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# Triangle inequality theorem

Intuition behind the triangle inequality theorem. Created by Sal Khan.

Video transcript

Let's draw ourselves a triangle. Let's say this
side has length 6. Let's say this side right
over here has length 10. And let's say that this side
right over here has length x. And what I'm going to
think about is how large or how small that
value x can be. How large or small
can this side be? So the first question
is how small can it get? Well, if we want
to make this small, we would just
literally have to look at this angle right over here. So let me take a look at this
angle and make it smaller. So let's try to make that
angle as small as possible. So we have our 10 side. Actually let me do it down here. So you have your 10 side,
the side of length 10, and I'm going to make this angle
really, really, really small, approaching 0. If that angle becomes 0, we end
up with a degenerate triangle. It essentially
becomes one dimension. We lose our
two-dimensionality there. But as we approach
0, this side starts to coincide or get closer
and closer to the 10 side. And you could imagine the
case where it actually coincides with it and you
actually get the degenerate. So if want this
point right over here to get as close as possible
to that point over there, essentially minimizing
your distance x, the closest way is if you
make the angle the way equal to 0, all the way. So let's actually-- let
me draw a progression. So now the angle
is getting smaller. This is length 6. x is getting smaller. Then we keep making that angle
smaller and smaller and smaller all the way until we get
a degenerate triangle. So let me draw that pink side. So you have the
side of length 10. Now the angle is essentially 0,
this angle that we care about. So this side is length 6. And so what is the
distance between this point and this point? And that distance is length x. So in the degenerate case, this
length right over here is x. We know that 6 plus x is
going to be equal to 10. So in this degenerate case,
x is going to be equal to 4. So if you want this to be a
real triangle, at x equals 4 you've got these points
as close as possible. It's degenerated into a
line, into a line segment. If you want this
to be a triangle, x has to be greater than 4. Now let's think about
it the other way. How large can x be? Well to think about
larger and larger x's, we need to make this angle bigger. So let's try to do that. So let's draw my 10 side again. So this is my 10 side. I'm going to make that
angle bigger and bigger. So now let me take my 6
side and put it like that. And so now our angle is getting
bigger and bigger and bigger. It's approaching 180 degrees. At 180 degrees, our
triangle once again will be turned into
a line segment. It'll become a
degenerate triangle. So let me draw the
side of length x, try to draw it straight. So we're trying to
maximize the distance between that point
and that point. So this is side of
length x and let's go all the way to
the degenerate case. In the degenerate case, at 180
degrees, the side of length 6 forms a straight line with
the side of length 10. And this is how you can get
this point and that point as far apart as possible. Well, in this situation,
what is the distance between that point
and that point, which is the distance which
is going to be our x? Well in this situation, x is
going to be 6 plus 10 is 16. If x is 16, we have a
degenerate triangle. If we don't want a
degenerate triangle, if we want to have two
dimensions to the triangle, then x is going to have
to be less than 16. Now the whole principle that
we're working on right over here is called the
triangle inequality theorem and it's a pretty basic idea. That any one side of a
triangle has to be less, if you don't want a
degenerate triangle, than the sum of the
other two sides. So length of a
side has to be less than the sum of the
lengths of other two sides. If you're willing to deal
with degenerate triangles-- where you essentially
form a line segment, you lose all your
dimensionality, you turn to a
one-dimensional figure-- then you could say
less than or equal, but we're just going to stick
to non-degenerate triangles. So the length of a
side has to be less than the sum of the lengths
of the other two sides. And just using
this principle, we could have come up with
the same exact conclusion. You could say, well look,
x is one of the sides. It has to be less than the sum
of the lengths of the other two sides. So it has to be
less than 6 plus 10, or x has to be less than
16-- the exact same result we got by visualizing
it like this. You want to say
how large can x be? Well you could say, well,
10 has to be less than-- Or how small can x be? You have to say 10
has to be less than 6 plus x, the sum of the lengths
of the other two sides. If you subtract 6 from
both sides right over here, you get 4 is less than x,
or x is greater than 4. So this is a, in some level,
it's a kind of a basic idea, but it's something that you'll
see definitely in geometry. And then you'll go far into
other types of mathematics and you'll see other versions
of what's essentially this triangle
inequality theorem.