hi everyone today I want to talk about
Pythagoras theorem which perhaps the
most famous theorem in mathematics oh
you know it the square on the hypotenuse
of a right angled triangle it's equal to
the sum of the squares on the other two
sides blue and pompom so everyone knows
it that what you might not know is it's
not just four squares so this is what I
want to do I want to experiment with
some other shapes
we'll look at circles and triangles and
Pentagon's and then we're going to take
what we learn and we're going to use it
to prove Pythagoras's theorem itself
using an idea from a young Albert
Einstein all true but let's start by
looking at the original Pythagoras's
theorem
so the original theorem says take a
right-angled triangle and then add a
square to each side of the triangle so
we're going to have a big square and
then two smaller squares then the
Pythagorean theorem says that the area
of the big square is equal to the total
area of the two smaller squares or if we
label the sides a b and c then the big
square will have an area of c squared
and the smaller squares will have areas
of a squared and B squared and then we
get a squared plus B squared equals C
squared but what about instead of
squares I use semi circles what is the
area of the big semi circle here well
the area of a circle is pi times the
radius squared we're going to use half
of that and the radius would be C over 2
so we're going to get 1/2 pi C over 2
squared which is PI over 8 C squared and
in the same way the other semi circles
will have areas of PI over 8 a squared
and PI over 8 B squared now because
Pythagoras's theorem is true
we know that a squared plus B squared
equals C squared
which automatically means that PI over 8
a squared plus PI over 8 B squared
equals PI over 8 C squared so the
semicircle on the hypotenuse is equal to
the sum of the semicircles on the other
two sides we have a Pythagorean theorem
for semicircles as well but what about
triangles so if we use equilateral
triangles here the area of the big
triangle it's going to be
half the base times the height well now
a triangle off side length one would
have a height of root 3 over 2 so the
area would be root 3 over 4 but we've
got a triangle here off side length C so
we're going to scale that up and you
will see the base is multiplied by C and
so is the height so now it's root 3 over
2 multiplied by C so the total area is
root 3 over 4 C squared in the same way
the other triangles have areas root 3
over 4 a squared and root 3 over 4 B
squared now because Pythagoras theorem
is true we automatically have root 3
over 4 a squared plus root 3 over 4 B
squared equal to root 3 over 4 C squared
so the area of the big triangle is the
sum of the two smaller triangles so
let's keep going what about Pentagon's
well by now you might be able to predict
what's going to happen let's say we took
a Pentagon off side length 1 well there
is a formula to work out the area of a
Pentagon it's quite complicated and
actually doesn't even matter let's say
for our purpose that the area is about
one point seven two but if we scale that
up by C then the width and the height is
multiplied by C which means the area is
multiplied by C squared so the area of
the big Pentagon here is going to be one
point seven two C squared and in the
same way the other Pentagon's are going
to have areas of one point seven 2a
squared and one point seven two B
squared
and because Pythagorean theorem is true
we have a squared plus B squared equals
C squared we automatically get one point
seven two a squared plus 1.7 two B
squared equals one point seven two C
squared we do have a Pythagorean theorem
for Pentagon's as well and some of you
may have noticed by now that this idea
will work for any shape even something
like this wobble where the wobbles here
are two scaled versions of themselves
then the wobble on the hypotenuse is
equal to the sum of the wobbles on the
other two sides and
because if we imagine a wobble of side
length one it would have some area let's
call it a and if we scale that up by a
factor of C the area is scaled by a
factor of C squared
so because Pythagoras's theorem is true
then the areas will be equal for any
shape but now here comes the clever part
we can actually prove Pythagoras's
theorem by running that argument in the
opposite direction and that's exactly
what a young Albert Einstein did when as
a child he came up with his own proof
for Pythagoras's theorem and the way
Einstein told the story he was 11 years
old he was learning classic Greek maths
like the Pythagorean theorem and he kind
of found it annoying it's a bit too
wordy kind of unnecessarily complicated
so he came up with his own proof for the
Pythagorean theorem which basically
comes down to drawing one line this is a
one line proof so here is the right
angle triangle and here is the line so
what has he done here well he's created
three triangles we've got the big
triangle and two smaller triangles you
might be able to see that better we kind
of flipped them out so we have the big
triangle is just the same as the
original and the two smaller triangles
in fact they're just scaled up versions
of themselves we know they're the same
because they actually have all the same
angles therefore they are the same
triangle but scaled up let's call the
area of these triangles a b and c then
simply by definition the two smaller
triangles add up to the bigger triangle
so we have a plus B equals C but now we
can think of each triangle that's just
been like some other triangle let's say
of area X but scaled up by factors of a
squared B squared + C squared and then a
simple cancellation will prove that a
squared plus B squared equals C squared
and that is a proof of the Pythagorean
theorem and is such a neat idea
so instead of showing that when
Pythagoras's theorem is true then the
areas
be equal then in this case if the areas
of equal then Pythagoras theorem must be
true and you happen thanks for watching