Hi everyone, welcome to another class on the Gis channel and today the long-awaited class by many students Pythagorean Theorem. So let's talk about the theorem, let's know a little about Pythagoras, who was Pythagoras? Pythagoras was a Greek mathematician, right, talking about mathematics, and he traveled.
It is said that he traveled through Egypt, through Babylon, to India, seeking mathematical knowledge, right, and then when Pythagoras returned to Greece, he founded a Pythagorean school, and the students of this Pythagorean school, when they made discoveries, what happened they attributed this discovery to Pythagoras, to the master, so it was never to the students, they always attributed it to Pythagoras. So guys, this theorem that I'm going to present to you today is called the Pythagorean Theorem, right, because it was discovered there in that Pythagorean school, ok. We don't know, right, within the history of mathematics, we don't know if it was really Pythagoras who discovered it or if it was his Pythagoreans, okay, but it came from the school of Pythagoras, so that's why it is called the Pythagorean Theorem, which is an important metric relationship that applied to right triangles, let's see what the theorem is?
Let's go well then as I said the Pythagorean Theorem applies in a right triangle what is a right triangle you know a right triangle is a triangle that has a right angle a 90 degree angle you might think that a right triangle is a triangle that is half of a rectangle look at the name, right triangle, but it can also be half of a square because a square is also a rectangle Ok guys? So what do I have in this right triangle, focus here on the triangle, in a right triangle the longest side of this triangle is the opposite side, opposite is the one facing the front, the side that is opposite the right angle goes being called the hypotenuse, okay, and the other two sides that remain are called legs Ok, so far, but what does this have to do with the Pythagorean Theorem? Now I'm going to show you what they discovered, look, I'm starting from the principle that we have a right-angled triangle, see that here a square was built on this side, here another square was built on this side, right, because side, how can you see cathetus and cathetus, cathetus and cathetus, and here another square was built on the hypotenuse.
So see that we have the construction of three squares, right, and do you remember how to calculate the area of a square, the area of a square is the multiplication of the two dimensions, right? of the class so you can come back and watch it there and clear up all your doubts, right, and come back here and crush the Pythagorean Theorem. So, see what I did here, I built a square here that has how many squares here, it has one two three four five six seven eight nine squares so we say that the area of this yellow square is 9, okay, let's say nine units We don't know the unit of measurement in meter centimeters, let's talk about units, okay, so the area of this square that was built on this side is 9 Ok, the area of this square that was built on this other side is, have you already counted 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16, and now the area of this red square that was built on the hypotenuse is how much has already been seen, which is 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25.
So look at the three areas of these squares, 9, 16 and this one 25. Can you think of any relationship between these three areas? of this square that was built on this side with the area of this other square that was built on this other side, the area of these two figures together results in the area of the square that was built on the hypotenuse.
So this is the explanation of the Pythagorean Theorem, the square of one side added to the square of the other side is equal to the square of the hypotenuse, which is the square built on the hypotenuse and how am I going to represent this every time I solve exercises I will have what to do with these little squares, no people, I made these squares here to represent to you that the combination of these two things gives the area of this other square and also for you to understand that Pythagoras' term he says is the squared leg because the squared leg because it comes from the area, the area of the square that was built on this leg, that's why we say side to the square which is related to the area that the area of the square look here look at the area of the square let's do it here look very cute here look so if here is L and here is L the area of this square will be represented by what, by L squared so we say the square of L, right, an example here but when I'm talking about the side here, right, this side here is the square of this side because it's related to the area, right. So now let's take this information from here and let's bring it here to this triangle, so this side here has how many units of measurement, I'm not talking about its area. How many units of measurement Oh, there's one two three units of measurement here, it's not three Ok, this other side here which is the part of this blue square has one two three four units of measurement, okay.
And here the square that is on the hypotenuse is how long is there in the unit there o one two three four five so it has 5 units of measurement. So see that we can form a triangle of three four and five that we said that forms the Pythagorean terne mark well Pythagorean terne okay because we have the three terms here oh so consecutive to 3 4 and 5 Okay. Now let's do a test and see if the Pythagorean theorem applies, take this side here and then we work with the measure of the side, take this side which is 3 squared , 3 squared by 9 which is the same thing as the area of this square here, because if here it measures three, here it measures three, the same thing I did here, three by three gives 9, right, so three squared, 9 takes four and squares it, four squared, which would be 4 x 4, which would form 16 which is the area of this square, add this 2, 9 to 16, it will give 25, it will not, it is not equal to the square of this side, side.
. . of this hypotenuse here, 5 squared 25, you see how cool it is, so this is the theorem of Pythagoras, in short, how are we going to do, in short, people like this in books depends on the author who wrote the books, he calls the side of one letter another side of another, the hypotenuse of another, you will find several ways, the most common way we meet out there is that this, let me write here, here, I'm going to write here that it already has the hypotenuse, right, this hypotenuse they represent by a, by the letter up to a, then by the measure a, this side we will represent by what has the measure b and this another side will have measurement c, ok?
So then the Pythagorean theorem generalizing the formula as it will be: squared, but we can change it here Gis, is it ok to put ob here and oc here? you want me to put it in order, so let's put it in order, I'll put it for you here, since most books say it like this ob, eoc here oh, that's it, it's the same, if you change it you won't make a mistake, the exercise will be fine, OK, because addition addition has the commutative property, you can add, for example, 5 plus 9, it's the same thing as 9 plus 5, right? So if I take this side and square it, you understand why I square it, right?
Because of the formation of this square here that I work on the area of this square formed, it is therefore elevated to the square because the area of a square is one side times the other, look how cool, right, you relate everything, you see. So then I'm going to add this here with the other side squared and this will result, for me, even though we can visualize the value of the hypotenuse squared, okay, that's the Pythagorean Theorem, which he says like this, oh sum of the squares of the legs the sum of the squares of the legs is equal to the square of the hypotenuse then the Pythagorean theorem makes perfect sense here looking at the drawings the sum of the squares the two squares here of the legs is equal to the square of the hypotenuse wanting to add the hypotenuse again squared by the hypotenuse. So the Pythagorean theorem is perfect.
So this would be the formula he brought to us in his theorem, okay, and like I said, you can find it out there. in some places it is written cat squared + cat squared which is cateto squared plus cateto squared, so students think it is a cat squared, it is because Cat in English here is cat, right, then they think it is cat and then it becomes It's confusing, it's like the hypotenuse squared, some places like this the students write like this to memorize, there's the hypotenuse, they say it's the hippopotamus squared and they invent a lot of things, a lot of relationships so that they can memorize the Pythagorean Theorem, But what matters is that you can solve the exercises , right ? and when you see a figure drawn with two collared peccaries, if we put it on Google and search for it, it's a kind of pig, a wild pig, there's an image that shows a collared peccary elevated to the square that looks like a mouse.
people, the students play, they make fun that it's a mouse, it's not a rat, that's a peccary, a kind of pig taking it to another language, the Portuguese language. Okay, so don't go, you won't play, you won't make fun if the teacher shows this No image there for you, because I showed it to my students, you don't believe what they said, that the one there that was the goat I drew, instead of writing it like that for them, I went there and drew the goat, right, then they said that It was a mouse, then it became a mess, everyone laughed, but finally they learned the Pythagorean Theorem and they will never forget it, they wanted to talk about how the goat was a capybara, but they learned, they solved the exercises and everyone did a great job, just as you will do it too when you decide to solve it. Pythagorean Theorem exercises Ok guys?
So, I hope you understood the wonderful explanation. Look how wonderful this is here, right, if you liked it, leave a thumbs up for Gis, leave the squares formed here for Gis, okay, and subscribe to the channel, I didn't ask you to subscribe to the channel, today take the opportunity to subscribe to receive notifications of upcoming classes, do you know why? Because after this demonstration that this class is an explanation of how we arrived at the Theorem, there will be a next class just about solving exercises applying the Pythagorean theorem, so I'll see you in the next class, bye.
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