okay as we all know when we have a right triangle like this we can talk about the famous pythagorean
theorem namely a squared plus b squared sequence c squared and yes that's what we are going to do
in this video i will show you guys the proof that i like the most although there are a lot of proofs
out there at least hundreds of them but i'll just show you guys the one like the most all right and
i'll also show you guys the inverse version of the pathagarian theorem so if you guys don't know
what it is be sure you guys watch till the end first of all let's take a look at this picture
if look at b as your base then a will be your height if you look at a as your base then b will
be your height if you look at c as your base then the height will be just going from here
perpendicular down here and i will just call that to bh and when we do that in fact we get one two
three three right triangles right so i'm going to draw them up for you guys but first let me
call this to be c1 and this right here to be c2 and of course c1 plus c2 will be the whole c
and let me just draw the smallest triangle right here first and c one will be the smallest side
so let me just put that down right here for you guys so c1 and this right here will be
h right so let me label c1 and h and in this case a is actually the hypotenuse of the smallest
triangle so this is the first one that we have and for the second one which is right
here h is actually the smaller side and i'll put it down on the side here
and c2 it's going to be the median side side which is right here and of course here's the right angle
and b is actually the hypotenuse so i'll just put that down right here
and i'll also put down the original for you guys so this is the biggest one and is
that one namely a b and c like that first of all all these two triangles are similar why because
angle angle so you can look at here you have this angle in the smallest triangle is right here right
this angle is the same as that and of course it's the same angle right here and both of these two
triangles they have 90 degree angles so angle angle the third angle will be the same as well
similarly for the second one this angle is right here and for the big one you have this angle so
they are all similar right this is similar to that this system is of course this and that are similar
as well by transitivity so they are all similar and with that being said we can actually set
up proportions to take care some business first i'm going to compare the smallest triangle here
and the biggest one namely the original one right here and i'm going to set proportions
but i don't want to involve h so to do that you see for the smallest one i have this and that
and i just have to say okay let's look at a over c1 this right here has to be the same as c over
a right hypotenuse to this hypotenuse to that so this right here is the first equation that we have
and you can of course cross multiply and you get a square and this times that is just c times c one
so that's it now for the second and the biggest one still to say i'll compare this and compare
that i want to set a proportion but i don't want h at all so i'll be looking at this and that this
that so i will have b over c2 that should be equal to c over b and of course let's cross multiply
we get b squared equals c times c2 so cc2 right here so that's cool and what we are going to do
next is well this is equation 1 this is equation 2. i'm just going to add them up i'll just say 1
plus 2 it's not 3 okay equation 1 plus equation on the left-hand side of a squared on the left-hand
side right here b squared so of course we get a squared plus b squared and for the right-hand side
here we have c c one here we have c c two notice that they both have a c so i can factor that out
so i get c1 plus c2 like this and now if you look at what's c1 plus c2
well of course if you look back c1 plus c2 is the whole c the original c so in another word we get a
squared plus b squared equals this c times another c and of course this is a squared plus b squared
equals c squared this right here is it i like this proof so much because to be honest i don't like
to draw that big triangle and cut this try i mean big rectangle and cut that into like four uh right
triangle all that stuff i think this is easier for me to draw and demonstrate in my opinion
and it can also practice the similar triangle business all right anyway in fact this picture
is also really cool because we can also talk about the inverse pythagorean theorem so let me
draw that again right here for you guys now this time we are going to talk about area again when we
have this right here if you take b as your base then a will be your height so we can talk about
the area right here the area of this triangle of course is just one half base times height
and let me just put on a times b right here no problem at all but if you look at c as your base
well the height will be you go from here to here and just like that i will call this to be h
well of course we're talking about the same right triangle the areas have to be the same we'll just
have the same area this has to be the same as one half base times height so c times h and now
we have this equation right here we can multiply both sides by two so that one half will be gone
and we'll just look at this as a b equals c h this time i don't want to you know end up with any c's
earlier we have the c this time we don't have the c earlier i didn't for the h this time we will get
the h so to do that i'm going to divide both sides by h right so if you do that you get a b over h
equals c and why do you want to do that because now we have that so we must
have a squared plus b squared equals c squared and what's our c now
namely a b over h so in other words this equation is the same as a squared plus b squared equals
a b over h and we square that well we can do some algebra real quick this is just a squared plus
b squared equals you can just put the powers right here a squared b squared over h squared that's
just so multiplying and dividing and i'm going to divide everybody by a squared m b squared so
let me just show you guys the word by putting this down in the front multiplied by 1 over a squared b
squared when you do this times that you get the a squared cancel and you have the b squared in the
denominator so we get 1 over b squared similarly when you do this times that the b squared will
cancel this time and we get 1 over a squared and that will be equal to this and that will
they will cancel out we just have the h squared in the denominator like this and ladies and gentlemen
let me put this down right here for you guys right here i will write down 1 over a squared
first and i'll put that down as a to the negative 2 power and then i'll write this down as plus b to
the negative 2 power and of course i will put this down as this is equal to h to the negative 2 power
and this right here it's inverse algorithm i think that's a legit name for this statement
which is really cool right you are involving the a b and h really really cool right so this
is it hopefully you guys like this video you guys can leave a comment down below let me know which
proof of the pythagorean room that you guys like the most as i said this is my what's yours