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# In mathematics, the number 45 (forty-five) is a natural number that follows 44 and precedes 46.

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Hey, guys. Climber here back with another lesson about special right triangles. We've already looked at 30 60 90 right triangles, Um, as a triangle that has very special sides and relationships between the lengths we're gonna get another one. We know this is 45 45 90 right triangle. So if you recall about the 30 60 90 right triangle, if you took an equilateral triangle which is 60 60 60 and you cut it right down the middle, you created two right triangles, each of which was a 30 60 90 right triangle. What happened was whatever this length was, this was double because of the relationship of the equilateral triangle. And this ended up being actually three, and we proved that with with a degree in therapy. Okay, so likewise, if you take a square, okay, which is another regular polygon. All sides and angles are equal, and you essentially draw Dagnall. You split it into two. I saw Seles, right triangles. Obviously the sides are equal because it's square. That's 90 degree angle because the square and you're gonna cut this angle, and basically you're gonna bisect it. So this is gonna be a 45 degree angle, and this is gonna be a 45 degree angle. So you end you end up with this a triangle looks like this disoriented a little differently. Okay? And this is your 45 this is your 45 and we can call this X. We can call this X. The question is, what's this site? Well again, based upon Pythagorean theorem. If we know this side is X and this side is X, then I could do Let's call this high partners. I know that by the Pythagorean theorem that X squared leg squared plus the other leg squared better be your iPod news squared. These were like terms, so I can call two x squared equals h squared, And then we can do is square root both sides and we're left with essentially x route to this simplifies because the squared of export is just x and then this house is simplifies. So we end up with a pot news. It's just gonna be literally radical two times a length of one of the legs. So this is a very straightforward look at what a 45 45 90 special right triangle looks like eso. Let's do a couple of examples and they'll be back with another video. They kind of like We'll break down some triangle. How to solve them, find sides, that kind of stuff. We'll do a couple of quick examples right now, so if we have a triangle and as long as you do that I know automatically, it's a 45 45 90 because you're labeling it as an isosceles triangle. If these sides are equal, then it forces the opposite angles to be equal, which forces them to 45 45. So I don't have to label them if I have labeled as a necessity strangle. So if I call this 10, this is automatically 10 because they're equal. This is automatically 10 route to because the iPod news is going to be radical two times one of the legs. Cool thing is this'll this discovery of a right triangle, particularly isosceles right triangle. They'll length off the high partners. Was the discovery essentially of irrational numbers In general, the Pythagorean is led by Pythagoras at first thought everything was or could be described as a rational number or ratio of two rational numbers and one of the Pythagorean, um, found that the iPod news wasn't April. You weren't able toe write it as a ratio of two. Rational Zip's irrational is a fairly simple proof of that for another day. But the high pot news of an isosceles right triangles something very special in the history of mathematics, and it's kind of like the birth of irrational numbers, and they've always existed. But in terms of the discovery of irrational numbers and making sense of it, some of the earliest proofs involved the length of high pot news of a 45 or 45 90 and discussing that to be what we call now or no now as an irrational number. So za kind of interesting. It has everything to do with this radical, too. That's attached to it. The funny thing is the Pythagorean theorem, which is based upon Pythagoras name who it first didn't believe in a rational numbers. It comes from that, so it's kind of a bit of a of irony there, Um, here's a few more examples, so let's say we have my 45 45 90 again. I don't have to label the angles because the tick marks indicate that. But let's say I said this was, you know, 15 route to I would automatically know this is 15 and I would automatically know this is 15 because of the x x ex route to situation that we know about when it comes to special right triangles, particularly apartment 45 45 90 triangle. So any time you encounter in geometry and equal lateral triangle just in the back of my head, I'm thinking maybe somewhere there's gonna be a 30 60 90 lurking. If we draw that median it remember, median is the segment that comes from the opposite Vertex to the midpoint of the opposite side. You're gonna get to 30 60 90 triangles if you do that. The median here, by the way, in this case, is also the altitude, which is also the angle by sector. They're all the same in that triangle because it's special. Likewise, if I think square, I see a square in the back of my head. I'm thinking there might be a 45 45 90 right triangle here, and I really think this applies when it comes to any kind of standardized test practices there's tons of equal lateral square 30 60 90 and 45 45 90 Situations that occur given that your you see a square, any collateral triangle or, um, you know, you just see the triangles outright. But these are very common questions. And to know the relationships between the sides, Um, namely x two X and x, Route three in this situation or x x and X route to in this situation, I would put that down on the definitely need to know list. Um, not only for every day, every day to day mathematics, but especially if you're you're standardized test practices. So I'll get back with some more. 45 45 90 right triangle questions. See you there. Webster University

#### Topics

Rigid Motions (Isometries)

Volume

Terminology

Relationships Within Triangles

##### Top Geometry Educators ##### Catherine R.

Missouri State University  ##### Heather Z.

Oregon State University ##### Samuel H.

University of Nottingham