Welcome back, everyone. So up to this point, we have spent a lot of time talking about trigonometric functions, the Pythagorean theorem, and how they all relate to the right triangle. Now what we're going to be learning about in this video is some of the special and common right triangles that you're going to see. Specifically, we're going to be talking about the 45-45-90 special triangle. The reason these triangles are special is that they show up relatively frequently, and there are actually some shortcuts that you can use to solve these triangles very fast. So if you don't like all the brute force work we've been doing with trigonometric functions and the Pythagorean theorem, you're going to learn some shortcuts for solving these triangles in this video. Without further ado, let's get right into things.

When you have a triangle with 45-degree angles, like this triangle down here, for example, this is going to be a situation where you have the special 45, 45, 90 triangle. In these triangles, the two legs of the triangle are always going to be the same length. So if you ever see a situation where you have a right triangle and 2 of the legs are the same, that means you're dealing with this special triangle. We can solve for the hypotenuse of the triangle by simply taking a multiple of the leg length. And the multiple that you're going to look for is the square root of 2. Because if you take a leg like 5, and you multiply it by the square root of 2, this will give you the hypotenuse. We just solved for the long side of this triangle. As you can see, this shortcut makes solving the sides of the triangle really straightforward and really fast.

If you didn't remember this relationship, there is another strategy you can use, which is simply the long version of using the Pythagorean theorem. Let's say that we set this side to $a$, that side to $b$, and then the hypotenuse equal to $c$, and we want to solve for the hypotenuse. You could say that $a^2+b^2$ equals $c^2$, that's the factoring theorem. In this case, we said $a$ and $b$ are both 5. So we have $5^2+5^2$ equals $c^2$, and $5^2$ is 25. So we have $25+25=c^2$. $25+25$ is 50, and what we can do is take the square root on both sides of this equation to get that $c$ is equal to the square root of 50. And the square root of 50 actually simplifies down to $5\times \sqrt{2}$. Using the long version of this problem-solving, we get to the same answer. This is what's nice about the shortcut; it lets you get this answer without having to go through this long process.

To ensure we know how to solve these types of triangles, let's see if we can solve some examples where we have this special case. For each of these examples, we're asked to solve for the unknown sides of each triangle. We'll start with example a. We have 2 45-degree angles and two legs that are the same length. That means we're dealing with a 45, 45, 90 triangle. Recall that to find the missing side or the hypotenuse, we just need to take one of the legs and multiply it by the square root of 2. So if one of the legs is 11, and then we multiply this by the square root of 2. That right there is the answer. Notice how quick it is using this method. It's very straightforward, and that's what's really nice about these special cases. But now let's take a look at example b. In this example, we have a 45-degree angle, and we are given the hypotenuse.

First off, we need to figure out if we are dealing with a special case triangle, and it turns out that we are. Because since we have a 45-degree angle here and a 90-degree angle there, we know by default this has to be a 45-degree angle. All the angles in a right triangle have to add to 180, and 90 plus 45 plus 45 equals 180, so this is a special case triangle. To solve for the missing sides, we use this relationship. In this situation, we're given the hypotenuse or the long side. So what I'll do is take the hypotenuse, set it equal to the number we have, which is 13, and say that that's equal to the leg multiplied by the square root of 2. To solve for the leg, I can divide $\sqrt{2}$ on both sides of this equation. That'll get the square roots to cancel, giving us that the leg of this triangle is equal to $\frac{13}{\sqrt{2}}$. What I can do is rationalize the denominator here by multiplying the top and bottom by the square root of 2. That'll get these square roots to cancel, giving us that the leg of this triangle is equal to $\frac{13\times \sqrt{2}}{2}$. So what we're going to end up with is $\frac{13\sqrt{2}}{2}$ for this side of the triangle, and then $\frac{13\sqrt{2}}{2}$ for that side of the triangle. Because again, for a 45, 45, 90 triangle, these two sides have to have the same length.

So that is how you can solve 45, 45, 90 triangles, and this is the shortcut that you can use. I hope you found this video helpful. Thanks for watching, and let me know if you have any questions.