Is this triangle possible? A right triangle with sides 1, x, and x^2

Hinzugefügt: 2026-05-11

[00:00:00]Today, I want to show you guys a very cool right triangle. Let's have a look.

[00:00:04]So, if I start with the base, let's say one right here.

[00:00:09]And you can always start with some lengths that you want. And what I'm going to do is I'm going to multiply the one by certain factor. I don't know what it is. I will just call it X.

[00:00:19]And I'm going to make that into my second leg like this. So, let's say here's X.

[00:00:25]Now, as I said, I want a right triangle, so I'm just going to connect the dots from what connect the ends right here and here.

[00:00:34]Okay.

[00:00:35]Here's the cool part. What if for the hypotenuse, I also wanted to multiply X with this right here to here. So, I will get X squared.

[00:00:45]Is this possible?

[00:00:47]Again, we start with one and we multiply by some number, we get to X, right? And then if we multiply by the same number again, which is X squared, and I'm trying to put it here, and we get X squared. Is that possible?

[00:01:01]Well, let's go ahead and figure that out.

[00:01:03]Since this is just a right triangle, so we can use the Pythagorean theorem.

[00:01:08]A squared plus B squared equals C squared.

[00:01:11]A and B are the sides or the legs, and C is the hypotenuse.

[00:01:17]So, based on A squared plus B squared equals C squared we get one squared plus X squared and make that C is X squared and then we square that again.

[00:01:36]Okay, and then solve it. This is one plus X squared equals X to the fourth power.

[00:01:43]Whoa.

[00:01:45]We end up with a fourth power equation, but it's actually not bad at all.

[00:01:49]Because the truth is we can look at this as a quadratic equation but in terms of X squared. Let me show you.

[00:01:57]First, let me bring these two terms to the other side. I'll write this down here first though.

[00:02:01]X to the fourth minus X squared and then minus one is equal to zero.

[00:02:08]And then for this right here, I will look at it as X squared and then squared and then minus X squared and then minus one.

[00:02:17]Have a look. Here we have the input X squared and here we have the same input like to the first power. This is a quadratic equation in terms of X squared.

[00:02:26]We can use the quadratic formula.

[00:02:29]Here, let me write down let's say T squared.

[00:02:32]A T squared.

[00:02:34]Because I have X right here already, let me use T. Plus some number B times T plus C.

[00:02:41]If this is equal to zero, then the input T is negative B plus or minus square root of B squared minus 4 AC all over 2 A.

[00:02:54]Here, our input is the X squared. So, I will say X squared is equal to B is negative one, so we have negative negative one.

[00:03:05]Plus or minus square root.

[00:03:08]B is negative one.

[00:03:10]Square that. Minus four. A is one.

[00:03:14]C is negative one.

[00:03:16]All over two times one.

[00:03:19]And then we see that's one plus minus square root.

[00:03:23]This is one plus four, which is five.

[00:03:27]And then over two.

[00:03:30]Keep in mind though this right here is just X squared.

[00:03:33]Now, here's the deal.

[00:03:35]X squared should not be ending with a negative number.

[00:03:39]If you do one minus square root of five, the result is negative, so we have to get rid of that.

[00:03:46]X squared is one plus square root of five over two. In fact, that's what we call the golden ratio.

[00:03:54]And of course, right here we can take the square roots both sides.

[00:04:00]Technically, you put a plus or minus, but X is the length of a triangle, so I don't want the negative either.

[00:04:07]Cancel this. Finally, in order for this to work X is the square root of the golden ratio.

[00:04:16]I will write it down like this.

[00:04:20]Yep.

[00:04:22]I think this is really really cool.

[00:04:24]And here's a question for you guys though.

[00:04:27]What if today I'm going to kind of just relabel the triangle a little bit. So, for the blue one, what if now I'm just going to do like this instead?

[00:04:36]What if I want the shortest side to be X squared and then the second shortest side is X and then the hypotenuse is one.

[00:04:45]Well, is that possible?

[00:04:48]Let me know. I actually have myself this here so, yeah, let me know. Here's a right angle.