I will tell you what the theorem is in a
minute it has something to do with how
all the sides of this triangle relate to
each other I'm going to show you how
that result is but I want to show you
how I get the result okay so all of this
that I'm about to do on the board you
will have your time to draw I actually
have like a little guide for you that
I'm going to put up on the screen in a
minute but for now I just want you to be
fully engaged with your eyes on what's
happening here so here's what I'm going
to do this theorem this property this
result relates these three sides
together so I'm just going to give them
some names do you remember in algebra we
gave numbers names when we didn't know
what they were what was the point of
doing that why is that useful yeah yeah
I can angle or a side or any kind of
unknown value I'm going to give these
guys names a B and C not very original I
know but at least that way it won't be
hard to remember okay now a B and C they
have three sides of this triangle and
what I'm going to do is I'm going to put
a few more copies of this same triangle
up on the board so I can make a bit of a
bit of a construction a bit of a shape
okay so all of these triangles are the
same I've got four of them okay so I'm
going to arrange them in a particular
way that I want you to notice so I'm
going to put this one over here good
okay
I've got two more copies that are
identical mmm that PP just sorry Tom
please the bear that's securely only
back for me now you put this fine I spin
around a little bit okay that's okay
and the last one thank you okay one
okay now I want you to look closely at
the shape that I just made by putting
these four triangles together I've
actually made two new shapes can you see
both of them there there's one that's
big all the way around what kind of
shape is that it's a square now just
pause for a moment don't answer yet just
think about it a perfect it looks like a
square how do you know that's a square
think for a moment think hmm Charlie I
saw your hand it got pretty quick why do
you think this is a sweat okay number
one the first like requirement for a
square it's gotta be a quadrilateral so
tick all right Nikhil give me something
else
ah now what makes a square square is
that the four sides have to be equal
right now hold on a second how did you
know that the force I mean I know
they're looking for how did you know
that they were equal Daniel what do you
think this one here okay now you
remember I said all four of these
triangles are the same yeah so I've got
a right angle here which means I've got
one here and here and here those are
important but they still don't tell me
that all of the sides right like this
one here this is a sign right it doesn't
tell me that that side is the same as
this one and this one and this one can
anyone see how we can know where your
eken friend is his square I'm trying to
shorten it to square there I know it
looks a lot like a square let me help
you out a little bit okay what do I need
these sides again a bit yeah so if this
is B here this one's slightly longer
than a can you see it right um
what's this side here this is also B
right so what's the height of my square
there's a napier eland Rihanna's
so you just add them together a
phosphate right you have you with that
now have a look at the bottom tell you
the bottom can you see I've got V here
what's this guy it's a so what's this
and you can do this little the way
around the four sides of the
quadrilateral you like they're all the
same
they've all got right angles that's the
square okay there's another square - do
you see where the other square is
it's right there stuck in the middle
right in fact it's so important I drew
it as well okay so you can see it's
offered an angle bit of a funny angle so
it's in here
okay now Pythagoras theorem has a lot to
do with this particular Blue Square
that's why I drew it for you okay I want
you to think for a moment what's the
area we've done area for what's the area
of this Blue Square you reckon to
remember all of these four triangles
around the edges are the same this is C
this side here that means this is C as
well in fact they're all set okay
in fact that's so important I'm gonna
label this square let's do it like this
C squared can you see that that's the
area now I don't know what C is
but algebra doesn't care it's like well
whatever it happens to be that square it
could be like 10 which would make this
100 it could be seven which would make
this what would that be 49 it could be
eight which would make bad for a no no
no no not could be anything it's my
point okay now let's just take this guy
and puts his square over here for a
minute just hold on to him in your mind
okay right I need a little space now
what I'm going to do here here comes the
proof that was just a setup I'm going to
just rearrange these same four triangles
okay I don't want you to watch the way
that I do it um do you remember back in
term 1 or 2 we learned about these
things called transformations we could
change shapes we could like move them
out in different ways and they stayed
the same do you remember what the three
transformations were any suggestions
Nikhil QD one slide or translate what's
another one Frank
reflect or flip and the third one was
rigidly rotate okay or spin or turn
right I'm going to use two of those I'm
going to use rotation and I'm going to
use translation that's sliding okay
watch I'm going to take this guy I'm
going to rotate him around like this
spin so he's over here okay are you
happy with that you see how I did it
okay I'm going to do another rotation
this one will rotate a similar way are
anti-clockwise
around like that no shapes have changed
there just move positions a little bit
okay now I'm going to do my
translational mudslide I'm going to take
this shape what's this shape now that
I've mayor it's a rectangle I'm gonna
take the whole rectangle and I'm going
to move it over okay now this is my new
set up still the same four triangles but
again I've made some other shapes that
you can kind of see you around there
right for starters can you see
I bet this guy over here you see this
guy what kind of shapes that it's a
square again but it done just look like
a square how do you know it's a square
hmm Krisha waiting yeah very good see I
think we've got this one labeled a and
because these triangles are all the same
this is also a okay and they're also all
those writings I'm talking about before
but that's the other square only square
this this one here right you see that
you with me now what are the areas come
strollin what are the areas of these two
squares well let's do the first one
first it's a times 8 which makes it
right how about this one over here yeah
B times B so it's B squared okay so I've
got these two cool extra squares but
then have a look now that I put
everything together what is this big
shape that I've made it's another square
in fact it's not just any square isn't
it the square I started with the big one
look once what's this width across the
bottom it's a plus B right and you can
say the same thing of the high country
so in other words the two squares I made
the two big ones were exactly the same
square okay with that okay now let me
just draw that swear over here it'd look
like this if you recall then what is
there we go remember that okay
here comes the finish these four
triangles they start up really important
okay but I'm going to take them away
from both of the squares that just they
do with those bits right now these are
the leftover parts right take away the
four triangles okay but look I've got
these two there with the leftover bits
in the first place the leftover bit was
this whole square do you agree so
therefore what Pike Babli theorem tells
us is that all these sides they're
related in this way if you add these two
gaps to get up the leftover bits of the
square they're the same as those
leftover bits of the square I just
rearranged them a little bit okay and
this one two three this is what we call
Pythagoras's theorem if you've got any
right-angled triangle okay now I want
you to open your book up again okay if
you didn't draw your triangle with a
ruler I'd like you to draw one with a
ruler maybe I need to borrow from a pen
and what I want you to do is I want you
to measure the side that's really
important that the sides meet at a right
angle like that if you measure them for
example online what I did was I measured
out these are all of these sides okay I
think what our memory I think this is
twelve point six centimeters and 16.8%
meters you can work out what the last
site is the longest side you can work it
out by just measuring it by putting a
ruler on it but you can also work it out
using this relationship here that a
squared plus b squared will equal this
last side here squared in this case it's
21 okay a really famous one if you
haven't if you have drawn like a
triangle and it's a bit messy you can
draw a new one make it like this this is
very famous one if you've got one that's
1 inches 3 centimeters and 1 inch that's
4 centimeters so long as they meet at a
right angle this last one should be 5
you can't help but make it 5 here's why
watch what's 3 squared 3 squared is 9
what's what's waiting with 16 and if you
add these two together you get a 5 which
is y squared right very good
so you can do this with any set of
trained another famous one is 5
centimeters and 12 centimeters and 13
you can do the squares yourself and you
can find out 25 plus 144 is equal to 169
okay and you can do this with any
triangle you like there has to be nice
neat numbers they can be weird decimal
numbers or fractions or anything
Pythagoras theorem tells us this is
always true no matter what right angle
triangle you have