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the triangle inequality theorem tells us
is that if you have you know three sides
that make up the sides of a triangle if
you add any two sides together they have
to add up to more than the third side if
they don't what you end up having is a
situation like this I kind of think of
it as a drawbridge but if these two
sides right here don't add up to the
length of this third side you can see a
triangle won't be formed because they
can't span that distance they can't
reach it's almost like even if you
rotate these sides down there they're
stretching but they can't span that
distance right so any two sides added
together have to be more than the third
side okay so that's something that we're
paying attention to in this example now
when we have these two sides five and
twelve let's see if we can just sketch
it out just to give us a visual way of
appreciating this so here's 12 and let's
just say that here is five right so the
third side could be like like that right
but you see this point right here this
is like a hinge okay so if we take this
and we rotate it okay like this the most
extreme situation would be kind of like
this where this is 12 and this is like
five right here and then the third side
would have to span that distance like so
right so the longest aside here could be
would be up to 17 not including 17
because if it was equal to 17 this would
actually be laying completely flat there
would be no internal space that wouldn't
be a triangle now the other scenario is
if we were to take this side here and
rotate it down like this so that five is
like right here now it only has to span
this distance from here to here which is
only going to be seven units so
essentially if you think about it as a
as a hinge okay the most extreme example
is when you make it like a hundred and
seventy nine point nine degree angle or
you make it like a 0.1 degree angle
right so see they're folding this way or
it's folding like that
so essentially what we have here for
this third side we'll just call it X it
has to be greater than twelve minus five
and less than twelve plus five so that's
the shortcut if you want to just get
down to it just a simple way of doing
this you can write it as a compound
inequality and it's going to
somewhere in between the sum of the two
sides and the difference of the two
sides so if we simplify you can see X is
greater than seven and less than
seventeen not equal to because otherwise
then it's going to collapse on down flat
so that's how you would use a triangle
inequality theorem to solve for the
possible lengths of the third side again
the shortcut is just to add and subtract
the two side lengths and that'll give
you an interval of the possible lengths
of the third side so I hope this helped
clear that up for you subscribe to the
channel check out more math tutoring
videos on my youtube channel Mario's
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seeing the future videos
