Pythagorean Theorem Proof Using Similarity

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

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

or has a right angel in it

Now we call the longest side of a right triangle,

we call that side and you can either view

the longest side of the right triangle

or the side opposite of 90 degree angle

it is called the hypotenuse

It is a very fancy word for a fairly simple idea

just the longest side of a right triangle

or the side opposite of the 90 degree angle

And it's good to know that because some might say hypotenuse

"oh, they're just talking about this side here,

the side longest, the side opposite of the 90 degree angle"

Now what I wanna do is prove a relationship,

a very famous relationship and you might see where this is going,

a very famous relationship between the lengths of the sides

of a right triangle

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

Let's call that length lowercase "a "

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

I'll use uppercases for points and lowercases for lengths

And let's call the length of the hypotenuse,

so length of AB, let's call that C

And let's see if we can come with a relationship between A, B and C

And to do that I'm first gonna construct another line

or a another segment, I should say, between C and the hypotenuse

And I'm gonna construct it so that they intersect at a right angle

And you can always do that, we'll call this point right over here,

we'll call this point capital "D "

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

You can imagine rotating this entire triangle like this

And this is proof but it just gives you the general idea

of how you can construct a point like this

So, if I've rotated it around, so now our hypotenuse

we're now sitting on a hypotenuse

This is point, this is now point B, this is point A,

so we've rotated the whole thing all the way around

This is point C, you can imagine just dropping a rock from point C

with maybe a string attached, and it would hit the hypotenuse

at a right angle

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

where we put our point D right over there

And the reason why I did that, is now we can do all sorts of

interesting relationships between similar triangles

Because we have 3 triangles here: we have triangle ADC,

we have triangle DBC and then we have the larger, original triangle

We can, hopefully, establish similarity between those triangles

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

Because both of them a have a right angle

ADC has right angle right over here

So, if this angle is 90 degrees,

this angle is gonna be 90 degrees as well

they're supplementary they have to add up to 180

And so they both have a right angle in them

The smaller one has a right angle,

the larger clearly has a right angle that's where we started from

And they both share, they also both this angle right over here

Angle DAC or BAC however you wanna refer to it

We can actually write down that triangle,

I'm gonna start with the smaller one:

ADC, maybe I'll shade it in, right over here

So, this is the triangle we're talking about, triangle ADC

and I went to the blue angle to the right angle

to the unlabelled angle from the point of view of ADC

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

It's applying to the larger triangle

So, we can say that triangle ADC, triangle ADC

is similar to, is similar to triangle,

once again you wanna start at the blue angle A

then we went to the right angle

So, we won't' have to go to the right angle again

To triangle- this was ACB

ACB

And because they're similar we can setup a relationship

between the ratios of their sides

For example we know the ratio of corresponding sides

we're gonna- well in general for similar triangles

we know that the ratios of the corresponding sides

are gonna be constant

So, we can take the ratios, the hypotenuse of this smaller triangle

So, the hypotenuse is AC or the hypotenuse of the larger one

which is AB

AC over AB is going to be the same thing as AD as one of this,

as one of the legs

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

on both similar triangles

This is AD over AC, over AC

You can look at these triangles yourself

and show "look, AD, point AD is between the blue angle

and red angle, and point- sorry Side AD

is between the blue angle and the red angle "

Side AC is between the blue angle and the red angle

of the larger triangle

So both of these are from the larger triangles

These are the corresponding sides of the smaller triangle

and if that is confusing, looking at them visually,

you can- as long as you wrote our similarity statement correctly

you can find the corresponding points

AC corresponds to AB on the larger triangle

AD on the smaller triangle corresponds to AC on the larger triangle

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

AC is lowercase "a "

AC is lowercase "a "

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

we do have a label for AB that is c over here

We don't have a label for AB, so let's just call that,

so AD, let's just call that lowercase "d "

So, lowercase "d" applies to that part right over there,

c applies to that entire part right over there

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

that'll just make things simpler for us

So, AD we'll just call d

And so we have A over C is equal to D over A

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

is equal to c times d, which is cd

So, that's a little bit of interesting result

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

So, this triangle right over here

So, once again it has right angle, the larger one has a right angle

and they both share this angle right over here

So, by angle, angle similarity the two triangles

are going to be similar

So, we can say triangle BDC, we went from pink to right,

to not labeled

So, triangle BDC, triangle BDC is similar to triangle,

now we're gonna look at the larger triangle,

now we're gonna start the pink angle B,

now we go to the right angle CA

BCA

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

at least from the point of view here before the blue

Now we setup some type of relationship here

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

side BC over BA

BC over BA

Once again we're taking the hypotenuses of both of them

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

Here's another color, BD, so this one of the legs

BD, the way I drew it as a shorter legs

BD over BC, I'm just taking the corresponding vertices, over BC

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

BC is lowercase "b "

BA is lower case "c "

And then BD we defined as lower case "e "

So, this is lowercase "e "

We can cross multiply here and we b times b

Which and I mentioned this in many videos cross multiplying

both sides by both denominators

b times b is equal to ce

And now we can do something kind of interesting

We can add these two statements down here

Let me rewrite this statement down here

So, b squared is equal to ce

So, if we add the left hand sides, we get a squared

plus b squared, a squared plus b squared is equal to cd, is eqaual to cd

plus ce

And then we have a ce in both of these terms so we can factor it out

So, this is gonna be equal to, we can factor out the c,

it's gonna be c times d plus e

c times d plus e, and close the parenthesis

Now what is d plus e?

d is this length

e is this length

So, d plus e is actually gonna be c as well

So, this is gonna be c

So, if c times c is the same thing as c squared

So, now we have an interesting relationship,

we have that a squared plus b squared is equal to c squared

Let me rewrite that

a squared, I'll do that- well let me just arbitrary new color

I deleted that by accident, so let me rewrite it

So, we've just established that A squared

plus B squared is equal to C squared

And this is just an arbitrary right triangle

this is for any two right triangles

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

is equal to the square of the hypotenuse

And this is probably one of the, what's easy one of the most famous theorems of

Mathematics, named after Pythagoras

Not clear if he was the first person to establish this

But it's called the Pythagorean Theorem

Pythagorean Theorem

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

but a lot of the Geometry that we're gonna do

And it forms the basis of all the Trigonometry that we're gonna do

And it's a really useful way if you know of the 2 sides

of a right triangle, you can always find the third