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CCSS.Math:

let's draw ourselves a triangle okay let's this side has length six let's say this side right over here has length ten 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 sawed 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 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 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 we end up with a degenerate triangle 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 you get so if you want this point right over here to get as close as possible to that point over there essentially minimizing your distance X the closest ways if you make the angle all the way equal to 0 all the way so you know let's actually let me draw a progression so now the angle is getting smaller these angle is getting smaller this is length 6 X is getting smaller X is getting smaller though we keep making that angle smaller and smaller 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 the side is length 6 side of 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 four so if you want this to be a real triangle at x equals four you've got these points as close as possible it's degenerate into a line into a line segment if you want this to be a triangle X has to be greater than four 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 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 10 side I'm going to make that angle bigger and bigger and so now let me make take my six side and put it like that and so now our angle is getting bigger and bigger and bigger it's approaching its approaching 180 degrees at 180 degrees our triangle once again will return into a line segment it will become a degenerate triangle so let me draw the side of length X try to draw it straight so we're trying to maximize it to that point and that point so this is side of length x and let's go all the way to the degenerate case at the degenerate case and 180 degrees the side of length six forms a straight line with the side of length 10 and this is how you can get this point and that point 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 to 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 to degenerate triangle than the sum of the other two sides so length length of a side has to be less than the sum of the lengths sum of lengths of other two sides of other two sides if you're willing to deal with degenerate triangles where you essentially form a line 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 triangle 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 6 plus 10 or X has to be less than 16 X has to be less than 16 the exact same result we'd got by visualizing it like this if you want to say how large can X be well you could say well 10 has to be less than you could say 10 10 has to be less let me do or how small can X be you have to say 10 has to be less than 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 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