are nearly perpendicular and itβs not possible to distinguish this
from the diagram. Generally, itβs not a very good
idea to just use a sketch to determine if lines are parallel or
perpendicular. In fact, we should perform some
sort of calculation.

We can recall that if two
straight lines have slopes of π sub one and π sub two, then they are
perpendicular if π sub two is equal to negative one over π sub one. Weβll first need to calculate
the slopes of each of the lines πΏ and π. The slope of the line passing
through two points with coordinates π₯ sub zero, π¦ sub zero and π₯ sub one, π¦
sub one is calculated as the slope π is equal to π¦ sub one minus π¦ sub zero
over π₯ sub one minus π₯ sub zero. For line πΏ then, its slope π
sub one is equal to six minus negative seven over negative nine minus negative
seven, which simplifies to negative 13 over two.

Now, letβs find the slope of
the line π. Its slope π sub two will be
calculated as three minus one over 14 minus one, and this is equal to two over
13. Now, we can check if π sub two
is equal to negative one over π sub one. If we didnβt know the value of
π sub two, we could find a perpendicular line to the line πΏ by taking π sub
two and setting it equal to negative one over negative 13 over two. And this would indeed give us a
value of two thirteenths for π sub two. We can therefore give the
answer that the statement which is true about the lines πΏ and π is option
(B). They are perpendicular.

We will now summarize the key
points of this video. The slope π of a straight line
passing through π₯ sub zero, π¦ sub zero and π₯ sub one, π¦ sub one is given by π
equals π¦ sub one minus π¦ sub zero over π₯ sub one minus π₯ sub zero. The angle πΌ is measured from the
horizontal axis, rotating counterclockwise until meeting the straight line. For π being the slope of a
straight line, the following results hold. For π is greater than or equal to
zero, the angle πΌ between this straight line and the horizontal axis is expressed
as πΌ equals arctan of π. For π is less than zero, the angle
πΌ between this straight line and the horizontal axis is expressed as 180 degrees
plus arctan of π. And for vertical lines, πΌ equals
90 degrees.

We also saw that if we take two
straight lines with slopes π sub one and π sub two and a π¦-intercept π one and
π two, then if π sub one equals π sub two and π sub one is not equal to π sub
two, then the two lines are distinct and parallel. This means that the lines never
meet and that they make the same angle with the horizontal axis. But if π sub one equals π sub two
and π sub one equals π sub two, then the lines are coincident or identical. But if π sub one times π sub two
equals negative one, then the two lines are perpendicular.