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# Proving slope is constant using similarity

Sal uses a clever proof involving similar triangles to show that slope is constant for a line. Created by Sal Khan.

Video transcript

We tend to be told in algebra
class that if we have a line, our line will have a
constant rate of change of y with respect to x. Or another way of
thinking about it, that our line will have
a constant inclination, or that our line will
have a constant slope. And our slope is literally
defined as your change in y-- this triangle is
the Greek letter delta. It's a shorthand
for "change in." It means change in y--
delta y means change in y-- over change in x. And if you're dealing with
a line, this right over here is constant for a line. What I want to do
in this video is to actually prove that
using similar triangles from geometry. So let's think about
2 sets of 2 points. So let's say that's
a point there. Let me do it in a
different color. Let me start at this point. And let me end up at that point. So what is our change in
x between these 2 points? So this point's x value
is right over here. This point's x value
is right over here. So our change in x is going
to be that right over there. And what's our change in y? Well, this point's y
value is right over here. This point's y value's
right over here. So this height or this
height is our change in y. So that is our change in y. Now, let's look
at 2 other points. Let's say I have this point
and this point right over here. And let's do the same exercise. What's the change in x? Well, let's see. If we're going-- this
point's x value's here. This point's x value's here. So if we start here
and we go this far, this would be the change
in x between this point and this point. And this is going
to be the change-- let me do that in
the same green color. So this is going to be the
change in x between those two points. And our change in y, well,
this y value is here. This y value's up here. So our change in y is going
to be that right over here. So what I need to
show-- I'm just picking 2 arbitrary points. I need to show that the
ratio of this change in y to this change of
x is going to be the same as the ratio
of this change in y to this change of x. Or the ratio of this purple
side to this green side is going to be the
same as the ratio of this purple side
to this green side. Remember, I'm just picking 2
sets of arbitrary points here. And the way that I will show
it is through similarity. If I can show that this triangle
is similar to this triangle, then we are all set up. And just as a reminder
of what similarity is, 2 triangles are similar--
and there's multiple ways of thinking about
it-- if and only if all corresponding--
or I should say, all three angles are the
same, or are congruent. And let me be careful here. They don't have to be
the same exact angle. The corresponding angles
have to be the same. So corresponding--
I always misspell it-- angles are
going to be equal. Or we could say
they are congruent. So for example, if I have
this triangle right over here. And this is 30, this
is 60, and this is 90. And then I have this
triangle right over here. I'll try to draw it--
so I have this triangle, where this is 30 degrees,
this is 60 degrees, and this is 90 degrees. Even though their side
lengths are different, these are going to
be similar triangles. They're essentially scaled
up versions of each other. All the corresponding
angles-- 60 corresponds to this 60, 30
corresponds to this 30, and 90 corresponds to this 1. So these 2 triangles
are similar. And what's neat about
similar triangles, if you can establish that
2 triangles are similar, then the ratio between
corresponding sides is going to be the same. So if these 2 are
similar, then the ratio of this side to
this side is going to be the same as
the ratio of-- let me do that pink color--
this side to this side. And so you can see
why that will be useful in proving that the
slope is constant here, because all we
have to do is look. If these 2 triangles
are similar, then the ratio between
corresponding sides is always going to be the same. We've picked 2 arbitrary
sets of points. Then this would be true,
really, for any 2 arbitrary set of points across the line. It would be true
for the entire line. So let's try to
prove similarity. So the first thing we
know is that both of these are right triangles. These green lines are
perfectly horizontal. These purple lines
are perfectly vertical because the green
lines literally go in the horizontal direction. The purple lines go in
the vertical direction. So let me make sure
that we mark that. So we know that these
are both right angles. So we have 1 corresponding
angle that is congruent. Now we have to show
that the other ones are. And we can show
that the other ones are using our knowledge
of parallel lines and transversals. Let's look at these
2 green lines. So I'll continue them. These are line segments,
but if we view them as lines and we just continue
them, on and on and on. So let me do that,
just like here. So this line is clearly
parallel to that 1. They essentially are
perfectly horizontal. And now you can view our
orange line as a transversal. And if you view it
as a transversal, then you know that this angle
corresponds to this angle. And we know from transversals
of parallel lines that corresponding
angles are congruent. So this angle is going to
be congruent to that angle right over there. Now, we make a very similar
argument for this angle, but now we use the
2 vertical lines. We know that this segment, we
could continue it as a line. So we could continue it,
if we wanted, as a line, so just like that,
a vertical line. And we could continue this
one as a vertical line. We know that these
are both vertical. They're just measuring--
they're exactly in the y direction,
the vertical direction. So this line is parallel to
this line right over here. Once again, our orange line
is a transversal of it. And this angle corresponds to
this angle right over here. And there we have it. They're congruent. Corresponding angles of the
transversal of 2 parallel lines are congruent. We learned that
in geometry class. And there you have it. All of the
corresponding-- this angle is congruent to this angle. This angle is congruent
to that angle. And then both of
these are 90 degrees. So both of these are
similar triangles. Just let me write
that down so we know that these are
both similar triangles. And now we can use the
common ratio of both sides. So for example, if we
called this side length a. And we said that this
side has length b. And we said this
side has length c. And this side has length d. We know for a fact
that the ratio, because these are
similar triangles, between corresponding
sides, the ratio of a to b is going to be equal
to the ratio of c to d. And that ratio is
literally the definition of slope, your change in
y over your change in x. And this is constant
because any right triangles that you generate
between these two points, we've just shown that they
are going to be similar. And if they are
similar, then the ratio of the length of
this vertical line segment to this horizontal
line segment is constant. That is the definition of slope. So the slope is
constant for a line.