Can we have a right triangle with sides 1, x and x^2 (but the hypotenuse is x)

Okay, so I came up with this interesting triangle question for you guys. Let's have a look. So we are going to start with a right triangle like this and what we want to do first is I want to pick one of the sides right here to be one. Let's say we have the one right here. And then we are going to multiply this one right here by a value x to get to the hypotenuse. So this is x now. And then I'm going to multiply this by x again to get to the other side. So here we'll get x squared. And now the question is that does this triangle exist in the real world? Well, let's take a look at the sound conditions. Firstly, in order for this to make sense, since we have this side being x already, so x has to be positive. Secondly, based on our labeling, this side has to be less than the hypotenuse. So the other condition is that x squared has to be less than x. How do we solve this inequality? Well, we know that x is positive, so we can legitimately divide both sides by x and that is very nice. Because that way we see x has to be less than one. So in order for this to be true, I was thinking that okay, x has to be in between of zero and one. So it kind of makes sense. But don't forget because I put this triangle on the right triangle, so we will have to ask ourself does this satisfy the Pythagorean theorem? Well, we don't know yet. Let's go ahead and find out. So first I will call this side a, this side b, and the hypotenuse has to be c. And then a squared plus b squared is equal to c squared, we must have one squared plus b which is x squared and then square that. That has to be equal to c squared which is x squared. Now work this out a little bit, we get one plus x to the fourth is equal to x squared and put this to the other side and reorganize it. x to the fourth minus x squared plus one is equal to zero. Now the question becomes does this equation have any real solutions? In fact, none at all. And you have a few ways to do it. I think the easiest way is to just solve it and firstly you will see that this equation has no real solutions so that this is actually not a possible triangle. But let me just show you what if you allow complex numbers and see the solutions. The key to solving this the easy way is realizing this as a quadratic equation in terms of x squared. Because this is the same as x squared and then minus one times x squared and then plus one is equal to zero. Let's use the quadratic formula. The input x squared is equal to negative b which is the negative one here plus or minus square root of b squared minus four ac. A is one, c is one all over two times a. And then work this out. That's one plus or minus square root. What's this? One minus three, we get negative one minus four, we get negative three. Oh no, we have a negative inside of the square root. Game over, right? But anyways, let's just continue for the fun of it. Later I'm going to put a negative on the outside and then we get the i. But before we do any of that, remember this is x squared. So we have x squared equals a like actually two different complex numbers. So how complex can x be? Very complex, I'll tell you that. Anyways, to continue, you take the square root of the whole thing on the right hand side. Put a plus or minus, get rid of this. Ladies and gentlemen, if you want to solve this equation, x will be plus or minus big square root of one plus or minus this thing will be the i keep the square root of three all over two. And now you might be wondering can we simplify this expression a little bit? I think so or maybe not, but doesn't really matter. I'm just going to leave the answer like this. Like why not? So if you allowed complex numbers, then you can work with this triangle. I think it's really cool because the idea is if you pick one of these right here, it actually works. You start with one side being one, multiply by an x value, you get to the hypotenuse and then you multiply by x again, you get to the other side. I just thought it was really cool. Yeah. Anyways, just that random thought and if

you allowed complex numbers on the side of a right triangle, you can do a lot of like fun stuff like crazy things. One thing I believe any of you guys have seen on the internet before, if you have a triangle like this, let's say this side is one, this side is i, then let me ask you, what will be the hypotenuse? You guys can leave your answer down below and let me know. That's it.