So what are we talking about today?
About my favourite problem in geometry when I was a fifth grader
and it is etched in my memory to the point of being able to recall it at any time
so we have three squares of the same sizes next to each other
the size does not matter
so what we will do you is pick one of the
top corners let's say the top left corner and connect it
to the bottom right corners of
each of those three squares.
Next, we will take this angle here at the bottom let's call it
alpha the next one at the bottom
let's call it well
beta and and that's incorrect
yeah this is upside down
and finally gamma alright
so the question is asking what
is the sum of these three angle so it's a simple enough problem
the interesting part about it is that it was given to me as a fifth grader
so I had to a attack it as fifth-grader
in fifth grade we learn how to use a protractor so
I measure the first angle it's about 45 degrees
and in fact then you can stop and think about it but of course it has to be 45
since the triangle we're looking at is right
and iscoclese this guy here
so it should not be a surprise to get a 45 degree angle there
and we go about measuring the others there perhaps we
run a little bit out of luck because the second angle
looks like it's pointing to 25
and a little bit maybe 26
and the third angle Wow where is it going
perhaps 15
20 so about 18 so when we
add this up 89
could that be it what do you think
it seems a bit arbitrary i don't know
Yeah and also there were all of these mistakes in our measurement
and why would they be asking a fifth grader how much this is
and ending up with such a random angle?
well if you try to make another measurement perhaps you will come up
with 91.3
and so you would say oh it's a little bit obtuse
but maybe it's a little bit acute so what would be
conjecture that people will leap to
if they believe this is a beautiful problem I would guess that
the correct answer they would think the correct answer is ninety
exactly ninety but how would we know really
is it ninety? Or some ugly angle nearby?
we cheat a little bit we say okay that problem was given to a fifth grader
it has got to be a nice answer we will believe for a moment it's ninety and
this is what mathematicians call a conjecture
and then put all of our effort into proving that it is ninety
and we are kind of a third of the way
to proving it because we know that one of the angles
for sure is forty five
so what should be left for the other two angles in order to make ninety?
well obviously forty five. so somehow we need to add up the other two angles beta
and gamma
and prove that they add up to up to forty five another experiment which a
fifth grader might try and you can also do at home
is cut up the whole thing with scissors
and eliminate
some of the error that is coming
from the protractor when you are rounding
the degrees
okay so let's try to put everything together
if we can
it does look like this is a ninety degree let's check
well yep it does look like 90 degrees
okay so now we truly believe the statement and we want to prove it
well the question is at what level of your mathematical career
or mathematical life you are because this problem has not only one
not only two but at least
54 solutions. 54 solutions?
No kidding. Are we gonna do all of them?
No. we will do the most brilliant one the most simple one that
a fifth grader may be able to grasp so the idea of our solution
at least the attempt when we cut with scissors can be carried forward
mathematically so we may want to move
all three angles into the same geometric location in our picture
so we need to find the best geometric location where to move those three
angles
well there is no best geometric location until you know what the solution is
so we might as well just try to move them all the way
in this corner next to gamma because we already have
a right angle here this guy we have gamma here
and so the other two angles alpha and beta must somehow fit
and here comes the brilliant idea for the solution
as I a said there is something missing in this picture
what's missing is half of the picture we will replicate
these 3 squares up and then we'll start seeing
the truth and now i think I'm going to start labeling things so people can
refer to them
later on I'm just labelling all of the
important points I will choose point
H and connect it to two other points
D and E
and I will say that no more constructions
will be done in this solution we somehow have to make sense of
what this picture is telling us well for starters
we see 3 angles exactly where we wanted them to be
where this ninety degree angle was we have our original gamma
and then 2 more could those two angles be indeed
alpha and beta well you can kinda guess that this
angle is bigger
than the next one
by the way it looks so perhaps that will be
the 45 alpha because alpha was bigger than beta
could this be forty five
and let's try as detectives figure out where it could come from this picture
it is involved in this
large triangle
did we see a similar situation earlier in the problem
certainly right here where we saw
our original alpha and where we decided that
alpha is 45 degrees because this was
ABE
was a right isosceles triangle
do we have a right isosceles triangle her
we might could EH
be equal to HD these are the two extra segments that we draw on top
the answer is yes because
these two segments are hypotenuses
Hypoteni or hypotenuses?
so EH and HD the two segments which we drew
are the hypotenuses of two right triangles
two that we also draw on top so I'm going to mark them
in blue
so these two sides are hypotenuses of two right triangles
E F H and H I D
the ones which I drew in blue now they are right because they have a right
angle here
coming from top but they are also exactly the same
size and shape this right triangle has one leg
two and the other leg 1 and this one also has a leg of two and leg of 1
which means that their third size, the two
hypotenuses must be the same so indeed
EH is equal to HD
and we are on our way to show that this big green triangle in the middle
is right isosceles it's already isoseles but the question is why is this
guy right
okay but what else is around this angle
there is this guy here
this angle in this other one
what do they add up to and the answer is also very simple
one of thise angles we already know I claim
this angle is our angle beta
because there is a third triangle which is precisely the same shape
and size as those because it is right here
again a right triangle
with legs 1 and two and so therefore this angle beta
must be equal to this single beta and guess what
it must be equal to this angle beta they all share exactly
the same angle so accidentaly
we have shown that we have moved angle beta to where we wanted it
gamma is there and if that is really 45 we will be done
and again backtracking we wanted to show that this big triangle was right
isosceles
we already know that this angle here is beta what is this guy
well this guy this second angle
participates in a right triangle where the other
acute angle is beta what could this guy be
we know that the angles in a triangle add up to 180
this guy is ninety therefore these 2 must add up to 90
simple arithmetic shows
this angle here must be ninety minus beta
ninety minus beta something and beta
must overall give you 180
what is the remaining angle here
this curly one if you start adding up let's do it
beta plus ninety minus beta
plus something must be 180
so beta goes away and that's something
turns out to the our 90 degrees
you've done it! that's right so our angle alpha is indeed moved here along with beta
and we can conclude the the three angles
add up to 90 degrees
and this is an amazing construction which
uses only squares
isosceles triangles in congruent
right triangle with legs 1 and 2
and that is indeed a brilliant approach
which mathematicians take to solve problems
they complete it to what's missing there
and then the problem becomes easier