this video was sponsored by brilliant
let's say you have a stick that you will
cut into three pieces by randomly and
independently selecting two points to be
locations of the cuts the question is
what are the odds the three pieces you
create can form a triangle you can see
for example with these three pieces I
can create one but if these were the two
cuts then it's impossible to make a
triangle now what I found to be an
extremely clever solution to this
requires one geometry theorem that I bet
a lot of people don't know if you have
an equilateral triangle and pick any
point on the interior then draw three
line segments from there such that
they're all perpendicular to the three
sides the some of those distances will
always equal the height of the triangle
regardless of where on the interior you
pick here's why this always works since
it's an equilateral triangle than every
side I can call X those line segments
will be labeled a B and C and the height
of the triangle will be H now the area
of the entire triangle would be base or
x times height all over 2 then you'll
notice I can break the bigger triangle
up into three smaller triangles using a
B and C as the respective heights the
area of this bottom one is base x times
height a all over to the next triangle
would be x times B all over two and the
last one would be X times C all over two
and the sum of all of those equals the
entire triangle canceling the twos and
the X's we get that a plus B plus C
equals H this is known as Vivien ease
theorem by the way but okay how does
this help us with our stick problem well
first off from geometry you should know
that for a triangle to be made the two
shortest legs combined in length must be
longer than the third if that criteria
is met then we can make a triangle so
now we'll just say that the stick we're
using matches the height of the triangle
thus any interior point with those three
perpendicular line segments corresponds
to cutting that stick in two locations
like this interior point corresponds to
these two cuts and as we move the
interior point around the overall length
stays the same but we change the
locations of those two cuts so all the
interior points of this triangle
correspond to every possible way you
could cut this stick in two locations
and at this point for example we see it
does correspond to a possible triangle
now I'm gonna draw another equilateral
triangle in the middle of the bigger one
to make a little Triforce here and look
at the top triangle for any point inside
there the longest line segment or
longest piece of the broken stick will
definitely be longer than half the
height or total length
thus the other two combined will be
shorter than that and when the two
shorter sides combined don't exceed the
third then you don't have a possible
triangle this applies to anywhere in
that upper triangle and by symmetry the
same thing can be said about the other
two the middle triangle is the one
exception of the longest line segment
won't exceed half the sticks length
thus the other two combine must and when
the two shorter sides combined exceed
the third then you can make a triangle
so of the four regions we see here all
equally likely to be the point
representation of the two cuts only one
of them yields lengths that can make a
triangle so the answer to our question
is 25% so now this ever comes up in a
future interview for you guys hopefully
you'll remember this video I decided to
make this a shorter one but if you like
these kinds of puzzles and
out-of-the-box thinking then I
definitely recommend checking out
brilliant org this video sponsor related
to this video they actually have an
entire series of geometry related
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concepts like that a beautiful geometry
course includes fun puzzles like the art
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folding or the use of picks theorem
which is an application of Euler's
formula and so on
or for those who like these advanced
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brilliant org is the perfect resource
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Zack star will get 20% off their annual
premium subscription and with that I'm
going to end that video there thanks as
always to my supporters on patreon
social media links to follow me or down
below and I'll see you guys in the next
video