Interesting facts about the Pythagorean theorem: we will learn new things about the well-known theorem (15 photos). Pythagorean pants Pythagorean pants Theorem proof

Pythagorean pants - equal on all sides.
To prove this, you need to film and show.

This rhyme has been known to everyone since high school, since the time when we studied the famous Pythagorean theorem in geometry class: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

To prove his theorem, Pythagoras drew a figure of squares on the sides of a triangle in the sand. The sum of the squares of the legs in a right-angled triangle is equal to the square of the hypotenuse A square plus B square is equal to C square. It was 500 BC. Today the Pythagorean theorem is being taught in high school. In the Guinness Book of Records, the Pythagorean Theorem is the theorem with the maximum number of proofs. Indeed, in 1940 a book was published containing three hundred and seventy proofs of the Pythagorean theorem. One of them was proposed by US President James Abram Garfield. Only one proof of the theorem is still unknown to any of us: the proof of Pythagoras himself. For a long time, it was believed that Euclid's proof was the Pythagorean proof, but now mathematicians think that this proof belongs to Euclid himself.

The classical proof of Euclid is aimed at establishing the equality of areas between rectangles formed by cutting a square above the hypotenuse with a height from a right angle with squares above the legs.

The construction used for the proof is as follows: for a right-angled triangle ABC with a right angle C, squares above the legs ACED and BCFG, and a square above the hypotenuse ABIK, build the height CH and the ray s extending it, splitting the square above the hypotenuse into two rectangles AHJK and BHJI. The proof is aimed at establishing the equality of the areas of the rectangle AHJK with the square above the leg AC; the equality of the areas of the second rectangle that makes up a square above the hypotenuse and the rectangle above the other leg is established in the same way.

The equality of the areas of the rectangle AHJK and ACED is established through the congruence of triangles ACK and ABD, the area of each of which is equal to half the area of rectangles AHJK and ACED, respectively, due to the following property: the area of a triangle is equal to half the area of a rectangle if the figures have a common side, and the height of the triangle is k the common side is the other side of the rectangle. The congruence of triangles follows from the equality of the two sides (sides of the squares) and the angle between them (composed of the right angle and the angle at A.

Thus, the proof establishes that the area of the square above the hypotenuse, made up of rectangles AHJK and BHJI, is equal to the sum of the areas of the squares above the legs.

German mathematician Karl Gauss proposed to cut giant Pythagorean trousers from trees in the Siberian taiga. Looking at these pants from space, aliens must make sure that intelligent beings live on our planet.

It's funny that Pythagoras himself never wore pants - in those days, the Greeks simply did not know about such a wardrobe item.

Sources:

sandbox.fizmat.vspu.ru
ru.wikipedia.org
kuchmastar.fandom.com

site, with full or partial copying of the material, a link to the source is required.

The Roman architect Vitruvius singled out the Pythagorean theorem "from the numerous discoveries that provided services to the development of human life," and called for it to be treated with the greatest respect. It was back in the 1st century BC. e. At the turn of the XVI-XVII centuries, the famous German astronomer Johannes Kepler called it one of the treasures of geometry, comparable to the measure of gold. It is unlikely that in all mathematics there will be a more weighty and significant statement, because in terms of the number of scientific and practical applications, the Pythagorean theorem has no equal.

Pythagoras' theorem for the case of an isosceles right triangle.

Science and Life // Illustrations

Illustration to the Pythagorean theorem from the "Treatise on the Measuring Pole" (China, III century BC) and the proof reconstructed on its basis.

Science and Life // Illustrations

S. Perkins. Pythagoras.

A blueprint for a possible proof of Pythagoras.

"Mosaic of Pythagoras" and an-Nayrizi tiling of three squares in the proof of the Pythagorean theorem.

P. de Hooch. A hostess and a maid in the courtyard. Around 1660.

J. Ohtervelt. Wandering musicians at the door of a rich house. 1665 year.

Pythagorean pants

Pythagoras' theorem is perhaps the most recognizable and undoubtedly the most famous in the history of mathematics. In geometry, it is used literally at every step. Despite the simplicity of its formulation, this theorem is by no means obvious: looking at a right-angled triangle with sides a< b < c, усмотреть соотношение a 2 + b 2 = c 2 невозможно. Однажды известный американский логик и популяризатор науки Рэймонд Смаллиан, желая подвести учеников к открытию теоремы Пифагора, начертил на доске прямоугольный треугольник и по квадрату на каждой его стороне и сказал: «Представьте, что эти квадраты сделаны из кованого золота и вам предлагают взять себе либо один большой квадрат, либо два маленьких. Что вы выберете?» Мнения разделились пополам, возникла оживлённая дискуссия. Каково же было удивление учеников, когда учитель объяснил им, что никакой разницы нет! Но стоит только потребовать, чтобы катеты были равны, - и утверждение теоремы станет явным (рис. 1). И кто после этого усомнится, что «пифагоровы штаны» во все стороны равны? А вот те же самые «штаны», только в «сложенном» виде (рис. 2). Такой чертёж использовал герой одного из диалогов Платона под названием «Менон», знаменитый философ Сократ, разбирая с мальчиком-рабом задачу на построение квадрата, площадь которого в два раза больше площади данного квадрата. Его рассуждения, по сути, сводились к доказательству теоремы Пифагора, пусть и для конкретного треугольника.

Figures shown in fig. 1 and 2, resemble the simplest ornament of squares and their equal parts - a geometric pattern known from time immemorial. They can completely cover the plane. A mathematician would call such a covering of a plane by polygons parquet, or tiling. What does Pythagoras have to do with it? It turns out that he was the first to solve the problem of regular parquets, which began the study of tilings of various surfaces. So, Pythagoras showed that the plane around a point can be covered without gaps by equal regular polygons of only three types: six triangles, four squares and three hexagons.

4000 years later

The history of the Pythagorean theorem goes back to ancient times. It is mentioned in the Babylonian cuneiform texts of the times of King Hammurabi (18th century BC), that is, 1200 years before the birth of Pythagoras. The theorem was used as a ready-made rule in many problems, the simplest of which is finding the diagonal of a square along its side. It is possible that the Babylonians obtained the ratio a 2 + b 2 = c 2 for an arbitrary right-angled triangle by simply "generalizing" the equality a 2 + a 2 = c 2. But it is forgivable for them - for the practical geometry of the ancients, which was reduced to measurements and calculations, no rigorous justification was required.

Now, almost 4,000 years later, we are dealing with a theorem that holds the record for the number of possible proofs. By the way, collecting them is a long tradition. The peak of interest in the Pythagorean theorem fell on the second half of the 19th - early 20th centuries. And if the first collections contained no more than two or three dozen proofs, then by the end of the 19th century their number was close to 100, and after another half a century it exceeded 360, and these are only those that were collected from various sources. Who has not undertaken the solution of this ageless problem - from eminent scientists and popularizers of science to congressmen and schoolchildren. And what is remarkable, in the originality and simplicity of the solution, some amateurs were not inferior to professionals!

The oldest surviving proof of the Pythagorean theorem is about 2300 years old. One of them - the strict axiomatic - belongs to the ancient Greek mathematician Euclid, who lived in the 4th-3rd centuries BC. e. In Book I of the Elements, the Pythagorean theorem is listed as Proposition 47. The most graphic and beautiful proofs are based on the reshaping of the "Pythagorean trousers". They look like a tricky square-cutting puzzle. But make the pieces move correctly - and they will reveal to you the secret of the famous theorem.

Here is an elegant proof obtained on the basis of a drawing from one ancient Chinese treatise (Fig. 3), and its connection with the problem of doubling the area of a square becomes immediately clear.

It was this proof that the seven-year-old Guido, the precocious hero of the short story "Little Archimedes" by the English writer Aldous Huxley, tried to explain to his younger friend. It is curious that the narrator, who observed this picture, noted the simplicity and persuasiveness of the proof, so he attributed it ... to Pythagoras himself. But the protagonist of Evgeny Veltistov's fantastic story "An Electronic - a Boy from a Suitcase" knew 25 proofs of the Pythagorean theorem, including those given by Euclid; true, he mistakenly called it the simplest, although in fact in the modern edition of the "Elements" it occupies one and a half pages!

First mathematician

Pythagoras of Samos (570-495 BC), whose name has long been inextricably linked with a remarkable theorem, in a sense can be called the first mathematician. It is with him that mathematics begins as an exact science, where any new knowledge is not the result of visual representations and rules derived from experience, but the result of logical reasoning and conclusions. This is the only way to establish once and for all the truth of any mathematical proposition. Before Pythagoras, the deductive method was used only by the ancient Greek philosopher and scientist Thales of Miletus, who lived at the turn of the 7th-6th centuries BC. e. He expressed the very idea of the proof, but applied it not systematically, selectively, as a rule, to obvious geometric statements such as “the diameter divides the circle in half”. Pythagoras went much further. It is believed that he introduced the first definitions, axioms and methods of proof, and also created the first course in geometry, known to the ancient Greeks under the name "The Tradition of Pythagoras". He also stood at the origins of the theory of numbers and stereometry.

Another important merit of Pythagoras is the founding of the glorious school of mathematicians, which for more than a century determined the development of this science in Ancient Greece. The term "mathematics" (from the Greek word μαθημa - doctrine, science) is also associated with his name, uniting four related disciplines of the knowledge system created by Pythagoras and his adherents, the Pythagoreans: geometry, arithmetic, astronomy and harmonics.

It is impossible to separate the achievements of Pythagoras from the achievements of his students: following custom, they attributed their own ideas and discoveries to their Teacher. The early Pythagoreans did not leave any compositions, they transmitted all information to each other orally. So 2500 years later, historians have no choice but to reconstruct the lost knowledge based on the transcriptions of other, later authors. Let's pay tribute to the Greeks: although they surrounded the name of Pythagoras with many legends, they did not ascribe anything to him that he could not discover or develop into a theory. And the theorem that bears his name is no exception.

Such a simple proof

It is not known whether Pythagoras himself discovered the relationship between the lengths of the sides in a right triangle or borrowed this knowledge. Ancient authors claimed that he himself, and loved to retell the legend of how, in honor of his discovery, Pythagoras sacrificed a bull. Modern historians tend to believe that he learned about the theorem by becoming acquainted with the mathematics of the Babylonians. We also do not know in what form Pythagoras formulated the theorem: arithmetically, as is customary today, - the square of the hypotenuse is equal to the sum of the squares of the legs, or geometrically, in the spirit of the ancients, - a square built on the hypotenuse of a right-angled triangle is equal to the sum of squares built on his legs.

It is believed that it was Pythagoras who gave the first proof of the theorem that bears his name. It, of course, has not survived. According to one version, Pythagoras could use the doctrine of proportions developed in his school. On it was based, in particular, the theory of similarity, on which the reasoning is based. Draw in a right-angled triangle with legs a and b the height to the hypotenuse c. We get three similar triangles, including the original one. Their respective sides are proportional, a: c = m: a and b: c = n: b, whence a 2 = c m and b 2 = c n. Then a 2 + b 2 = = c · (m + n) = c 2 (Fig. 4).

This is just a reconstruction proposed by one of the historians of science, but the proof, you see, is quite simple: it takes only a few lines, you do not need to complete, redraw, calculate anything ... It is not surprising that it was rediscovered more than once. It is contained, for example, in the "Practice of Geometry" by Leonardo of Pisa (1220), and it is still quoted in textbooks.

This proof did not contradict the ideas of the Pythagoreans about commensurability: initially they believed that the ratio of the lengths of any two segments, and hence the areas of rectilinear figures, can be expressed using natural numbers. They did not consider any other numbers, did not even allow fractions, replacing them with ratios 1: 2, 2: 3, etc. However, ironically, it was the Pythagorean theorem that led the Pythagoreans to the discovery of the incommensurability of the diagonal of a square and its side. All attempts to numerically represent the length of this diagonal - for the unit square it is equal to √2 - have led nowhere. It turned out to be easier to prove that the problem is unsolvable. For such a case, mathematicians have a proven method - proof by contradiction. By the way, he is also attributed to Pythagoras.

The existence of a relationship that is not expressed in natural numbers put an end to many ideas of the Pythagoreans. It became clear that the numbers they knew were not enough to solve even simple problems, let alone all geometry! This discovery was a turning point in the development of Greek mathematics, its central problem. First, it led to the development of the doctrine of incommensurable quantities - irrationalities, and then - to the expansion of the concept of number. In other words, the centuries-old history of the study of the set of real numbers began with him.

Mosaic of Pythagoras

If you cover the plane with squares of two different sizes, surrounding each small square with four large ones, you get the "Pythagoras mosaic" parquet floor. Such a pattern has long adorned stone floors, recalling the ancient proofs of the Pythagorean theorem (hence its name). By applying a square grid to the parquet in different ways, you can get the partitions of squares built on the sides of a right-angled triangle, which were proposed by different mathematicians. For example, if you arrange the grid so that all its nodes coincide with the upper right vertices of the small squares, fragments of the drawing will appear for the proof of the medieval Persian mathematician al-Nayrizi, which he placed in the comments to Euclid's Beginnings. It is easy to see that the sum of the areas of the big and small squares, the original elements of the parquet, is equal to the area of one square of the grid superimposed on it. And this means that the specified partition is really suitable for laying parquet: by connecting the resulting polygons into squares, as shown in the figure, you can fill the entire plane with them without gaps and overlaps.

Some discussions amuse me immensely ...

Hi what are you doing?
-Yes, I solve problems from the magazine.
-Wow! Didn't expect from you.
-What did you not expect?
-That you stoop to problems. It seems clever, after all, but you believe in all sorts of nonsense.
-Sorry I dont understand. What do you call nonsense?
-Yes, all that math of yours. After all, it is obvious that the garbage is complete.
-How can you say that? Mathematics is the queen of sciences ...
- Just come on without this pathos, right? Mathematics is not a science at all, but one continuous heap of stupid laws and rules.
-What?!
-Oh, well, don't make such big eyes, you yourself know that I'm right. No, I do not argue, the multiplication table is a great thing, it played a significant role in the formation of the culture and history of mankind. But now all this is already irrelevant! And then, why complicate things? There are no integrals or logarithms in nature, these are all inventions of mathematicians.
-Wait a minute. Mathematicians did not invent anything, they discovered new laws of interaction of numbers, using proven tools ...
-Yes of course! And do you believe that? Can't you yourself see what nonsense they are constantly talking about? Can you give an example?
-Yes, be kind.
-Yes please! Pythagorean theorem.
-What's wrong with her?
-Yes, it's not like that! "Pythagorean pants are equal on all sides," you see. Did you know that the Greeks did not wear pants in the days of Pythagoras? How could Pythagoras even reason about what he had no idea about?
-Wait a minute. What do the pants have to do with it?
-Well, they seem to be Pythagorovs? Or not? Do you admit that Pythagoras had no pants?
- Well, actually, of course, it wasn't ...
-Yeah, it means that the very title of the theorem has an obvious discrepancy! After that, how can you take what it says seriously?
- Wait a minute. Pythagoras said nothing about pants ...
-You admit it, right?
-Yes ... So, can I continue? Pythagoras did not say anything about pants, and there is no need to ascribe other people's nonsense to him ...
-Yeah, you yourself agree that this is all nonsense!
- I didn't say that!
- I just said. You're contradicting yourself.
-So. Stop. What does the Pythagorean theorem say?
-That all pants are equal.
- Damn, have you ever read this theorem ?!
-I know.
-Where?
-I read.
-What did you read?!
-Lobachevsky.
*pause*
-Sorry, what does Lobachevsky have to do with Pythagoras?
-Well, Lobachevsky is also a mathematician, and he seems to be an even cooler authority than Pythagoras, say no?
*sigh*
-Well, what did Lobachevsky say about the Pythagorean theorem?
-That the pants are equal. But this is nonsense! How can you wear pants like that? And besides, Pythagoras did not wear pants at all!
-Lobachevsky said so ?!
* second pause, with confidence *
-Yes!
-Show me where it's written.
-No, well, it is not written so directly there ...
-What name has this book?
-Yes, this is not a book, this is an article in a newspaper. About the fact that Lobachevsky was in fact an agent of German intelligence ... well, this is beside the point. Anyway, he probably said so. He is also a mathematician, so he and Pythagoras are at the same time.
-Pythagoras did not say anything about pants.
-Well, yes! About that and speech. It's all bullshit.
-Let's come in order. How do you personally know what the Pythagorean theorem says?
-Oh, come on! Everyone knows that. Ask anyone, they will immediately answer you.
-Pythagorean pants are not pants ...
-And, of course! This is an allegory! Do you know how many times I've heard this before?
-The Pythagorean theorem says that the sum of the squares of the legs is equal to the square of the hypotenuse. AND EVERYTHING!
-Where are the pants?
-Yes, Pythagoras didn't have any pants !!!
-Well, you see, I'm talking about that. All your math is bullshit.
-And that's not bullshit! Take a look yourself. Here is a triangle. Here is the hypotenuse. Here are the legs ...
-And why all of a sudden this is the legs, and this is the hypotenuse? Maybe the other way around?
-Not. The legs are two sides that form a right angle.
-Well, here's another right angle for you.
- He's not straight.
-What is he, crooked?
-No, it's spicy.
-So this one is also sharp.
- It's not sharp, it's straight.
- You know, don't fool me! You just name things as you like, just to adjust the result to the desired one.
-The two short sides of a right-angled triangle are legs. The long side is the hypotenuse.
-And who is shorter - that leg? And the hypotenuse, then, is no longer rolling? You yourself, listen to yourself from the outside, what nonsense you are talking about. It's the 21st century, the flourishing of democracy, and you have some kind of Middle Ages. His sides, you see, are unequal ...
-A right-angled triangle with equal sides does not exist ...
-Are you sure? Let me draw for you. Look. Rectangular? Rectangular. And all sides are equal!
-You drew a square.
-So what?
-The square is not a triangle.
-And, of course! As soon as it does not suit us, immediately "not a triangle"! Do not fool me. Count it yourself: one corner, two corners, three corners.
-Four.
-So what?
-It's a square.
-And a square, not a triangle? He's worse, isn't he? Just because I drew it? Are there three corners? There is, and even here is one spare. Well, there is nothing here, you know ...
- Okay, let's leave this topic.
-Yeah, giving up already? There is nothing to argue with? Do you admit that math is bullshit?
- No, I don't.
-Well, again, great! I just proved everything to you in detail! If all your geometry is based on the teachings of Pythagoras, and, I apologize, it is complete nonsense ... then what can you talk about further?
-The teaching of Pythagoras is not nonsense ...
-Well, how! And then I have not heard about the school of the Pythagoreans! They, if you want to know, indulged in orgies!
-What does it have to do with it ...
-And Pythagoras was generally a fagot! He himself said that Plato is his friend.
-Pythagoras?!
-You didn `t know? Yes, they were all fagots. And three on the head. One was asleep in a barrel, the other was running around the city naked ...
-Diogenes slept in a barrel, but he was a philosopher, not a mathematician ...
-And, of course! If someone climbed into the barrel, then they are no longer a mathematician! Why do we need extra shame? We know, we know, passed. But you explain to me why all sorts of fagots who lived three thousand years ago and ran without pants should be an authority for me? Why on earth should I accept their point of view?
-Okay, leave ...
-No, listen! In the end, I also listened to you. These are your calculations, calculations ... You all know how to count! And ask you something in essence, right there at once: "this is a quotient, this is a variable, and these are two unknowns." And you tell me in o-o-o-general, without particulars! And without any unknown, unknown, existential ... It makes me sick, you know?
-Understand.
- Well, explain to me why twice two is always four? Who invented this? And why am I obliged to take it for granted and have no right to doubt?
-Yes, doubt as much as you want ...
-No, you explain to me! Only without these things of yours, but it is normal, humanly, so that it is clear.
-Two times two equals four, because two times two equals four.
-Oil oil. What new did you tell me?
-Two times two is two times two. Take two and two and add them ...
-So add or multiply?
-This is the same...
-Both on! So if I add and multiply seven and eight, it’s the same thing too?
-Not.
-And why?
-Because seven plus eight does not equal ...
-And if I multiply nine by two, it turns out four?
-Not.
-And why? I multiplied two - it worked, but with a nine suddenly a bummer?
-Yes. Twice nine - eighteen.
-And twice seven?
-Fourteen.
-And twice five?
-Ten.
- That is, four turns out only in one particular case?
-Exactly.
-Now think for yourself. You say that there are some strict laws and rules for multiplication. What laws can we talk about here at all, if in each specific case a different result is obtained ?!
-It's not entirely true. Sometimes the result may be the same. For example, twice six equals twelve. And four times three - too ...
-Even worse! Two, six, three, four - nothing at all! You can see for yourself that the result does not depend in any way on the initial data. The same decision is made in two radically different situations! And this despite the fact that the same two, which we take constantly and do not change for anything, always gives a different answer with all the numbers. Where, one wonders, is the logic?
-But this is, once again, logical!
-For you - maybe. You mathematicians always believe in all sorts of outrageous crap. And these calculations of yours do not convince me. And do you know why?
-Why?
-Because I I know why is your math really needed. What does it all boil down to? "Katya has one apple in her pocket, and Misha has five. How many apples must Misha give to Katya so that they have equal apples?" And you know what I'll tell you? Misha owes nothing to anyone give away! Katya has one apple - that's enough. Is it not enough for her? Let him go to work, and honestly earn herself at least for apples, at least for pears, at least for pineapples in champagne. And if someone wants not to work, but only to solve problems - let him sit with his one apple and not show off!

Pants - get a valid ridestep promo code on Akademik or buy pants at a discount at a sale at ridestep

Zharg. shk. Shuttle. Pythagoras' theorem, which establishes the relationship between the areas of squares built on the hypotenuse and legs of a right triangle. BTS, 835 ... A large dictionary of Russian sayings

Pythagorean pants- The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of the rectangle and diverging in different directions resemble the cut of pants. I loved geometry ... and even got it from ... ... Phraseological dictionary of the Russian literary language

pythagorean pants- The humorous name of the Pythagorean theorem, which establishes the relationship between the areas of the squares built on the hypotenuse and the legs of a right triangle, which outwardly looks like the cut of pants in the figures ... Dictionary of many expressions

Inosk .: about a gifted person Cf. This is an undoubted sage. In ancient times, he probably would have invented the Pythagorean pants ... Saltykov. Colorful letters. Pythagorean pants (geom.): In a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Big Explanatory Phraseological Dictionary

Pythagorean pants are equal on all sides- The number of buttons is known. Why dick cramped? (roughly) about pants and the male genitals. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants ... Live speech. Dictionary of colloquial expressions

Piѳagorov trousers (invent) a sock. about a man gifted. Wed This is undoubtedly a wise man. In antiquity, he probably would have invented Piѳagor's pants ... Saltykov. Variegated letters. Piѳagorov trousers (geom.): In the rectangle square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)

Pythagorean pants are equal in all directions- A humorous proof of the Pythagorean theorem; also joking about buddy's baggy trousers ... Dictionary of folk phraseology

Ex., Rude ...

PYTHAGOR'S PANTS ON ALL SIDES ARE EQUAL (THE NUMBER OF BUTTONS IS KNOWN. WHY IS THE FUCK TIGHT? / TO PROVE IT, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Explanatory dictionary of modern colloquial phraseological units and sayings

Noun., Pl., Uptr. cf. often Morphology: pl. what? pants, (no) what? pants, why? pants, (see) what? pants what? pants about what? about pants 1. A pants is a piece of clothing that has two short or long legs and covers the lower part ... ... Dmitriev's Explanatory Dictionary

Books

Pythagorean pants,. In this book you will find fantasy and adventure, miracles and fiction. Funny and sad, ordinary and mysterious ... What else is needed for an entertaining reading? The main thing is to have ...
Miracles on Wheels, Markusha Anatoly. Millions of wheels revolve all over the earth - they roll cars, measure the time in clocks, knock under trains, perform countless jobs in machine tools and various mechanisms. They…

One can be one hundred percent sure that when asked what the square of the hypotenuse is, any adult will boldly answer: "The sum of the squares of the legs." This theorem is firmly rooted in the minds of every educated person, but it is enough to ask someone to prove it, and then difficulties can arise. Therefore, let's remember and consider different ways of proving the Pythagorean theorem.

Brief biography

The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who gave birth to it is not so popular. This is fixable. Therefore, before studying the different ways of proving the Pythagorean theorem, you need to briefly get acquainted with his personality.

Pythagoras is a philosopher, mathematician, thinker originally from Today it is very difficult to distinguish his biography from the legends that have developed in memory of this great man. But as follows from the writings of his followers, Pythagoras of Samos was born on the island of Samos. His father was an ordinary stone cutter, but his mother came from a noble family.

According to legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, a born boy should have brought many benefits and goodness to mankind. Which he actually did.

The birth of the theorem

In his youth, Pythagoras moved to Egypt to meet there with famous Egyptian sages. After meeting with them, he was admitted to study, where he learned all the great achievements of Egyptian philosophy, mathematics and medicine.

Probably, it was in Egypt that Pythagoras was inspired by the majesty and beauty of the pyramids and created his great theory. This may shock readers, but modern historians believe that Pythagoras did not prove his theory. He only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.

Be that as it may, today not one method of proving this theorem is known, but several at once. Today, it remains only to guess how exactly the ancient Greeks made their calculations, so here we will consider different ways of proving the Pythagorean theorem.

Pythagorean theorem

Before starting any calculations, you need to figure out which theory is to be proven. The Pythagorean theorem reads as follows: "In a triangle, in which one of the angles is 90 °, the sum of the squares of the legs is equal to the square of the hypotenuse."

In total, there are 15 different ways to prove the Pythagorean theorem. This is a fairly large figure, so let's pay attention to the most popular of them.

Method one

First, let's designate what is given to us. These data will also apply to other methods of proving the Pythagorean theorem, so you should immediately remember all the available designations.

Suppose a right-angled triangle is given, with legs a, b and a hypotenuse equal to c. The first method of proof is based on the fact that you need to draw a square from a right-angled triangle.

To do this, you need to draw a segment equal to the leg b to the leg of length a, and vice versa. This should create two equal sides of the square. It remains only to draw two parallel lines, and the square is ready.

Inside the resulting figure, you need to draw another square with a side equal to the hypotenuse of the original triangle. To do this, from the vertices ac and sv, you need to draw two parallel segments equal to c. Thus, we get three sides of the square, one of which is the hypotenuse of the original right-angled triangle. It remains only to finish the fourth segment.

Based on the resulting figure, we can conclude that the area of the outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, it contains four right-angled triangles. The area of each is 0.5 av.

Therefore, the area is equal to: 4 * 0.5av + s 2 = 2av + s 2

Hence (a + b) 2 = 2ab + c 2

And therefore c 2 = a 2 + b 2

The theorem is proved.

Method two: similar triangles

This formula for the proof of the Pythagorean theorem was derived on the basis of a statement from the geometry section about similar triangles. It says that the leg of a right-angled triangle is the proportional average for its hypotenuse and the segment of the hypotenuse emanating from the apex of the 90 ° angle.

The initial data remain the same, so let's start right away with the proof. Let's draw a segment of SD perpendicular to the side AB. Based on the above statement, the legs of the triangles are:

AC = √AB * HELL, SV = √AB * DV.

To answer the question of how to prove the Pythagorean theorem, the proof must be completed by squaring both inequalities.

AC 2 = AB * HELL and SV 2 = AB * DV

Now you need to add up the resulting inequalities.

AC 2 + SV 2 = AB * (HELL * DV), where HELL + DV = AB

It turns out that:

AC 2 + SV 2 = AB * AB

And therefore:

AC 2 + CB 2 = AB 2

The proof of the Pythagorean theorem and various ways to solve it require a versatile approach to this problem. However, this option is one of the simplest.

Another calculation technique

The description of different ways of proving the Pythagorean theorem may not say anything, until you start to practice on your own. Many techniques provide not only mathematical calculations, but also the construction of new figures from the original triangle.

In this case, it is necessary to complete another right-angled triangle of the VSD from the leg of the BC. Thus, now there are two triangles with a common leg BC.

Knowing that the areas of such figures have a ratio as squares of their similar linear dimensions, then:

S awd * s 2 - S awd * in 2 = S awd * a 2 - S awd * a 2

S abc * (s 2 -v 2) = a 2 * (S awd -S vd)

s 2 -w 2 = a 2

c 2 = a 2 + b 2

Since this option is hardly suitable from different methods of proving the Pythagorean theorem for grade 8, you can use the following technique.

The easiest way to prove the Pythagorean theorem. Reviews

Historians believe that this method was first used to prove a theorem back in ancient Greece. It is the simplest one, as it does not require absolutely any calculations. If you draw the figure correctly, then the proof of the statement that a 2 + in 2 = c 2 will be clearly visible.

The conditions for this method will be slightly different from the previous one. To prove the theorem, suppose that the right-angled triangle ABC is isosceles.

We take the AC hypotenuse as the side of the square and subdivide its three sides. In addition, it is necessary to draw two diagonal lines in the resulting square. So that inside it there are four isosceles triangles.

To the legs AB and CB, you also need to draw in a square and draw one diagonal line in each of them. The first line is drawn from vertex A, the second from C.

Now you need to take a close look at the resulting drawing. Since there are four triangles equal to the original one on the AC hypotenuse, and two on the legs, this speaks of the veracity of this theorem.

By the way, thanks to this method of proving the Pythagorean theorem, the famous phrase was born: "Pythagorean pants are equal in all directions."

J. Garfield's proof

James Garfield is the 20th President of the United States of America. In addition to leaving his mark on history as the ruler of the United States, he was also a gifted self-taught person.

At the beginning of his career, he was an ordinary teacher in a folk school, but soon became the director of one of the higher educational institutions. The desire for self-development and allowed him to propose a new theory of proof of the Pythagorean theorem. The theorem and an example of its solution are as follows.

First, you need to draw two right-angled triangles on a sheet of paper so that the leg of one of them is a continuation of the second. The vertices of these triangles need to be connected to ultimately form a trapezoid.

As you know, the area of a trapezoid is equal to the product of the half-sum of its bases and the height.

S = a + b / 2 * (a + b)

If we consider the resulting trapezoid as a figure consisting of three triangles, then its area can be found as follows:

S = av / 2 * 2 + s 2/2

Now you need to equalize the two original expressions

2av / 2 + s / 2 = (a + b) 2/2

c 2 = a 2 + b 2

More than one volume of a textbook can be written about the Pythagorean theorem and the methods of its proof. But does it make sense when this knowledge cannot be applied in practice?

Practical application of the Pythagorean theorem

Unfortunately, modern school curricula provide for the use of this theorem only in geometric problems. Graduates will soon leave the school walls without knowing how they can apply their knowledge and skills in practice.

In fact, everyone can use the Pythagorean theorem in their daily life. And not only in professional activities, but also in ordinary household chores. Let's consider several cases when the Pythagorean theorem and methods of its proof may be extremely necessary.

The connection between theorem and astronomy

It would seem how stars and triangles can be connected on paper. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.

For example, consider the motion of a light beam in space. It is known that light moves in both directions at the same speed. The trajectory AB, which the light beam moves, will be called l. And half the time it takes for light to get from point A to point B, let's call t... And the speed of the beam - c. It turns out that: c * t = l

If you look at this very ray from another plane, for example, from a space liner, which moves with a speed v, then with such an observation of bodies their speed will change. In this case, even stationary elements will begin to move with speed v in the opposite direction.

Let's say the comic liner is sailing to the right. Then points A and B, between which the ray is tossed, will move to the left. Moreover, when the beam moves from point A to point B, point A has time to move and, accordingly, the light will already arrive at a new point C. To find half the distance by which point A has shifted, you need to multiply the speed of the liner by half the travel time of the beam (t ").

And in order to find how much distance a ray of light could travel during this time, you need to designate half of the path with a new letter s and get the following expression:

If we imagine that the points of light C and B, as well as the space liner, are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two right-angled triangles. Therefore, thanks to the Pythagorean theorem, you can find the distance that a ray of light could travel.

This example, of course, is not the best one, since only a few can be lucky enough to try it out in practice. Therefore, we will consider more mundane applications of this theorem.

Radius of transmission of a mobile signal

Modern life is already impossible to imagine without the existence of smartphones. But would they be of much use if they could not connect subscribers via mobile communications ?!

The quality of mobile communication directly depends on the height at which the mobile operator's antenna is located. In order to calculate how far the phone can receive a signal from the mobile tower, you can apply the Pythagorean theorem.

Let's say you need to find the approximate height of a stationary tower so that it can propagate a signal within a radius of 200 kilometers.

AB (tower height) = x;

Aircraft (signal transmission radius) = 200 km;

OS (radius of the globe) = 6380 km;

OB = OA + ABOV = r + x

Applying the Pythagorean theorem, we find out that the minimum height of the tower should be 2.3 kilometers.

The Pythagorean theorem in everyday life

Oddly enough, the Pythagorean theorem can be useful even in everyday matters, such as determining the height of a wardrobe, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements using a tape measure. But many are surprised why certain problems arise during the assembly process, if all measurements were taken more than accurately.

The fact is that the wardrobe is assembled in a horizontal position and only then it rises and is installed against the wall. Therefore, the side of the cabinet in the process of lifting the structure must pass freely both in height and diagonally of the room.

Suppose you have a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 2600 mm. An experienced furniture maker will tell you that the height of the cabinet should be 126 mm less than the height of the room. But why exactly 126 mm? Let's look at an example.

With ideal dimensions of the cabinet, we check the action of the Pythagorean theorem:

AC = √AB 2 + √BC 2

AC = √2474 2 +800 2 = 2600 mm - everything converges.

Let's say the height of the cabinet is not 2474 mm, but 2505 mm. Then:

AC = √2505 2 + √800 2 = 2629 mm.

Therefore, this cabinet is not suitable for installation in this room. Since lifting it to an upright position can damage its body.

Perhaps, having considered different ways of proving the Pythagorean theorem by different scientists, we can conclude that it is more than true. Now you can use the information received in your daily life and be completely sure that all calculations will be not only useful, but also correct.