Teorema de TALES | Prof. Gis/

Hi guys, welcome to another class on the Gis channel and today you will learn the Tales theorem, speaking of Tales, Tales of what? Thales of Miletus and do you know who Thales of Miletus was? Let 's tell you, he was a mathematician, astronomer, philosopher and merchant from ancient times, approximately 625 years before Christ, that's right and do you know what contributions he left for us in mathematics?

Thales was considered the father of descriptive geometry, a descriptive mark there, and he also contributed to the advancement of ratio and proportion calculations that are used to this day, that's right, look at how many things people, so today we are going to study the theorem that he left it for us, so if you want to learn more about this content, I invite you to watch this class, let's go? Do you already have any idea what Thales' theorem is? So Thales' theorem tells us that every time I have a bundle of parallel lines, bundle?

Beam means a set of several parallel lines, parallel lines, look here in our image, we have here three lines parallel to the line R the line S and the line P the line R parallel straight S parallel straight P, so see what I have there represented by three parallel lines, we can then say that it is a bundle of parallel lines, and what does it have there, so cut because they are cut by two straight lines, straight U and straight V, which are called transversal lines, okay? And what happens every time I have a bundle of parallel lines that are cut by two transversal lines, they form proportional segments, like so, Gis proportional segments, well, you can see here that where my transversal line crossed with my parallel line here formed a point and this point I called point A here in the same way I called point B here points C and on this side the same case point D point E point F, okay so you can see there that I have the formation of segments, okay, but before talking about these segments, can you see in this image in general a lesson that we have already studied about angles? That's right, the relationships between alternate and external angles, angles that are internal and external collaterals, corresponding angles, angles opposed by the vertex, if you watched this class, right about angles, you're remembering this image, because here we also have, in addition to looking, for the segment we can see angles, not that the angles are the focus of the class now, okay, but just to show you that there is the formation of an angle, for example here there is the formation of an angle and this angle that if formed here is equal to the measure of this angle that is formed here which is equal to the measure of this angle that is formed here, so just so you understand, I have here, for example, an angle here and an angle here that are angles opposite the vertex, I I have angles that are alternate internal angles, for example this angle here and this angle that forms here are alternate internal angles, okay, but just to show you that here I can also see that concept there of angles, and if you want to review this angles class OK, I left you a recommendation, ok?

So, what are the segments that we can obtain here, so at the intersection of these straight lines there is also something that we can imagine a lot here, let's use our imagination, hey, it's the streets, right ? A city, for example, doesn't have its own. they don't intersect, the three streets here are being crossed, I don't know, by two avenues here, let's think like this, this also happens, there are also exercises that talk about determining the distance between the block and the distance from the street, okay, so we'll see in other classes, we'll see also applied exercises, okay, so let's learn more about the theorem here.

So here I have the formation of a segment that from this point to this point I then have the segment ab and from this point to this point or segment cd, okay? in the same way that on this side here I have segment de and the segment ef remembers what a straight line segment is oh, I can't believe you don't remember. A straight line segment is that part of the straight line that has a beginning and an end, okay, what more explanations about this?

It also indicates that you have access to this class, right? Oh guys and one more thing and now take advantage of the fact that you are going back there, take advantage of Subscribe to Gis's channel and if you like the class, leave your thumbs up at the end, okay, at the end don't leave, collaborate with Gis. So guys, we're going to do it like this, remember that I said every time I have parallel lines, what's the name, bundle of parallel lines cut by transversals by transversals, right guys, I have two transversals here, in this case, I then have segments that are proportional and Do you remember at the beginning of the class that I said that he contributed to the advancement of the calculation of ratio and proportion, so now we work with ratio and proportion because as I have segments here that are proportional, right, let's work with ratio and proportion, so let's go enough talking and let's do the point, right?

more about ratio where you should go to the class in which I also explained in detail about ratio and proportion ok so going back the segment ab is to the segment dc as well as the segment de, the segment e is to the segment e ef is since that this is going to happen every time , right ? having exercises for example you will find exercises that the segment ab here it will have its respective measurement o bc o de o ef, they will have the measurements and of course there will be unknown terms for us to discover, right? So let's take advantage, let's go and exercise?

Guys, now look at the example I brought, it's applied to real life, ok, so here you must imagine that the mathematics, it's very good to imagine things, okay, we have a condominium here, the lots are going to start being sold, okay? the lots are here, here's a lot and here's another lot, all regulars, here's a lot, everyone, here's a lot, okay? and then they will start dividing this lot to sell, right, so I went there, I also wanted to buy a lot to build, right, my house so I can relax at the weekend, okay?

But then guess which lot I wanted to buy, folks, the lot that has a lake here, that's the one, so you can go fishing in the lake in the afternoon, right? oh, what happened, the owners of that condominium there, they already had the measurements, oh, he already had a measurement here of this part here, you know, of this lot, this distance here, he already had a measurement here, he already had a measurement here, but as they didn't have much technology, so they couldn't do the distance from here because there was a lake right in the middle of the way, what's it like to cross the lake by boat, so it wouldn't be possible and that's when they hired Gis to carry out these calculations for them, right? So I'm going to apply Tales' idea, you see how Tales was very important in helping people who don't have technology to carry out these calculations, so what did you learn from Tales' theorem?

That every time I have a bundle of parallel straight lines that are cut by two transversals, what happens is the ratio of the segments formed here in this transversal to the ratio of the segment formed by the other side of the transversal, so this means that this piece here where it goes to be composed of one side of my land, I'm going to call it x, it's logical, right, people, that x has to appear in the resolution of these things, so what 's the assembly like in this exercise? Here is the segment ab, so it measures 18, now we can work directly with the value but if you want to put it there then to remember the segment ab is to the segment bc as well as the one is to ó, here is the ratio of the segments and here is the ratio of the other two to the segment ef, if you If you want to do this here, you can, but you don't need to, go directly to your account, so segment ab measures 18, so it will be 18, which is 45, which is the segment bc, as well as x, Gis is giving me trouble, so I wanted to buy one without the lake, right? It wasn't easier, it's 63, so here we have a proportion and how do I calculate it then a proportion?

I multiply the terms here, 45 go times crossed multiplication crossed what will it look like here then guys? Let's start on the x side? 45 x = 18 x 63 and then we're going to do the math here 45 x = 18 x 63 which will give 1134 oh, I already knew it in my head, you see what I do in my head, and now how do I do it same?

that here I have an equation, right now this 45 that I was multiplying will be divided so we will have x = 1134 divided by 45 and then what is the value of x now x will be equal to let's think 25. 2 so that's the measure of the land there that I want to buy on one side of the land that I want to buy one is the lake it's good so 25. 2 remember that the units of measurement here are all given in meters, right Gis wrote the units of measurement, meters meters and meters so that means that my land there the distance from this part of my land measures 25.

2 so you saw that here I applied Thales' theorem everything related to what? reason and proportion, okay? and another way if you also wanted to do the math, many prefer to do it straight from 45 to 63.

I have to find out how much the number was, the multiplier, let's see, to find out how much this multiplier was, I divide 63 by 45, so this multiplier was 1. 4 I also did the math in advance, 45 is there to check 45 x 1. 4 of 63, ok?

In the same way, if you do 18 x 1. 4, it will give you 25. 2, so it's also another way for you to apply it there more quickly, I don't know your math skills to solve, so this type of situation, right, so it helped Gis to resolve it so now we can go fishing in the Gis lake, agreed guys?

One more example for you to apply better, let's go guys, now I don't want to know the size of the land that I already bought, right? So I want to build a shelf in my office, the studio where I record classes, look at my shelf, what it will look like, I had to make this shelf here, this slope here because if I made it a little straighter I would catch the water pipe that was running through behind the wall, imagine if the water pipe got punctured, right? So I had to build it in this structure, my shelf was modern, right, to put Gis's math books, okay, so look at the situation that I now brought the shelf, so here from gis is there here I already know that this segment here that goes from a to b measures 30 o from b to c 50 od to x and o from e to f 2x minus 12, a lot of people think it here, right there in the other example of the lake I didn't mention that but a lot of people think that this segment here is equal, this segment here is not equal, people, they are proportional, this segment here is proportional to this one, which means that from this to this one it was multiplied by some number, so they are not equal, they would be equal if these two transversal lines were perpendicular, they were straight here, forming an angle of 90 degrees, so if it is exactly the same as the measurement here and how it is inclined, right, look here at the inclination that is here on my shelf, so just to clarify this doubt that They already asked me here is it equal to and the value they didn't ask in the first example, I didn't say it so I'm specifying it here now, okay?

And then what will the assembly look like? So for us to find out the first value of but there it is for baby for c ok, so 30 is for 50 cm, right now I put the measured units as I have here 30 times 2x - 12 = 50 x directly proportional, simple rule of three directly proportional, if you want to attend the class in which I explained it, I will leave the indication for you so you can remember this content, it's continuing here so, I will distribute it 30 times two are 60 x okay and 30 times 12 will give minus 360 because three times 12 = 36 put the 0 here = 50 x now here in the equation I must separate the letter on one side and the number on the other 60x - 50x which is equal to 360 positive is here subtracting it will be 10x = 360 and finally I do the division so x = 360 / 10 and what is 360 / 10 is 36, so the 36 people here that I found is the value of who is the value of x so this x here is worth 36 centimeters ok and how do I find out the value of this other measurement here from my height of the shelf below here o two times x so it would be two times 36 two times 36 let's do it here so you can understand, right? I'm doing this that goes here twice 36 - 12 is twice 36 = 72 takes 12 = 60 so that means that this segment of mine here on my shelf will measure 60 centimeters, right?

so see that I figured out that the ratio was not between the segments, I solved it by cross-multiplying, but if you wanted to solve it, then as I said at the time, this segment here is proportional, this segment is proportional to this one, I could then find out who the proportionality constant was. what number what number I have here which is 50 which I multiplied by 50 which I arrived at 60, in this specific case of this exercise it will not be possible to do it like this because you see here on both sides here we have the unknown but there in that first example from the lake it was perfect, I took the number from here, multiplied it by the constant of proportionality, I found the number from here, okay? So what number is this, could you identify now that I know the value of x here, from 30 to 36 I multiplied it by how much I multiplied it by 1.

2 30 x 1. 2 it will give 36 so if I multiply 50, 50 by 1 ,2 will give my 60, so it shows you that this proportionality exists, okay, and in cases like this where there is an unknown here, it's a little more difficult to do, so what do I advise you to put together here, so the proportion sets up proportion multiplies crossed and find the value of x right guys so then you managed to figure out Gis to find the measurement of the shelf there where I'm going to put the math books for the studio right to record the classes for you guys okay so if you liked the class Once again, we're going to ask you to subscribe to my channel, go there and leave a thumbs up if you liked this class and don't forget to watch the next Tales theorem classes, okay, then I'll bring other cases a little more difficult combined people? So don't miss the next class folks, and I'll see you there!

bye and see you later. . .