WEBVTT

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This triangle that we have right
over here is a right triangle.

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And it's a right triangle
because it has a 90 degree

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angle, or has a
right angle in it.

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Now, we call the longest
side of a right triangle,

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we call that side,
and you could either

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view it as the longest side of
the right triangle or the side

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opposite the 90 degree angle,
it is called a hypotenuse.

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It's a very fancy word
for a fairly simple idea,

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just the longest side of a
right triangle or the side

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opposite the 90 degree angle.

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And it's just good to
know that because someone

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might say hypotenuse.

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You're like, oh, they're just
talking about this side right

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here, the side longest, the side
opposite the 90 degree angle.

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Now, what I want
to do in this video

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is prove a relationship, a
very famous relationship.

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And you might see
where this is going.

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A very famous relationship
between the lengths

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of the sides of
a right triangle.

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So let's say that the length of
AC, so uppercase A, uppercase

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C, let's call that
length lowercase a.

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Let's call the length of BC
lowercase b right over here.

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I'll use uppercases for
points, lowercases for lengths.

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And let's call the length of the
hypotenuse, the length of AB,

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let's call that c.

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And let's see if we can come
up with the relationship

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between a, b, and c.

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And to do that I'm
first going to construct

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another line or
another segment, I

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should say, between
c and the hypotenuse.

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And I'm going to
construct it so that they

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intersect at a right angle.

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And you can always do that.

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And we'll call this point
right over here we'll.

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Call this point capital D.

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And if you're wondering,
how can you always do that?

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You could imagine rotating
this entire triangle like this.

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This isn't a rigorous proof,
but it just kind of gives you

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the general idea of
how you can always

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construct a point like this.

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So if I've rotated it around.

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So now our hypotenuse, we're
now sitting on our hypotenuse.

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This is now point
B, this is point A.

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So we've rotated the whole
thing all the way around.

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This is point C. You
could imagine just

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dropping a rock from point C,
maybe with a string attached,

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and it would hit the
hypotenuse at a right angle.

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So that's all we did here to
establish segment CD into where

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we put our point D
right over there.

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And the reason why
did that is now we

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can do all sorts of
interesting relationships

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between similar triangles.

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Because we have
three triangles here.

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We have triangle ADC,
we have triangle DBC,

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and then we have the
larger original triangle.

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And we can hopefully
establish similarity

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between those triangles.

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And first I'll show you that ADC
is similar to the larger one.

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Because both of them
have a right angle.

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ADC has a right angle
right over here.

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Clearly if this
angle is 90 degrees,

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then this angle is going
to be 90 degrees as well.

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They are supplementary.

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They have to add up to 180.

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And so they both have
a right angle in them.

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So the smaller one
has a right angle.

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The larger one clearly
has a right angle.

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That's where we started from.

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And they also both
share this angle right

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over here, angle
DAC or BAC, however

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you want to refer to it.

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So we can actually write
down that triangle.

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I'm going to start with
the smaller one, ADC.

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And maybe I'll shade
it in right over here.

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So this is the triangle
we're talking about.

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Triangle ADC.

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And I went from the blue
angle to the right angle

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to the unlabeled angle from the
point of view of triangle ADC.

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This right angle isn't applying
to that right over there.

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It's applying to
the larger triangle.

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So we could say triangle
ADC is similar to triangle--

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once again, you want to
start at the blue angle.

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A. Then we went to
the right angle.

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So we have to go to
the right angle again.

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So it's ACB.

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And because they're
similar, we can set up

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a relationship between
the ratios of their sides.

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For example, we know the
ratio of corresponding sides

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are going to do, well, in
general for a similar triangle,

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we know the ratio of
the corresponding sides

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are going to be a constant.

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So we could take the ratio of
the hypotenuse of the smaller

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triangle.

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So the hypotenuse is AC.

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So AC over the hypotenuse
over the larger one, which

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is a AB, AC over AB is going
to be the same thing as AD

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as one of the legs, AD.

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And just to show that, I'm just
taking corresponding points

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on both similar triangles,
this is AD over AC.

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You could look at these
triangles yourself and show,

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look, AD, point AD, is
between the blue angle

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and the right angle.

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Sorry, side AD is between the
blue angle and the right angle.

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Side AC is between the blue
angle and the right angle

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on the larger triangle.

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So both of these are
from the larger triangle.

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These are the corresponding
sides on the smaller triangle.

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And if that is confusing
looking at them visually,

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as long as we wrote our
similarity statement correctly,

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you can just find the
corresponding points.

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AC corresponds to AB
on the larger triangle,

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AD on the smaller
triangle corresponds

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to AC on the larger triangle.

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And we know that AC, we can
rewrite that as lowercase a.

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AC is lowercase a.

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We don't have any
label for AD or for AB.

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Sorry, we do have
a label for AB.

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That is c right over here.

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We don't have a label for AD.

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So AD, let's just
call that lowercase d.

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So lowercase d applies to
that part right over there.

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c applies to that entire
part right over there.

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And then we'll call DB,
let's call that length e.

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That'll just make things a
little bit simpler for us.

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So AD we'll just call d.

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And so we have a over
c is equal to d over a.

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If we cross multiply, you have
a times a, which is a squared,

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is equal to c times
d, which is cd.

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So that's a little bit
of an interesting result.

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Let's see what we can do
with the other triangle

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right over here.

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So this triangle
right over here.

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So once again, it
has a right angle.

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The larger one
has a right angle.

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And they both share this
angle right over here.

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So by angle, angle
similarity, the two triangles

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are going to be similar.

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So we could say triangle
BDC, we went from pink

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to right to not labeled.

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So triangle BDC is
similar to triangle.

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Now we're going to look
at the larger triangle,

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we're going to start
at the pink angle.

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B. Now we're going to
go to the right angle.

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CA.

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BCA.

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From pink angle to right
angle to non-labeled angle,

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at least from the
point of view here.

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We labeled it before
with that blue.

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So now let's set up some
type of relationship here.

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We can say that the ratio on
the smaller triangle, BC, side

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BC over BA, BC over
BA, once again,

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we're taking the
hypotenuses of both of them.

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So BC over BA is going
to be equal to BD.

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Let me do this in another color.

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BD.

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So this is one of the legs.

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BD.

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The way I drew it
is the shorter legs.

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BD over BC.

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I'm just taking the
corresponding vertices.

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Over BC.

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And once again, we know BC is
the same thing as lowercase b.

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BC is lowercase b.

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BA is lowercase c.

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And then BD we defined
as lowercase e.

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So this is lowercase e.

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We can cross
multiply here and we

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get b times b, which, and I've
mentioned this in many videos,

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cross multiplying is really
the same thing as multiplying

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both sides by both denominators.

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b times b is b squared
is equal to ce.

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And now we can do something
kind of interesting.

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We can add these two statements.

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Let me rewrite the
statement down here.

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So b squared is equal to ce.

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So if we add the
left hand sides,

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we get a squared plus b squared.

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a squared plus b squared
is equal to cd plus ce.

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And then we have a c
both of these terms,

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so we could factor it out.

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So this is going to be equal
to-- we can factor out the c.

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It's going to be equal
to c times d plus e.

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c times d plus e and
close the parentheses.

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Now what is d plus e?

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d is this length,
e is this length.

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So d plus e is actually
going to be c as well.

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So this is going to be c.

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So you have c times c,
which is just the same thing

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as c squared.

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So now we have an
interesting relationship.

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We have that a squared plus b
squared is equal to c squared.

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Let me rewrite that.

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a squared.

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Well, let me just do
an arbitrary new color.

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I deleted that by accident,
so let me rewrite it.

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So we've just established
that a squared plus b squared

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is equal to c squared.

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And this is just an
arbitrary right triangle.

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This is true for any
two right triangles.

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We've just established that
the sum of the squares of each

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of the legs is equal to the
square of the hypotenuse.

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And this is probably
what's easily

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one of the most famous
theorem in mathematics, named

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for Pythagoras.

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Not clear if he's the first
person to establish this,

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but it's called the
Pythagorean Theorem.

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And it's really the basis of,
well, all not all of geometry,

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but a lot of the geometry
that we're going to do.

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And it forms the basis of a
lot of the trigonometry we're

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going to do.

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And it's a really
useful way, if you

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know two of the sides
of a right triangle,

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you can always find the third.