Triangle inequality theorem

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.