Welcome back, everyone. So in recent videos, we've been talking about special case right triangles. And we recently learned about the 45, 45, 90 triangle, and some shortcuts you can use for solving this special case. Well, in this video, we're going to be taking a look at the 30, 60, 90 triangle, which is another special case. And for these triangles, there are shortcuts you can use to solve them as well. And this is a type of triangle that is going to show up a lot, not only in this course but also in future math courses and possibly other courses too, like physics or other science-related courses because this situation just tends to happen a lot.

So without further ado, let's get right into some shortcuts you can use to solve this triangle. Now the thing that's special about this triangle is that you can relate the side lengths to the shortest leg on the triangle. Now keep in mind for the 30, 60, 90 triangle, it's a bit different than the 45, 45, 90 triangle. Because for the 45, 45, 90 case, we learned that two of the side lengths are the same for the triangle, and so you can use one relationship to relate all the sides. Well, for the 30, 60, 90 case, the two side lengths are not the same. We have two different lengths.

So what we need to do here is keep track of which side relates to which other side, but we can do that using the shortcuts. So, for example, if you want to find the hypotenuse of a 30, 60, 90 triangle, you can take the short leg of your triangle, the shortest side, and you can multiply it by 2. And if you want to find the long leg of your triangle, you can take the short leg of the triangle again, and you can multiply it by the square root of 3. Now again, I want to emphasize, these shortcuts only work if you're dealing with a 30, 60, 90 triangle, and this is the special case we have in this example.

So let's see if we can solve for the missing sides. Well, I can see that the hypotenuse is going to be the shortest leg multiplied by 2, I can see the shortest leg is 5, and if we multiply that by 2 we're going to get 10, meaning the hypotenuse is 10. Now I can see that the long leg of the triangle is equal to the short leg, which we have down here, and we said that that's 5 multiplied by the square root of 3. So the long leg of the triangle is going to be 5×3. So this is how you can find the missing sides of a 30, 60, 90 right triangle.

So as you can see, it's still very straightforward when you have the shortcuts available. But to make sure we know how to use these shortcuts well, let's try some other examples that are a little bit more complicated. So for these examples, we're asked to solve for the unknown side of each triangle, and we're going to start with example a. Now in this example, I see that we have the hypotenuse and a 30-degree angle. So what we first have to ask ourselves is, is this a 30, 60, 90 triangle?

Well, it actually is. Because if this is 30 degrees and this is 90 degrees, then the other missing angle has to be 60 degrees. Because 60 plus 30 gives you 90, and then you add another 90, you get to 180. And all angles in the triangle have to add to 180 degrees. So this is a special case. So that means we can use the shortcuts we learned about. First, I'm going to try solving for the short leg, and I'm going to do this using this relationship up here. Because we have that the hypotenuse is equal to the short leg multiplied by 2. Now the hypotenuse, we can see, is the longest side, which is 8, and so that's going to equal the short leg of the triangle multiplied by 2.

So if I want to find the short leg, all I need to do is divide both sides of this equation by 2. That's going to get the twos to cancel on the right side, giving us that the short leg of our triangle is equal to 8 divided by 2, which is 4. So that means this missing side is 4. Now if I want to find the long side of the triangle, I could use the Pythagorean theorem from here, or I could use this relationship that says that the long leg is the short leg multiplied by the square root of 3. So we have that the long leg is 4×3. Meaning, this missing side of the triangle is 4×3. So that is how you can use shortcuts to find the missing sides of a 30, 60, 90 triangle.

But now let's try another example, in this case, example b. And for this example, we're asked to find the missing sides as well. And what I can see is that we have a 60-degree angle in the long side of the triangle given to us. Now because we have a 60-degree angle here, by default, this other angle has to be 30 degrees, Because all angles in the triangle have to add to 180, so this is a special case. So let's try using some of these relationships.

Well, since I see that we have the long leg, I'm going to use this relationship right here, which says that the long leg is going to be equal to the short leg multiplied by the square root of 3. Now I can see that we have that the long leg of the triangle is 1. So we'll have 1, the long side, is equal to the short side, multiplied by the square root of 3. So what I can do from here is divide the square root of 3 on both sides of this equation, which will give the square root of 3's to cancel on the right side, giving us that the short leg of this triangle is equal to 1 divided by the square root of 3.

And I can go ahead and rationalize this denominator by multiplying the top and bottom by the square root of 3, that'll get the squares to cancel, giving us that the short leg of this triangle is equal to the square root of 3 over 3. So that's going to be the short leg of the right triangle. Now our last step for solving this right triangle is to find the hypotenuse or the longest side. But I can see based on the relationship up here that the hypotenuse is going to be the short leg, which we already determined is square root 3 over 3 multiplied by 2.

So that means that this missing side of the triangle is going to be 2 times the square root of 3 over 3. So this is how you can find all the missing sides of a 30, 60, 90 triangle using some simple shortcuts.

So that's how you solve these types of problems. Hope you found this video helpful. Thanks for watching.