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30-60-90 Right Triangles

In mathematics, the natural numbers are used to count (or enumerate) objects, such as the objects in a collection or set. These have been defined by the Peano axioms. Natural numbers can be used to count objects (e.g. number of students in a class), to measure lengths (e.g. number of meters in a yard), or to measure rates (e.g. number of miles per hour). The first three numbers (1, 2, and 3) are the first three counting numbers.


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Video Transcript

all right. We're here today to talk about a very special right triangle, though there's lots of right triangles and, you know, depending on who you are, you might consider one triangle more supposed than the other. But there are two very special right triangles, and I'm gonna talk about one of them today, and that is the 30 60 90 right triangle. It's got a very special property. If you have a 30 60 90 right triangle drama, do my best to draw one here. Let's call this the 30 degree angle here, and this is the 60 degree angle here then, first of all, we gotta talk about the how the angles and sides relate to each other in a triangle, the smaller the angle of the small of the opposite side and vice versa. So the smallest side is going to be this side because 30 degrees is the smallest angled triangle. So if I called that X, what's very special and unique about a 30 60 90 right triangle is that given the small aside, the hypotenuse is always double it, and the other leg is always going to be X Times Route three. So if you know the smallest side or actually any of the sides. You can use this pattern to figure out the other sides. So this also comes from Trig. So if you if you know of your unit circle, you'll see a lot of times one half and route three over to show up. And it's because of this triangle. So So let's talk about why a 30 60 90 right triangle special where the side relationships come from. So if we look at and equal lateral triangle, which we've talked about previous videos way have a triangle has all three sides equal, it has a triangle that every angle is 60 degrees. So all three of these air equal Well, if you draw a perpendicular down to the opposite side from that vertex than by the nature of the equilateral triangle, you're basically gonna get to congruent triangles. You're going to get this left triangle. Congrats into this right triangle, and they're gonna be right triangles because of the right angle. So essentially, what you have is the right triangle, and because of how you are basically cutting that equals triangle in half, this site here is now gonna be half of the base. And since all the sides are equal, we know that this half is gonna be half of what now is the high pot news. So now if I know this has some length of X, it forces the high pot news to be double that, which is two X. I also know that this is 60 degrees because that was this angle in the corner. And now I know that this is 30 degrees because we cut the bisected, the 60 degree angle, the Vertex. Now it's 30. So again, looking at this relationship that we have just created, here's my 32 year angle. Here's my 60 degree angle. Here's my right angle. We're gonna call this, um, our trail length x of Just seeing this is gonna be double that with the thuggery. And the're, um, that we've seen in another video is going to say that this missing side, which I'll call l for the leg we know that X squared plus l squared equals quantity two x squared. Well, that would be X squared. Plus else squared equals four x squared. Or, in other words, l squared equals three x squared. If we go ahead and then take the square root of both sides. We get l equals Route three X. And that's where our relationship that I mentioned at the very beginning or ex Route three is coming from. So we have these forced relationships between the sides. Um, but nice. Where you have X the smallest side, always opposite the 30 degree angle. Hi, Partners is double that. And the leg is going to be radical three times that. So let me give you an example. If you have a 30 60 90 right triangle, and I know, for example, this is four units, I automatically know this is gonna be eight units. This is gonna be eight route three units that quick or if I have again, Ah, 30 60 90 right Triangle. And I know let's say this is 20 because this is the high pot news, the smaller side, because it's opposite the 30 degrees notice I brought it. I did not write 30 degrees, but I wrote six degrees in the other angle, which is gonna force the other angle of 30 degrees. We automatically this is gonna be 10 because it's half the high partners automatically. This is gonna be 10. Route three. Likewise again. 30 60 90 right Triangle 30 60 90. Let's say I know that the leg over here is 12 or three. Well, now I know that the base or the other leg that's opposite the 30 degree side is 12 and that the iPod news is 24 all because of the special relationship that we get by cutting the collateral triangle in half, you get 2, 30 60 90 right triangles, and then what happens is the special triangles. Besides have a relationship of essentially some value, some value doubled and that value times Route three. So it given one side you have all the other side's. Specifically in a 30 60 90 right triangle, check out my next video that has several examples of how we could use in practice with 30 60 90 right triangles.