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Video transcript

if someone walks up to you on the street and says alright I have a challenge for you I want you to construct a triangle that has sides that have sides of length two so sides of length let me write this a little bit neater sides of length two two and five can you do this let's try to do it and we'll start with the longest side the side of length five so the side of length five that's that side right over there and now let's try to draw the sides of length two every side of the triangle obviously connects with every other side so that's one side of length two and then this is another side of length two another side of length two and you might say fine these aren't touching right now these two points in order to make a triangle we have to touch them so let me move them closer to each other but we have to remember we have to keep these side lengths the same and we have to keep touching the side of length five at its endpoint so we could try to move them in we could try to move them in but what's going to happen well you could rotate them all the way down and they're still not going to touch because two plus two is still not equal to five if they rotate all the way down they're still going to be one apart so you cannot construct this triangle you cannot construct this triangle and I think you're noticing a property of triangles to the longest side cannot be longer than the sum of the other two sides here the sum of the other two sides is four two plus two is four and the other side is longer and even if the other side was exactly equal to the sum of the other two sides you're going to have a degenerate triangle let me draw that so this would be sides say two two and four so let's draw the side of length four side of length four side of length four let me draw it a little bit shorter so that's your side of length four and then in order to make the two sides of length two touch in order to make them touch you have to rotate them all the way inward you have to rotate them all the way inward so that both this angle and this angle essentially have to be zero degrees and so your resulting triangle if you rotate this one all the way in and you rotate this all the way the points will actually touch but this triangle will have no area anymore this will become a degenerate degenerate triangle and it really looks more like a line segment so let me write that down this is a degenerate in order for you to draw a non degenerate triangle the sum of the other two sides have to be longer than the longest side so for example you could definitely draw a triangle with sides of length three three and five so if that's the side of length five and then this if you were to rotate all the way and those two points would if you were to rotate all the way in if you were to rotate let me draw this a little bit neater so let's say that's where they connect and we know that we could do that because if you think about if you were to keep rotating these they're going to have to pass each other at some point they're going to have to overlap if they were if you tried to make a degenerate triangle these points wouldn't touch they'd actually overlap by one unit right over here so you could rotate them out and actually form a non degenerate triangle so this one you absolutely could and then there's another interesting question is this the only triangle that you could construct that has sides of length three three and five well you can't change this length you can't so you can't change that point in that point and then you can't change these two lengths so the only place where they will be able to touch each other is going to be right over there so this right over here is the only triangle that meets those constraints you could rotate it in whatever else but if you rotate this it's still the same triangle this is the only triangle that has sides of like three three and five you can't change any of the angle somehow to get a different triangle