How to Use the Pythagorean Theorem (a² + b² = c²)

0:03 Hey everybody, welcome back to Math with Mr. DJ. Today we're going to be talking about the Pythagorean theorem. So, let's get started. Let's jump right in. Every right triangle has a secret. Two short sides, one long side. I know, crazy, but just know what we're talking about with triangles is we're talking about a right triangle. a triangle that has a 90 degree angle. Now, what you need to know is the long side is called the hypotenuse. It's always across from the right angle. So, you can just take a little arrow, point it directly across from that right angle, and that is the hypotenuse. Always. And it's always the longest

0:50 side. You're going to have the A side, the B side, and the C side. The C is going to be your hypotenuse. It's always your longest. And the two short sides are the legs. That's what we call them. The A and the B are going to be your legs. So, here's the rule. A 2 + B 2 = C^2. Now, remember when we're labeling the legs of the right triangle, one leg will be A and one leg will be B and the hypotenuse will be C. the leg squared plus leg squared equals hypotenuse squared. That's basically all it is in squaring a number is just multiplying it by itself. All right, let's start off with example

1:36 one. We're going to find the hypotenuse to start. Let's say the legs of this right triangle are three and four. Now again remember we have to find the hypotenuse the longest side and the formula is we have a^2 + b^2 = c^2. So what we're going to do is we're going to take the 3 replace the the a with that 3 to make it 3 squared. Then you take the 4 cuz that's the b and you're going to put that in where you see the b. So now we have 32 + 4^2 = c^2. Now you have nothing you can put in for the c right now because that's what we're looking for. We're looking for the longest side. We're

2:21 looking for the hypotenuse which is the c. 3^2 is 9. 4^2 is 16. So now we have 9 + 16 = c^ 2. 9 + 16 is 25. So 25 = c^2. and C is going to equal 5 because the square of 25 is equal to 5. So when you are looking for what C is cuz we're not looking for what C squared is, we want to know what C is. So 5^2AR is 25 or the of 25 equ= C. That's how we get C is equal to 5. In this triangle, we have a 345

3:09 triangle, which is actually a special type of triangle. And I did this problem on purpose so that we could talk about the 345 triangle. And it comes up often when you're doing problems like these. So keep an eye out for it. So for example, it might not be the legs are three and four and the hypotenuse will be five. It could be 6 8 and 10. Right? They could be proportional to each other. It's it might not just be three, four, five, but if you do get legs three and four, you should know the hypotenuse is going to be five. If you have one leg that's three and a hypotenuse that's five, you know the other leg has to be four. You're going to see it everywhere. Memorize it. The 345. All right, let's

3:56 move on to example two. We just did a problem finding the hypotenuse. Now, let's do one where we're finding the leg. All right, one leg is five and the hypotenuse is 13. I always find that having a visual when I'm trying to solve these helps me a lot. So, at home right now, make sure you're drawing a right triangle. Make one of the legs five and make the hypotenuse the long side opposite the right angle 13. And you can make the other leg B. Now remember the hypotenuse always goes in the C spot. Always goes in that long side opposite the right angle. So now we have 5^2 + b ^2 = 13^2.

4:41 5^2 is 25. We're just going to rewrite the b ^2 and 13^2 is 169. So now what you need to do is you're going to subtract 25 from both sides so that we can get b ^2 all alone to solve for b. So subtract 25, subtract 25 and we're going to get b ^2 is equal to 144. Now what you're trying to do is you're trying to find b * b = 144. b ^2 = 144. So one way you can do it is you can think about that or you can find the square root of 144. The square root of 144 is 12 because 12 × 12 or 12 squared

5:26 equals 144. Nice. All right, example three. The ladder problem. Woo. These ones show up a lot on standardized testing and it this type of problem always trips up students. So let's walk through this one. So, the ladder problem, we're going to imagine there's a 10-ft ladder leaning against a wall. So, if you ever have this type of problem, write in your brain, it should say, "Oh, this must be a Pythagorean theorem problem." Remember a^2 + b²= c^2. And in these problems, when that 10ft ladder is leaning up against the wall, it's almost that the ladder itself is the hypotenuse, the c. So, let's walk through it again like I was just saying. All right. So, it says its base is 6 feet out. So, that's

6:12 going to be the bottom part of this triangle. That'll be six. How high does it reach? Let's take a look. The ladder is the hypotenuse. So, we're going to have 6^2 6 2 is the base of how high the ladder is coming out. The ladder itself was 10 ft. They told us. So, uh 6 2 6 2 is 36 = 10 2 10 × 10 is 100. Then what you're going to have to do is you have to subtract the 36 from both sides because again we want to get the that variable all alone on one side so that we can solve for it. Now we have h2 equals 64. So you have to do one more step. You have to find the square of 64. What × what is going to get us 64? 8. Yeah. Because 8 × 8 equals 64.

7:00 Now it's your turn. The legs of 8 and 15. Find C. Take a moment. Pause the video, try it out yourself. The best way to learn this is to actually put in some time and try to work it out yourself. So, if you haven't paused the video yet, I really encourage you to pause it and try it out. Remember to draw the right triangle. And for your legs, you're going to put 8 and 15. And then your hypotenuse is going to be C. And don't forget, A2 + B2= C^2. See if you can do it. All right. Answer is going to be pinned in the comments. So, take a look in the comments. See if you got it done correctly. If you did, nice job. Make sure you leave your own comment. Tell me what you got. If you

7:45 didn't get it done correctly, where did you go wrong? It's okay. Remember, making mistakes in math is okay as long as you're learning from them. All right, we'll see you in the next video. Thank you so much. Don't forget to like, subscribe, tell your friends. See you soon. Take care. Bye.