 and welcome back today in this video we're going to talk about special right triangles in this video we're going to talk about our 30, 60 and 90 degree special right triangles and what I've provided down here at the bottom is I've provided kind of a little cheat sheet for us for our 30, 60, 90 triangles now X in this case is going to be the short side the short leg x-root 3 is going to be the long leg and then 2x is going to be the high-pot news okay so those those specific words we need to know for our 30, 60, 90 triangles okay so in this example or in this video excuse me what I'm going to do is I'm going to do two different examples of using our 30, 60, 90 triangles okay so the directions are basically going to be the same for both for both examples find the values of X and Y give your answers in simplest radical form okay so no decimals or any that kind of messy stuff we're just going to use radicals alright so in this case looking at this triangle for this example the only the only side that they give me is my hypotenuse is this 22 right here so if I go down to my little cheat sheet here my hypotenuse is right here now whenever you're looking at a 30, 60, 90 triangle the one side you always want to try to find first is your short side your short leg so this X that I have down here that's the one you should find first now if I'm going from the hypotenuse down to the shorter leg I have to divide by two notice here 2x and X to get from 2x to X I would have to divide by two so that's what I have to do in this case if I want to get from the hypotenuse down to the shorter leg I'm gonna have to divide by two but the first thing I have to identify first is which one is my shorter leg now this cheat sheet yes helps us with some of these triangles but notice that this one's in a different orientation so now the shorter or the shortest leg of a triangle is always going to be opposite of the smallest angle so notice this 30 degrees over here is going to give me this that's the smallest angle opposite of that is always going to give me the smallest side okay the smaller the angle the smaller the opposite side so what this does is this tells me that X right here is in fact my shortest leg so in this case X is going to be my short leg is going to be half of what excuse me it's gonna be half of what the hypotenuse is so in this case X is equal to 11 okay now that that short leg is usually again usually one of the first ones you always want to find because it's usually the simplest one to find and if you want to find anything else from there it's usually easiest to use the short leg to do that okay now change things up a little bit now what we got to find is we got to find why over here why is a cross from the 60 degree angle which means it's my longer leg it's going to be the biggest leg of the triangle not to say that it's the biggest leg overall 22 is still going to be the largest side it's the hypotenuse it's always going to be the biggest side we're just saying why is the larger leg X is the smaller leg why is the larger leg all right so to get from and again we're always we're always trying to use our smaller leg to get from the small leg to the long leg what all we have to do is multiply by the square root of 3 so in this case why is going to be pretty easy to find all I have to do is take 11 and multiply times the square root of 3 okay and now in this case we want our answers in simplest radical form so that's it that's as good as I get I can't actually multiply those numbers together 11 is on the outside of the radical and 3 the square root of 3 that's on the inside of the radical so you kind of have to leave it there you can't cross this boundary here you can't cross that okay so that's right there's as good as a gift now 11 times the square root of 3 11 times the square root of 3 I'm plugging into my calculator now is equal to 19.0525 bunch of the decimals after that okay so to get a better idea of how long that leg is it's about 19 so we have sides of 11 19 and 22 okay now what that also tells us is that yes it's the longer leg but it is still smaller than the hypotenuse so I know that I did do that correctly okay so there's the values of x and y for that triangle for that example we'll do another one to kind of do some variety here okay so same directions this time what we're doing is we're finding the values of x and y give your answers in simplest radical form okay so here's our example that we have first thing we got to do is identify what side they did give us I do have our cheat sheet down here a little down here so we can kind of refer to that I have this 15 here now I got to figure out what side is that now the only angle that they give me is this 60 which means this angle down here is 30 degrees this 30 degrees is the smallest which makes x this is going to be my short leg y is going to be my hypotenuse okay so the side that they did give me is they gave me the 15 they gave me the longer leg this is the longer leg this is the shorter leg this is my hypotenuse okay so again we got a little bit of work ahead of us this is what I would say is is the problem that's probably going to give us the most trouble is going to give us the most work but that's okay so notice here we are starting at the down here a little cheat sheet we are starting with the longest leg to get from the longest leg to the shorter leg all I have to do is divide by the square root of three so what I have to do is I have to take 15 and I have to divide by the square root of three okay now just like the previous video that we did with 45 45 90 triangles you we do not like this square root of three on the bottom that is not simplest radical form so we have to get rid of that radical on the bottom to do that what we have to do is we have to multiply by that radical that we want to get rid of so I'm going to multiply by the square root of three on top and bottom okay now what that does on top doesn't really do much 15 root 3 again numbers and radicals they do not mix together so I'm just going to put them next to each other yes they're still multiplying by one another but it's not really going to affect one another because one's on the outside one's on the inside on the bottom though is a different different story both of those are underneath the radical so I can multiply those together and I get the square root of nine okay and then this simplifies to 15 root 3 over well I know what the square root of 9 is that's going to be 3 and one other thing that I see is 15 and 3 both of those are outside of the radical which means that I can divide those so that's going to give me 5 root 3 nice a lot of simplifying there okay what that means is that what we just did is we took 15 and we divided by the square root of 3 and then that will give us our smaller leg so in this case x is equal to 5 square root of 3 okay now to get a better idea of what that is what I'm going to do is on my calculator here I'm gonna take 5 times the square root of 3 and I get x is about 8.66 and some decimals after that okay what that does that just gives me a kind of a better idea so this is about eight and a half a little bit more than eight and a half okay what that does tell me though that 15 is my longer leg 8.66 that's going to be my smaller leg that makes sense it's gonna be a little bit smaller but now what I'm gonna do is I'm gonna find the hypotenuse now going from the shorter leg to the hypotenuse all I need to do is multiply by 2 so for y here in this case what I'm gonna do is actually not write y first I gotta take I gotta take 5 root 3 and I have to multiply times 2 okay now this actually isn't really confusing at all when you when you take something like this and multiply times 2 it's only the numbers on the outside that are going to affect one another this 2 is not gonna is not gonna multiply with the square root of 3 inside so actually y is simply just going to be 10 root 3 okay 10 root 3 okay we just basically take the 5 multiply times 2 we get 10 now again in my calculator what I'm gonna do is I'm gonna plug in 10 times the square root of 3 and I get that y is about now 17.3205 it's more decimals after that okay so basically what that tells me is this gives me a little bit better idea of how long the hypotenuse is so my hypotenuse is about 17.3 which is so I have a 15 an 8.6 and a 17.3 so my hypotenuse is in fact the longest side so that makes sense okay just double check in to make sure I did this right all righty and and again finding the values of x and y and give your answers in simplest radical form again when you divide by radicals sometimes that can be a little bit of a process because you have to do a little bit of simplifying there but I showed you right there how to what's called rationalize the denominator if you ever have a radical on the bottom of your fraction just multiply the top and the bottom by that radical and afterwards it'll simplify to something nice and neat all righty that's it that is our 30 60 90 triangles just a couple of examples I've had to find all the sides all the all the legs all the hypotenuses of our 30 60 90 triangles all righty and yep that was 30 60 90 triangles thank you for watching and we'll see you next time