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 because they show up relatively frequently, 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. So without further ado, let's get right into things.

Now, 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 two of the legs are the same, that means you're dealing with this special triangle. What we can do with this is we can actually 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. And that's the answer. We just solved for the long side of this triangle. So as you can see, this shortcut right here makes solving for the sides of the triangle really straightforward and fast.

Now, 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. So let's say that we set this side to a, that side to b, and then the hypotenuse is equal to c, and we want to solve for the hypotenuse. Well, you could say that \( a^2 + b^2 = c^2 \), that's the Pythagorean theorem. And in this case, we said a and b are both 5. So we have \( 5^2 + 5^2 = c^2 \), 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 the square root of 2. So, notice when using the long version of this problem solving, we get to the same answer. But this is what's nice about the shortcut; it lets you get this answer without having to go through this long process.

Now 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. So for each of these examples, we're asked to solve for the unknown sides of each triangle. And we'll start with example a. Now notice we have two 45-degree angles and two legs that are the same length. So that means we're dealing with a 45-45-90 triangle. Now recall that if we want 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. Well, one of the legs is 11, and then we multiply this by the square root of 2. And that right there is the answer. Notice how quick it is using this method. See, 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. So how can we go about solving this? Well, first off, we need to figure out if we are actually 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 cusp there, we know by default this has to be a 45-degree angle. You see, 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.

Now to solve for the missing sides, what we can do is use this relationship. Notice in this situation, we're given the hypotenuse or the long side. So I'm going to do is take the hypotenuse, set it equal to the number we have, which is 13, and I'll say that that's equal to the leg multiplied by the square root of 2. Now to solve for the leg, what I can do is divide the square root of 2 on both sides of this equation. That'll get the square root of twos to cancel, giving us that the leg of this triangle is equal to 13 over the square root of 2. Now 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 13 times the square root of 2 over 2. So what we're going to end up with is 13 radical 2 over 2 for this side of the triangle and then 13 radical 2 over 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.

So I hope you found this video helpful. Thanks for watching, and let me know if you have any questions.