PYTHAGOREAN PANTS ON ALL SIDES ARE EQUAL

This caustic remark (which has a continuation in full: to prove it, you need to remove and show), invented by someone, apparently shocked by the inner content of one important theorem of Euclidean geometry, perfectly reveals the starting point from which the chain completely simple reflections quickly leads to the proof of the theorem, as well as to even more significant results. This theorem, attributed to the ancient Greek mathematician Pythagoras of Samos (6th century BC), is known to almost every schoolchild and sounds like this: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. Perhaps many will agree that geometric figure , called the encryption "Pythagorean pants are equal on all sides", is called a square. Well, with a smile on your face, let's add a harmless joke for the sake of what was meant in the continuation of encrypted sarcasm. So, "to prove it, you need to remove and show." It is clear that "this" - the pronoun meant directly the theorem, "remove" - ​​this is to get in hand, take the named figure, "show" - meant the word "touch", bring some parts of the figure into contact. In general, "Pythagorean pants" were dubbed a graphic construction that looked like pants, which was obtained on the drawing of Euclid during a very difficult proof of the Pythagorean theorem. When a simpler proof was found, perhaps some rhymer made up this tongue twister-hint so as not to forget the beginning of the approach to the proof, and popular rumor already spread it around the world like an empty saying. So if you take a square, and place a smaller square inside it so that their centers coincide, and rotate the smaller square until its corners touch the sides of the larger square, then on the larger figure 4 identical right-angled triangles will be highlighted by the sides of the smaller square. From here, there is already a straight line a way to prove a well-known theorem. Let the side of the smaller square be denoted by c. The side of the larger square is a + b, and then its area is (a + b) 2 \u003d a 2 + 2ab + b 2. The same area can be defined as the sum of the area of ​​\u200b\u200bthe smaller square and the areas of 4 identical right triangles, that is, as 4 ab/2+c 2 =2ab+c 2. We put an equal sign between two calculations of the same area: a 2 +2ab+b 2 =2ab+c 2. After reducing the terms 2ab, we get the conclusion: the square of the hypotenuse of a right triangle is equal to the sum squares of the legs, that is, a 2 + b 2 \u003d c 2. Not everyone will immediately understand what is the use of this theorem. From a practical point of view, its value lies in serving as a basis for many geometric calculations, such as determining the distance between points on a coordinate plane. Some valuable formulas are derived from the theorem, and its generalizations lead to new theorems that bridge the gap between calculations in the plane and calculations in space. The consequences of the theorem penetrate into number theory, revealing individual details of the structure of a series of numbers. And many more, you can't list them all. A view from the point of view of idle curiosity demonstrates the presentation of entertaining problems by the theorem, which are formulated to the extreme understandable, but are sometimes tough nuts. As an example, it suffices to cite the simplest of them, the so-called question of Pythagorean numbers, which is asked in everyday terms as follows: is it possible to build a room, the length, width and diagonal on the floor of which would be simultaneously measured only in whole values, say, in steps? Just the slightest change in this question can make the task extremely difficult. And accordingly, there are those who wish, purely out of scientific enthusiasm, to test themselves in splitting the next mathematical puzzle. Another change to the question - and another puzzle. Often, in the course of searching for answers to such problems, mathematics evolves, acquires fresh views on old concepts, acquires new systematic approaches, and so on, which means that the Pythagorean theorem, however, like any other worthwhile doctrine, is no less useful from this point of view. Mathematics of the time of Pythagoras did not recognize other numbers than rational ones (natural numbers or fractions with a natural numerator and denominator). Everything was measured in whole values ​​or parts of wholes. Therefore, the desire to do geometric calculations, to solve equations more and more in natural numbers is so understandable. Addiction to them opens the way to the incredible world of the mystery of numbers, a number of which in geometric interpretation initially appears as a straight line with an infinite number of markings. Sometimes the relationship between some numbers in the series, the "linear distance" between them, the proportion immediately catches the eye, and sometimes the most complex mental constructions do not allow us to establish what laws the distribution of certain numbers is subject to. It turns out that in the new world, in this "one-dimensional geometry", the old problems remain valid, only their formulation changes. For example, a variant of the task about Pythagorean numbers: "From home, the father takes x steps of x centimeters each, and then walks at steps of y centimeters. The son walks behind him z steps of z centimeters each. What should be the size of their steps in order to at the z-th step did the child step into the footprint of the father? For the sake of fairness, it is necessary to note some difficulty for a novice mathematician of the Pythagorean method of thought development. This is a special kind of mathematical thinking style, you need to get used to it. One point is interesting. The mathematicians of the Babylonian state (it arose long before the birth of Pythagoras, almost one and a half thousand years before him) also apparently knew some methods for finding numbers, which later became known as Pythagorean ones. Cuneiform tablets were found, where the Babylonian sages wrote down the triplets of such numbers that they identified. Some triples consisted of too large numbers, in connection with which our contemporaries began to assume that the Babylonians had good, and probably even simple, ways of calculating them. Unfortunately, nothing is known about the methods themselves, or about their existence.

Pythagorean pants The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of pants. I loved geometry ... and at the entrance exam to the university I even received praise from Chumakov, a professor of mathematics, for explaining the properties of parallel lines and Pythagorean pants without a blackboard, drawing with my hands in the air(N. Pirogov. Diary of an old doctor).

Phraseological dictionary of the Russian literary language. - M.: Astrel, AST. A. I. Fedorov. 2008 .

See what "Pythagorean pants" are in other dictionaries:

Pythagorean pants- ... Wikipedia

Pythagorean pants- Zharg. school Shuttle. The Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and the legs of a right triangle. BTS, 835... Big dictionary of Russian sayings

Pythagorean pants- A playful name for the Pythagorean theorem, which establishes the ratio between the areas of squares built on the hypotenuse and the legs of a right-angled triangle, which looks like the cut of pants in the drawings ... Dictionary of many expressions

Pythagorean pants (invent)- foreigner: about a gifted person Cf. This is the certainty of the sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley 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 is the dick cramped? (roughly) about pants and the male sexual organ. 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

Pythagorean pants invent- Pythagorean pants (invent) foreigner. about a gifted person. Wed This is the undoubted sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)

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

Adj., rude...

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (NUMBER OF BUTTONS IS KNOWN. WHY IS IT CLOSE? / TO PROVE THIS, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Dictionary modern colloquial phraseological units and sayings

pants- noun, pl., use comp. often Morphology: pl. what? pants, (no) what? pants for what? pants, (see) what? pants what? pants, what? about pants 1. Pants are a piece of clothing that has two short or long legs and covers the bottom ... ... Dictionary of Dmitriev

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• How the Earth was Discovered Svyatoslav Vladimirovich Sakharnov. How did the Phoenicians travel? What ships did the Vikings sail on? Who discovered America and who first circumnavigated the world? Who compiled the world's first atlas of Antarctica, and who invented ...

famous the Pythagorean theorem - "In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs"- everyone knows from the school bench.

Well do you remember "Pythagorean pants", which "equal in all directions"- a schematic drawing explaining the theorem of the Greek scientist.

Here a and b- legs, and With- hypotenuse:

Now I will tell you about one original proof of this theorem, which you may not have known about ...

But first, let's look at one lemma- a proven statement that is useful not in itself, but for proving other statements (theorems).

Take a right triangle with vertices X, Y and Z, where Z- right angle and drop the perpendicular from the right angle Z to the hypotenuse. Here W- the point where the altitude intersects the hypotenuse.

This line (perpendicular) ZW splits the triangle into similar copies of itself.

Let me remind you that similar triangles are called, the angles of which are respectively equal, and the sides of one triangle are proportional to the similar sides of the other triangle.

In our example, the formed triangles XWZ and YWZ are similar to each other and also similar to the original triangle XYZ.

It is easy to prove this.

Starting with triangle XWZ, note that ∠XWZ = 90 and so ∠XZW = 180-90-∠X. But 180–90-∠X -  is exactly what ∠Y is, so triangle XWZ must be similar (all angles equal) to triangle XYZ. The same exercise can be done for triangle YWZ.

Lemma proven! In a right triangle, the height (perpendicular) dropped to the hypotenuse splits the triangle into two similar ones, which in turn are similar to the original triangle.

But, back to our "Pythagorean pants" ...

Drop the perpendicular to the hypotenuse c. As a result, we have two right triangles inside our right triangle. Let's denote these triangles (in the picture above in green) letters A and B, and the original triangle - letter FROM.

Of course, the area of ​​the triangle FROM is equal to the sum of the areas of the triangles A and B.

Those. BUT+ B= FROM

Now let's break the figure at the top (“Pythagorean pants”) into three house figures:

As we already know from the lemma, triangles A, B and C are similar to each other, therefore the resulting house figures are also similar and are scaled versions of each other.

This means that the area ratio A and a², -  is the same as the area ratio B and b², as well as C and c².

Thus we have A / a² = B / b² = C / c² .

Let us denote this ratio of the areas of the triangle and the square in the figure-house by the letter k.

Those. k- this is a certain coefficient connecting the area of ​​the triangle (the roof of the house) with the area of ​​the square below it:
k = A / a² = B / b² = C / c²

It follows that the areas of triangles can be expressed in terms of the areas of the squares below them in this way:
A = ka², B = kb², and C = kc²

But we remember that A+B=C, which means ka² + kb² = kc²

Or a² + b² = c²

And this is proof of the Pythagorean theorem!

The potential for creativity is usually attributed to humanities, naturally scientific leaving the analysis, practical approach and dry language of formulas and figures. Mathematics cannot be classified as a humanities subject. But without creativity in the "queen of all sciences" you will not go far - people have known about this for a long time. Since the time of Pythagoras, for example.

School textbooks, unfortunately, usually do not explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and feel its fundamental principles. And at the same time, try to free your mind from clichés and elementary truths - only in such conditions are all great discoveries born.

Such discoveries include the one that today we know as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be fun. And that this adventure is suitable not only for nerds in thick glasses, but for everyone who is strong in mind and strong in spirit.

From the history of the issue

Strictly speaking, although the theorem is called the "Pythagorean theorem", Pythagoras himself did not discover it. The right triangle and its special properties have been studied long before it. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.

Today you can no longer check who is right and who is wrong. It is only known that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid's Elements may belong to Pythagoras, and Euclid only recorded it.

It is also known today that problems about a right-angled triangle are found in Egyptian sources from the time of Pharaoh Amenemhet I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise Sulva Sutra and the ancient Chinese work Zhou-bi suan jin.

As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. Approximately 367 various pieces of evidence that exist today serve as confirmation. No other theorem can compete with it in this respect. Notable evidence authors include Leonardo da Vinci and the 20th President of the United States, James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or, in one way or another, connected with it.

Proofs of the Pythagorean Theorem

AT school textbooks mainly give algebraic proofs. But the essence of the theorem is in geometry, so let's first of all consider those proofs of the famous theorem that are based on this science.

Proof 1

For the simplest proof of the Pythagorean theorem for a right triangle, you need to set ideal conditions: let the triangle be not only rectangular, but also isosceles. There is reason to believe that it was such a triangle that was originally considered by ancient mathematicians.

Statement "a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs" can be illustrated with the following drawing:

Look at the isosceles right triangle ABC: On the hypotenuse AC, you can build a square consisting of four triangles equal to the original ABC. And on the legs AB and BC built on a square, each of which contains two similar triangles.

By the way, this drawing formed the basis of numerous anecdotes and cartoons dedicated to the Pythagorean theorem. Perhaps the most famous is "Pythagorean pants are equal in all directions":

Proof 2

This method combines algebra and geometry and can be seen as a variant of the ancient Indian proof of the mathematician Bhaskari.

Construct a right triangle with sides a, b and c(Fig. 1). Then build two squares with sides equal to the sum of the lengths of the two legs - (a+b). In each of the squares, make constructions, as in figures 2 and 3.

In the first square, build four of the same triangles as in Figure 1. As a result, two squares are obtained: one with side a, the second with side b.

In the second square, four similar triangles constructed form a square with a side equal to the hypotenuse c.

The sum of the areas of the constructed squares in Fig. 2 is equal to the area of ​​the square we constructed with side c in Fig. 3. This can be easily verified by calculating the areas of the squares in Fig. 2 according to the formula. And the area of ​​​​the inscribed square in Figure 3. by subtracting the areas of four equal right-angled triangles inscribed in the square from the area of ​​\u200b\u200ba large square with a side (a+b).

Putting all this down, we have: a 2 + b 2 \u003d (a + b) 2 - 2ab. Expand the brackets, do all the necessary algebraic calculations and get that a 2 + b 2 = a 2 + b 2. At the same time, the area of ​​the inscribed in Fig.3. square can also be calculated using the traditional formula S=c2. Those. a2+b2=c2 You have proved the Pythagorean theorem.

Proof 3

The very same ancient Indian proof is described in the 12th century in the treatise “The Crown of Knowledge” (“Siddhanta Shiromani”), and as the main argument the author uses an appeal addressed to the mathematical talents and powers of observation of students and followers: “Look!”.

But we will analyze this proof in more detail:

Inside the square, build four right-angled triangles as indicated in the drawing. The side of the large square, which is also the hypotenuse, is denoted With. Let's call the legs of the triangle a and b. According to the drawing, the side of the inner square is (a-b).

Use the square area formula S=c2 to calculate the area of ​​the outer square. And at the same time calculate the same value by adding the area of ​​​​the inner square and the area of ​​\u200b\u200ball four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options to calculate the area of ​​a square to make sure they give the same result. And that gives you the right to write down that c 2 =(a-b) 2 +4*1\2*a*b. As a result of the solution, you will get the formula of the Pythagorean theorem c2=a2+b2. The theorem has been proven.

Proof 4

This curious ancient Chinese proof is called the "Bride's Chair" - because of the chair-like figure that results from all the constructions:

It uses the drawing we have already seen in Figure 3 in the second proof. And the inner square with side c is constructed in the same way as in the ancient Indian proof given above.

If you mentally cut off two green right-angled triangles from the drawing in Fig. 1, transfer them to opposite sides of the square with side c and attach the hypotenuses to the hypotenuses of the lilac triangles, you will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will see that the "bride's chair" is formed by two squares: small ones with a side b and big with a side a.

These constructions allowed the ancient Chinese mathematicians and us following them to come to the conclusion that c2=a2+b2.

Proof 5

This is another way to find a solution to the Pythagorean theorem based on geometry. It's called the Garfield Method.

Construct a right triangle ABC. We need to prove that BC 2 \u003d AC 2 + AB 2.

To do this, continue the leg AC and build a segment CD, which is equal to the leg AB. Lower Perpendicular AD line segment ED. Segments ED and AC are equal. connect the dots E and AT, as well as E and FROM and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tested: we find the area of ​​the resulting figure in two ways and equate the expressions to each other.

Find the area of ​​a polygon ABED can be done by adding the areas of the three triangles that form it. And one of them ERU, is not only rectangular, but also isosceles. Let's also not forget that AB=CD, AC=ED and BC=CE- this will allow us to simplify the recording and not overload it. So, S ABED \u003d 2 * 1/2 (AB * AC) + 1 / 2BC 2.

At the same time, it is obvious that ABED is a trapezoid. Therefore, we calculate its area using the formula: SABED=(DE+AB)*1/2AD. For our calculations, it is more convenient and clearer to represent the segment AD as the sum of the segments AC and CD.

Let's write both ways to calculate the area of ​​​​a figure by putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify right side records: AB*AC+1/2BC 2 =1/2(AB+AC) 2. And now we open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having finished all the transformations, we get exactly what we need: BC 2 \u003d AC 2 + AB 2. We have proved the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proved using vectors, complex numbers, differential equations, stereometry, etc. And even physicists: if, for example, liquid is poured into square and triangular volumes similar to those shown in the drawings. By pouring liquid, it is possible to prove the equality of areas and the theorem itself as a result.

A few words about Pythagorean triplets

This issue is little or not studied in the school curriculum. Meanwhile, it is very interesting and has great importance in geometry. Pythagorean triples are used to solve many mathematical problems. The idea of ​​them can be useful to you in further education.

So what are Pythagorean triplets? So called natural numbers, collected in threes, the sum of the squares of two of which is equal to the third number squared.

Pythagorean triples can be:

• primitive (all three numbers are relatively prime);
• non-primitive (if each number of a triple is multiplied by the same number, you get a new triple that is not primitive).

Even before our era, the ancient Egyptians were fascinated by the mania for the numbers of Pythagorean triples: in tasks they considered a right-angled triangle with sides of 3.4 and 5 units. By the way, any triangle whose sides are equal to the numbers from the Pythagorean triple is by default rectangular.

Examples of Pythagorean triples: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20) ), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (10, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (27, 36, 45), (14 , 48, 50), (30, 40, 50) etc.

Practical application of the theorem

The Pythagorean theorem finds application not only in mathematics, but also in architecture and construction, astronomy, and even literature.

First about construction: the Pythagorean theorem finds in it wide application in tasks of different levels of complexity. For example, look at the Romanesque window:

Let's denote the width of the window as b, then the radius of the great semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed in terms of b: r=b/4. In this problem, we are interested in the radius of the inner circle of the window (let's call it p).

The Pythagorean theorem just comes in handy to calculate R. To do this, we use a right-angled triangle, which is indicated by a dotted line in the figure. The hypotenuse of a triangle consists of two radii: b/4+p. One leg is a radius b/4, another b/2-p. Using the Pythagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Next, we open the brackets and get b 2 /16+ bp / 2 + p 2 \u003d b 2 / 16 + b 2 / 4-bp + p 2. Let's transform this expression into bp/2=b 2 /4-bp. And then we divide all the terms into b, we give similar ones to get 3/2*p=b/4. And in the end we find that p=b/6- which is what we needed.

Using the theorem, you can calculate the length of the rafter for gable roof. Determine how high a mobile tower is needed for the signal to reach a certain settlement. And even steadily install a Christmas tree in the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

As far as literature is concerned, the Pythagorean theorem has inspired writers since antiquity and continues to do so today. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired by her to write a sonnet:

The light of truth will not soon dissipate,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
Will not cause doubts and disputes.

The wisest when it touches the eye
Light of truth, thank the gods;
And a hundred bulls, stabbed, lie -
The return gift of the lucky Pythagoras.

Since then, the bulls have been roaring desperately:
Forever aroused the bull tribe
event mentioned here.

They think it's about time
And again they will be sacrificed
Some great theorem.

(translated by Viktor Toporov)

And in the twentieth century, the Soviet writer Yevgeny Veltistov in his book "The Adventures of Electronics" devoted a whole chapter to the proofs of the Pythagorean theorem. And half a chapter of a story about a two-dimensional world that could exist if the Pythagorean theorem became the fundamental law and even religion for a single world. It would be much easier to live in it, but also much more boring: for example, no one there understands the meaning of the words “round” and “fluffy”.

And in the book “The Adventures of Electronics”, the author, through the mouth of the mathematics teacher Taratara, says: “The main thing in mathematics is the movement of thought, new ideas.” It is this creative flight of thought that generates the Pythagorean theorem - it is not for nothing that it has so many diverse proofs. It helps to go beyond the usual, and look at familiar things in a new way.

Conclusion

This article was created so that you can look beyond the school curriculum in mathematics and learn not only those proofs of the Pythagorean theorem that are given in the textbooks "Geometry 7-9" (L.S. Atanasyan, V.N. Rudenko) and "Geometry 7 -11” (A.V. Pogorelov), but also other curious ways to prove the famous theorem. And also see examples of how the Pythagorean theorem can be applied in everyday life.

Firstly, this information will allow you to claim higher scores in math classes - information on the subject from additional sources is always highly appreciated.

Secondly, we wanted to help you get a feel for how interesting mathematics is. To be convinced by specific examples that there is always a place for creativity in it. We hope that the Pythagorean theorem and this article will inspire you to do your own research and exciting discoveries in mathematics and other sciences.

Tell us in the comments if you found the evidence presented in the article interesting. Did you find this information helpful in your studies? Let us know what you think about the Pythagorean theorem and this article - we will be happy to discuss all this with you.

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A playful proof of the Pythagorean theorem; also in jest about a buddy's baggy trousers.

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"Pythagorean pants are equal in all directions" in books

11. Pythagorean pants