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# 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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you guys, Mr clambered here and we're gonna do some practice with some 30 60 90 right triangles. Recall that in a 30 60 90 right triangle. Here's your 30 degree angle. Here's your 60 degree angle, not drawn to scale. But opposite the smaller angle is the smaller side. We label the X. The high pot news is double that length, and the other leg is radical three times that length. Remember that all comes from taking basically an equilateral triangle kind of get into. And then you get that relationship of one to to. So we're gonna use this information to basically solve some triangles. Find some missing sides, etcetera. So if you look at question one were given the iPod news, and it's always important to know where the 32 year angle is because that's half the iPod news that automatically means we got six, which automatically we have a six. Route three. So simple is that if you have the sides relationships memorize Okay, here's another example. This time, knowing that this is 60 tells us this is 30 and then that tells us that this is the smallest side. Here's the high pot news That's double the small side. That's 10. And this is Route three, the smallest size. Pardon me. So this is gonna be five, 33 Okay, so here's an interesting problem, because we look at the six Route three. We think that's the route three side. So this is gonna be six, and this is gonna be 12. But that's not the case, actually, because the 30 degree angle is opposite the smaller side, which is usually the quote unquote Exide, not the expert three side. So here's what we know. We know that the hypotenuse M is gonna be double the smallest side, which is two times six or three. Or, in other words, 12 3. The longer leg that's opposite the 60 degree angle is usually whatever this site is, it's Route three it It's route three times that size that side. So in our case, it's gonna be six or three times an additional Route three Route three times. Route three. The radicals cancel and you're left with six times three, which is 18. So don't be fooled by a side that's labeled with a radical three and think it's the longer leg, the legs opposite the 60 degree angle because it's not guaranteed to be that way. Let's look at another example. So here, this is an appropriate example of Look at this. This is the 60 degree angle. This is the site opposite, which should be the X Route three side, which automatically tells me then then this is five. And since this is the high partners, this is 10. That's an okay situation. You haven't really careful about which side you're labeled. How so? These are the ones that really get people. This is the 62 year angle. It's opposite the ex parte me the X Route three side. I don't see a Route three. I deceived nine. Okay, so what that means is, in order for that leg to be nine, why must have had a route three attached to it? In order for this to be a value that's none not irrational, not attached to a radical three. Something must have happened to where whatever, why is we had to multiply it by radical three, obviously to get to the longer leg, and that ended up being a really nice into your value. So that means why it's not a nice energy value. And so here's how you can look at this, you can say, Well, I know the expert three site is you will tonight. That would mean that X equals nine over route three. Okay, Now, I'm actually confusing you, so I'm actually gonna apologize. I'm gonna take away this. I'm gonna take away this only because of the way the triangles labeled. And this is technically should be. Why Route three? Because whatever. Why is times route through? You get the longer leg. So technically, you could say that. Why is nine over route three? We're not gonna see it that way that often, Because for lots of reasons, we don't like to have rash radicals and the denominator. So you've heard of this process called rationalizing a fraction. And you basically multiplied by a very fancy form of one. And you get nine. Route 3/3. Because don't forget that a radical temps itself cancels the radical out. The 9/3 becomes just three Route three. So this smaller leg opposite the 30 degree angle is three. Route three, which makes the pot news, which is double the smaller leg two times 33 or in other words, six or three. A little shortcut for this is notice that nine divided by three is three. And that's actually the shortcut for all of these problems. So if you have the larger leg, the longer leg and it's not attached to a Route three. So, for example, let's say I had 15. Is the mhm the longer leg? My shortcut is? Take 15 divided by three. That's five and then put a radical three on it. That's how you can get a shortcut to the shorter side if this doesn't have a radical three attached to it, meaning it's not just the simple example where everything is nice and neat based upon your X two X and X Route three. Of course, if we didn't know that, we could always say, All right, let's call this X. That means 15 better be X Route three, because whatever this site is, Ida multiplied by route through to get here. That means that X is equal to 15 over Route three, which means we're gonna multiply by Route 3/3 again. And remember, this becomes a 3 15 divided by three is five. That's where I get this shortcut from so just something to think about their in terms of If you do enough of these. Is there a a shortcut? Okay, so we could do a few. All right, so here's another one. Where this is the 60 degree angle. This is the opposite side. Typically, this is the uh huh. Smaller leg times route three. So in this case, this would be like be times route three. We can do this two ways. We could do the shortcut, which is basically just taking this side divided by three, which is 12, and then throwing a radical three on it, which makes this 24 Route three. Because it's double or remembering that 36 has to equal be route three. Because whatever this site is radical three times it is the longer leg that would make be equal to 36 over three. If you multiply by Route three of the Route three to rationalize it, you'll get 12 or three. Okay, So the nice thing about these videos, you can pause, rewind. You can digest these problems at your own pace. And so hopefully you can see why the long way works. But then Why the short where the shortcut works as well. Let's do a few more problems. All right, Here's an example of we have the 30 degree angle. Opposite side is the quote unquote smaller side. So the hypothesis double that. So two times four, Route three. Is this simply eight? Route three? That's nice. And remember that the longer leg is Times Route three, its route three times this. So this equals four route three times Route three and route three times Route threes three because the radicals cancel and you get 12. So here is a hopefully more than a handful of examples of how you could use a special right triangles toe, find missing side lengths. Um, by using the pattern of x two X and extra three. I'll see you next time we're here. You pardon me? 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