Everyone, in a previous video, we discussed the basics of triangles, and I mentioned that we would very commonly be working with right triangles. But one of the most common situations that you'll see is where you have two sides of a right triangle that are known, but you have an unknown side. For example, we have 34 that are known here, but this side x here is missing. Well, don't worry because, in these kind of situations, we can always solve for this missing side by using something called the Pythagorean theorem. It's probably something that you've heard before in a math class, but we're going to be using it a lot in this course, and you'll need to know it. So I'm going to go ahead and explain it to you. What I'm going to show you is that it's really just an equation relating the three sides of a right triangle. Let's go ahead and get started. We'll do some examples together. Alright?
The first thing you need to know about the Pythagorean theorem is you can only use it when you have a right triangle. Alright? So you can only use it when you assume or know that one of the angles over here is 90 degrees. If you don't know that, then this equation won't work. So what is the equation? Well, it's really just a2+b2=c2. Again, you've probably heard that before, but it really means that if I take these two numbers over here, a and b, they'll just be numbers, and I square them and add them together, that's the same exact value as this side over here squared as well.
Let's take a look at our first example, so we can actually just get some practice with this and do it together. Alright? So we have 34 that are known over here, and you have x that's unknown. This is a right triangle, so I'll be able to use the Pythagorean theorem to solve that missing side. I just have my equation over here, a2+b2=c2. Alright?
So how does this work? Well, what's really, really important about the Pythagorean theorem as well is that you always have to keep in mind that your a and b need to be the shorter legs of the triangle. Always set your a and b as the shorter legs that form the corner, the 90-degree angle, and then you want to set c as the hypotenuse. The hypotenuse of a triangle is always the longest side, which usually is going to be the diagonal. Not always, but almost always. So in this right triangle over here, what we can see is that these two form the sort of corner like this, that's a and b, and c is going to be the diagonal, the longer one. That's what we set as c. Now, when it comes to a and b, it actually doesn't matter which one you pick as a or b.
So what this equation says is that a2+b2=c2. In other words, if I take 4 and square it, and I add it to 3 and square it, and I figure that out, that's going to give me this missing side squared. That's going to be x squared. Alright. So 422+322 actually just ends up being 16 plus 9. That's going to equal x squared. And if you actually just go ahead and work that out, that's going to be 25. So are we done here? Is the answer just 25?
Well, no. A lot of students will mess this up. You have one last step to do here, which is you have to take the square root of both sides because you want x, not x squared. If you do that, what you're going to get is that x is equal to 5, and that is the answer. So the hypotenuse of this triangle is equal to 5. Alright? One really quick way to check your work is to remember the hypotenuse has to be the longest side. So notice how 5 is longer than 3 and 4. If I got something that was lower than 3 or 4, then I know I would have messed something up.
Alright? But that's really all it is. Alright? That’s the Pythagorean theorem. Go ahead and pause the video and see if you can find the answer to this problem over here, example B. Alright? So let's go ahead and work it out.
So we've got this triangle over here. Notice how it's actually a little bit different because we have a right triangle just like we did over here. But this time, we actually know what the hypotenuse is. That diagonal longest side is something we already know. And, in fact, one of the shorter legs is actually one of our unknown values. But we can still use the Pythagorean theorem because remember, it's just we know 2 sides out of 3. I'm going to start off with my equation over here, a2+b2=c2. I set my c to be the hypotenuse. In other words, c is 10.
Again, it doesn't matter which one is my missing variable. Right? Or sorry, which one is my a and b. Either my a and b will just make up that 90-degree angle. So I'm just going to go ahead and set this one to be a and this one to be b. We'd get the exact same answer if you did it the other way. Alright? So this is b equals 6 and a equals 1. So what the Pythagorean theorem says is that a2+b2=c2. So in other words, y2+62=102. Now, you just plug in the values that you already know and calculate.
This is going to be y squared plus 36, that's 36 over here, that's going to equal 10 squared, which is 100. You can subtract 36 from both sides like this. Subtract 36. What you're going to get over here is that y squared is equal to 100 minus 36, which equals 64. That's equal 64. Now the last thing you have to do is just take the square root of both sides. You have y is equal to the square root of 64, and that's equal to 8.
So that is equal to 8 over here. So that means we go back into our diagram and this is going to equal 8. Notice how again, the hypotenuse, the 10 is still longer than both of the other shorter sides over here, and so that perfectly makes sense. Alright. So that's it for this one, folks. Thanks for watching and I'll see you in the next one.