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Pythagorean Examples

In mathematics, a Pythagorean triple consists of three positive integers "a", "b", and "c" such that "a" + "b" = "c" and "a" ? "b" (mod "c"), i.e., "a", "b", and "c" are mutually Pythagorean. The term "Pythagorean triple" is also used to refer to the set of all such triples with a given integer value of "c".

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Video Transcript

Hey, guys, we're here to talk about some Pythagorean theorem and Pythagorean examples how to use the Pathet. You're in there when I'm working on. People always tell my students, Man, there are some math agrarian triples. We call them Pythagorean triples that are very common on. But what I mean by that is numbers that make the Pythagorean theorem true. So a classic example would be like 345 because three squared plus four squared equals five squared. So this is an example of Pythagorean triple. Some other common examples would be 68 10 please note. It's just this doubled also 5 12 13 and another one is common. It's 7 24 25. You'll see this on standardized tests quite a bit, those four pathetic Korean triples. And if you have to memorize because you've seen them enough, then well, these problems become real quick. So, for example, if you look at thes problems that are asking you to find the missing side, well, I can already tell you that Let's say on, um, this problem, for example, that the hypotenuse see has to be 13. And by the way, remember, the Pythagorean theorem is gonna say a square apartment a squared plus B squared equals C squared. So seize assumed be the high powered news And as you can see, 5 12 13 was one of the examples I mentioned. That is very common. Well, there you go. I also mentioned 68 10 here. So we know in this case, this SBS six. So there's super super duper common. Now, if they're not or if you don't have them memorized, just do the Pythagorean here. So, for example, on let's say question too, this is a trap because you see a five, you see a 13, and I did mention that 5 12 13 was a Pythagorean triple Onley of 13 of the iPod news. And here 13 is one of the legs. So we actually have to do five squared plus 13 squared equals our question Mark squared. That's gonna be 25 plus 1. 69 25 plus 1. 69 equals 1 94. So 1 94 is equal to our question. Mark square. Well, we're gonna square root both sides, but the square root of 1 94 isn't pretty. Now you can break down 1 94 maybe into something smaller. It's got a perfect square in it. But you can leave this as the square tonight, 1 94 and they'll be fine where you can go ahead and get the approximate decimal and its 13.9 ish. So again, I'm either knowledgeable of some very common Pythagorean triples. Or if I'm not, I'm just gonna actually apply the Pythagorean theorem and solve for the missing side. Question three is a working backwards question because we have the high partners. So this would be like saying All right, question Mark squared plus 10 squared equals 20 squared. Well, this is 100. This is 400. So it's like saying question marks squared equals 300 and you can go ahead and to say that the question mark we're looking at is equal to the square root of 303 100 Does divide by four nicely, actually divides by more than that. Divides by 100 obviously, so we could call this 300 is the same thing is three times 100. So you're gonna get 10 Route three so you could call the missing leg 10. Route three. You call it radical 300. But either way, you're setting up the Pythagorean theorem, and then you're just solving for what you're missing. So the same thing goes for if I don't give me the triangle, but I give you two of the sides. If you want to draw a picture, you can keep in mind that you have the hypotenuse see and all three of these cases. We have the iPod news, and we're working backwards to find one of the legs. So let's work this first one. This would be like saying 15 squared plus B squared equals 17 squared. Well, 15 squared is 2 25 17. Squared is to 89. If you take 29 minus 2 25 you get 64 in the square root of 64 his eight. So here's a very nice, but that you're in triple. What I mean by Nice is it's essentially three integers that behave very nicely with each other. Okay, let's get a few more examples. How about some problems? So if it says let's say, for example, question love it, Miss Green tells you that the right triangles I partners are 13 in the leg of five. So this is not much of a word problem. Here is your hypotheses of 13. One of your legs is five. That's one of the ones I mentioned. Five, 12, 13. I automatically know that 12 is my other leg. Five squared plus 12 squared is 13 squared. In other words, 25 plus 1. 44 is 1 69. That one's pretty plain simple, cause it's basically talking about a right triangle. Looking question. 12. 2 joggers run eight miles north, so north is up five miles. Five miles west is right. Apartment left. So we have a triangle is basically doing this. So this is eight. This is five. The question is, what's the shortest distance from when they started to the when they stopped? Okay, because you've gone straight north and then West, you have a right triangle and we're looking for your iPod news. So, in other words, five squared plus eight squared equals C squared. There would be 25 plus 64 which is 93. Seeing it with 93 equals C squared or, in other words, Route 93 equals C. In the square root of 93 is roughly 9.6. So they'll be roughly 96 miles, um, away from where they started. So on any pathetic Yuri in question, you're essentially going to, um, set up a triangle. You're either gonna have both legs and find the iPod news, or you're gonna have the high pot news in one of the legs, and you're gonna work backwards to figure out the other leg. But that's essentially, um, the examples that you would see with the Pythagorean. Yeah, a few other problems might be I'm gonna add a page here. And you three sides. And I said, All right, you have three side lengths. One is, let's say, four, seven and nine. And I wanted to know, Does this make a right triangle? Do we get a right triangle out of those three sides and to test that? Let's see if the Pythagorean theorem holds notice. I'm picking the two smaller side because it's gonna be right. Triangle. The nine is gonna be the iPod news. I'm gonna put a question mark here. So is 16 plus 49 equal to 81. Well, 16 plus 49 equal 65. 65 dozen equal. Anyone? No we don't have the right track. So if it is the right triangle, the Pythagorean theorem will hold. If the Pythagorean theorem holds, you have a right triangle. So this would be an example of another example of how you would use a Pythagorean theorem to test whether or not you have Russia angle. Okay, I'll see you next time. Thanks.