in this video we're going to focus on Pythagorean triples so let's say we have a right triangle and we're familiar with Pythagorean theorem which gives us an equation relating to the three sides of the right triangle and that is a squared plus b squared is equal to c squared now the Pythagorean triples are basically whole numbers that fit this equation so for instance 345 is considered a Pythagorean triple because 3 squared plus 4 squared is equal to 5 squared 3 squared is 9 4 squared is 16. 5 squared is 25. 9 plus 16 is 25. so what this means is if you know two sides of the right triangle you can easily know the third side without doing any math so if this side is 3 and this side is 4 automatically you know this side is 5. and so being familiar with the Pythagorean triples helps you to solve right triangles quickly now any multiple of the three four five ratio will work as well if you multiply this by two you're going to get the 6 8 10 right triangle if you multiply by three you'll get the 9 12 15 right triangle if you multiply by 4 you'll get the 12 16 20 right triangle if you were to multiply by 5 you'll get the 15 20 25 right triangle now all of these you'll likely see this in school I've seen it doing homework problems and they're very common so make sure you're familiar with the multiples of the Pythagorean triples if you could divide it by a common number then you can easily identify the base Pythagorean triple so this is the first one we need to be familiar with the next Pythagorean triple is five 12 13. and you could test it in this formula 5 squared plus 12 squared is 13 squared 5 times 5 is 25. 12 times 12 is a 144. 13 times 13 is 169. now 25 plus 144 that's 169 so you can see that works as well and of course the multiples of this Pythagorean triple will apply as well so we also have the 10 24 26 right triangle the 15 36 39 right triangle and so forth but now from here on let's focus on just the base Pythagorean theorem so the smallest most common one which we've covered is the three four five right triangle next we have the 512 13 triangle after that the next common one that you'll see in school is the seven 24 25 right triangle and then there's the 8 15 17 right triangle so these four are the most common base Pythagorean triples that you're going to see in a typical high school or college math course now there are some other less common Pythagorean triples but you're going to counter some of these as well the next four is the nine 40 41 triangle and you could check it if you type in in your calculator 9 squared plus 40 squared 9 squared is 81. 40 squared 40 times 40 That's 1600. and if you type in 41 squared you'll get 1681. so you could see that this particular Pythagorean triple checks out after the 9 40 41 triangle the next one is the 11 60 61 right triangle after that it's the 12. 35 37 right triangle and there's also the 20 21 29 right triangle so for these you're typically not expected to memorize these but I've seen them in certain problems so you can add these to your lists so those are the most common Pythagorean triples that you're going to encounter in school now there's a lot more that are very uncommon you really don't have to memorize these but for those of you who are curious I'm going to write them down so the next one the next few that I'm going to write are the ones that are under a hundred so we have the 1384 85 right triangle next is the 16 3 65 right triangle after that we have the 28 45 53 right triangle and then there's the 33 56 65 right triangle next is 36 77 85. then there's the 39 80 89 the 48 55. 73 and then the 65 72 97. and you could find these numbers online if you do a Google search just type in you know a list of Pythagorean triples and there are websites out there that will give you this but for the most part you really don't need to know the ones highlighted in green I've never seen them in a problem the first date the ones highlighted in blue you should definitely memorize those the one highlighted in red they're less common but you may see them in a problem so if you memorize the first eight you're going to be okay uh for almost any problem that involves Pythagorean theorem now there are some other ones that go well past 100. I'll write a few of them so we have the 15 112 113 right triangle next is the 17 5 I mean 144 145 right triangle after that we have the 19 0 181 20 99 101 and then there's the 21 220. 221 and then 23 264 265. and I'm gonna do two more 24 143 145. and 25 312 313. so this more than that but I mean you can go forever with these numbers but as you can see there's a lot of Pythagorean triples but for the most part if you memorize the first eight in blue and red you should be okay