Message about Pythagorean pants. The Pythagorean theorem: background, evidence, examples of practical application. From the history of the issue

Almost Equal 12 Chairs Club Almost Equal IRONIC PROSE Death came to a poor man. And he was deaf. So normal, but a little deaf ... And he saw badly. I saw almost nothing. - Oh, we have guests! Please pass. Death says: - Wait to rejoice,

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MBOU Bondarskaya secondary school Student project on the topic: “Pythagoras and his theorem” Prepared by: Ektov Konstantin, student of grade 7 A Head: Dolotova Nadezhda Ivanovna, mathematics teacher 2015

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Annotation. Geometry is a very interesting science. It contains many theorems that are not similar to each other, but sometimes so necessary. I became very interested in the Pythagorean theorem. Unfortunately, one of the most important statements we pass only in the eighth grade. I decided to lift the veil of secrecy and explore the Pythagorean theorem.

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Tasks To study the biography of Pythagoras. Explore the history of the emergence and proof of the theorem. Find out how the theorem is used in art. Find historical problems in which the Pythagorean theorem is used. To get acquainted with the attitude of children of different times to this theorem. Create a project.

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Research progress Biography of Pythagoras. Commandments and aphorisms of Pythagoras. Pythagorean theorem. History of the theorem. Why are "Pythagorean pants equal in all directions"? Various proofs of the Pythagorean theorem by other scientists. Application of the Pythagorean theorem. Poll. Conclusion.

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Pythagoras - who is he? Pythagoras of Samos (580 - 500 BC) Ancient Greek mathematician and idealist philosopher. Born on the island of Samos. Got a good education. According to legend, Pythagoras, in order to get acquainted with the wisdom of Eastern scientists, went to Egypt and lived there for 22 years. Having mastered all the sciences of the Egyptians, including mathematics, he moved to Babylon, where he lived for 12 years and got acquainted with the scientific knowledge of the Babylonian priests. Traditions attribute to Pythagoras a visit to India. This is very likely, since Ionia and India then had trade relations. Returning to his homeland (c. 530 BC), Pythagoras tried to organize his philosophical school. However, for unknown reasons, he soon leaves Samos and settles in Croton (a Greek colony in northern Italy). Here Pythagoras managed to organize his own school, which operated for almost thirty years. The school of Pythagoras, or, as it is also called, the Pythagorean Union, was at the same time philosophical school, and a political party, and a religious fraternity. The status of the Pythagorean union was very severe. By their own philosophical views Pythagoras was an idealist, defender of the interests of the slave-owning aristocracy. Perhaps this was the reason for his departure from Samos, since supporters of democratic views had a very large influence in Ionia. In public matters, by "order" the Pythagoreans understood the rule of the aristocrats. They condemned ancient Greek democracy. Pythagorean philosophy was a primitive attempt to justify the dominance of the slave-owning aristocracy. At the end of the 5th century BC e. a wave of democratic movement swept through Greece and its colonies. Democracy won in Croton. Pythagoras leaves Croton with his disciples and goes to Tarentum, and then to Metapont. The arrival of the Pythagoreans at Metapont coincided with the outbreak of a popular uprising there. In one of the night skirmishes, almost ninety-year-old Pythagoras died. His school has ceased to exist. The disciples of Pythagoras, fleeing persecution, settled throughout Greece and its colonies. Earning their livelihood, they organized schools in which they taught mainly arithmetic and geometry. Information about their achievements is contained in the writings of later scientists - Plato, Aristotle, etc.

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Commandments and aphorisms of Pythagoras Thought is above all between people on earth. Do not sit down on a grain measure (i.e., do not live idly). When leaving, do not look back (that is, before death, do not cling to life). Do not go down the beaten road (that is, follow not the opinions of the crowd, but the opinions of the few who understand). Do not keep swallows in the house (i.e., do not receive guests who are talkative and not restrained in language). Be with the one who takes the load, do not be with the one who dumps the load (that is, encourage people not to idleness, but to virtue, to work). In the field of life, like a sower, walk with even and steady steps. The true fatherland is where there are good morals. Do not be a member of a learned society: the wisest, making up a society, become commoners. Revere sacred numbers, weight and measure, as a child of graceful equality. Measure your desires, weigh your thoughts, number your words. Be astonished at nothing: astonishment has produced gods.

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Statement of the theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

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Proofs of the theorem. On the this moment in scientific literature 367 proofs of this theorem were recorded. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. Of course, all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs.

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Pythagorean theorem Proof Given a right triangle with legs a, b and hypotenuse c. Let's prove that c² = a² + b² Let's complete the triangle to a square with side a + b. The area S of this square is (a + b)². On the other hand, the square is made up of four equal right triangles, each S equal to ½ a b, and a square with side c. S = 4 ½ a b + c² = 2 a b + c² Thus, (a + b)² = 2 a b + c², whence c² = a² + b² c c c c c a b

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The history of the Pythagorean theorem The history of the Pythagorean theorem is interesting. Although this theorem is associated with the name of Pythagoras, it was known long before him. In Babylonian texts, this theorem occurs 1200 years before Pythagoras. It is possible that at that time they did not yet know its evidence, and the very relationship between the hypotenuse and the legs was established empirically on the basis of measurements. Pythagoras apparently found proof of this relationship. An ancient legend has been preserved that in honor of his discovery, Pythagoras sacrificed a bull to the gods, and according to other testimonies, even a hundred bulls. Over the following centuries, various other proofs of the Pythagorean theorem were found. Currently, there are more than a hundred of them, but the most popular theorem is the construction of a square using a given right triangle.

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Theorem in Ancient China "If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4."

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Theorem in Ancient Egypt Kantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonapts, or "stringers", built right angles using right triangles with sides 3, 4 and 5.

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About the theorem in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and substantiation. In their hands, computational recipes based on vague ideas have become an exact science.

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Why are "Pythagorean pants equal in all directions"? For two millennia, the most common proof of the Pythagorean theorem was that of Euclid. It is placed in his famous book "Beginnings". Euclid lowered the height CH from the vertex of the right angle to the hypotenuse and proved that its continuation divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs. The drawing used in the proof of this theorem is jokingly called "Pythagorean pants". For a long time he was considered one of the symbols of mathematical science.

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The attitude of children of antiquity to the proof of the Pythagorean theorem was considered by students of the Middle Ages to be very difficult. Weak students who memorized theorems without understanding, and therefore called "donkeys", were not able to overcome the Pythagorean theorem, which served for them like an insurmountable bridge. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill”, composed poems like “Pythagorean pants are equal on all sides”, and drew caricatures.

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Proofs of the theorem The simplest proof of the theorem is obtained in the case of an isosceles right triangle. Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true. For example, for triangle ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two.

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"Chair of the bride" In the figure, the squares built on the legs are placed in steps one next to the other. This figure, which occurs in evidence dating no later than the 9th century CE, e., the Hindus called the "chair of the bride."

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Application of the Pythagorean theorem At present, it is generally recognized that the success of the development of many areas of science and technology depends on the development of various areas of mathematics. An important condition for increasing the efficiency of production is the widespread introduction of mathematical methods in technology and the national economy, which involves the creation of new, effective methods qualitative and quantitative research, which allow us to solve the problems put forward by practice.

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Application of the theorem in construction In buildings of the Gothic and Romanesque styles, the upper parts of the windows are divided by stone ribs, which not only play the role of an ornament, but also contribute to the strength of the windows.

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Historical tasks To fix the mast, you need to install 4 cables. One end of each cable should be fixed at a height of 12 m, the other on the ground at a distance of 5 m from the mast. Is 50 m of rope enough to secure the mast?

One thing you can be sure of one hundred percent, 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 planted in the minds of every educated person, but it is enough just 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 overview of the biography

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

Pythagoras - 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 was to bring many benefits and good to mankind. Which is what he actually did.

The birth of a theorem

In his youth, Pythagoras moved to Egypt to meet the famous Egyptian sages there. 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. But 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 technique for proving this theorem is known, but several at once. Today we can only guess how exactly the ancient Greeks made their calculations, so here we will consider different ways of proving the Pythagorean theorem.

Pythagorean theorem

Before you start any calculations, you need to figure out which theory to prove. The Pythagorean theorem sounds like this: "In a triangle in which one of the angles is 90 o, the sum of the squares of the legs is equal to the square of the hypotenuse."

There are 15 different ways to prove the Pythagorean Theorem in total. This is a fairly large number, so let's pay attention to the most popular of them.

Method one

Let's first define what we have. This data will also apply to other ways of proving the Pythagorean theorem, so you should immediately remember all the available notation.

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

To do this, you need to draw a segment equal to the leg in to the leg length a, and vice versa. So it should turn out 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 draw the fourth segment.

Based on the resulting figure, we can conclude that the area of ​​\u200b\u200bthe outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, it has four right-angled triangles. The area of ​​each is 0.5 av.

Therefore, the area is: 4 * 0.5av + s 2 \u003d 2av + s 2

Hence (a + c) 2 \u003d 2av + c 2

And, therefore, with 2 \u003d a 2 + in 2

The theorem has been proven.

Method two: similar triangles

This formula for the proof of the Pythagorean theorem was derived on the basis of a statement from the section of geometry about similar triangles. It says that the leg of a right triangle is the mean proportional to its hypotenuse and the hypotenuse segment emanating from the vertex of an angle of 90 o.

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

AC=√AB*AD, SW=√AB*DV.

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

AC 2 \u003d AB * HELL and SV 2 \u003d AB * DV

Now we need to add the resulting inequalities.

AC 2 + SV 2 \u003d AB * (AD * DV), where AD + DV \u003d AB

It turns out that:

AC 2 + CB 2 \u003d AB * AB

And therefore:

AC 2 + CB 2 \u003d AB 2

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

Another calculation method

Description of different ways of proving the Pythagorean theorem may not say anything, until you start practicing on your own. Many methods involve not only mathematical calculations, but also the construction of new figures from the original triangle.

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

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

S avs * s 2 - S avd * in 2 \u003d S avd * a 2 - S vd * a 2

S avs * (from 2 to 2) \u003d a 2 * (S avd -S vvd)

from 2 to 2 \u003d a 2

c 2 \u003d a 2 + in 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 in ancient Greece. It is the simplest, since it does not require absolutely any calculations. If you draw a picture correctly, then the proof of the statement that a 2 + b 2 \u003d 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 triangle ABC is isosceles.

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

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

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

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

Proof by J. Garfield

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.

At the beginning of his career, he was an ordinary teacher at a folk school, but soon became the director of one of the higher educational institutions. The desire for self-development and allowed him to offer 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 piece 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 end up with a trapezoid.

As you know, the area of ​​a trapezoid is equal to the product of half the 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 \u003d av / 2 * 2 + s 2 / 2

Now we need to equalize the two original expressions

2av / 2 + s / 2 \u003d (a + c) 2 / 2

c 2 \u003d a 2 + in 2

More than one volume of a textbook can be written about the Pythagorean theorem and how to prove it. But does it make sense when this knowledge cannot be put into practice?

Practical application of the Pythagorean theorem

Unfortunately, in modern school programs This theorem is intended to be used 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, use the Pythagorean theorem in your Everyday life everyone can. 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 can be extremely necessary.

Connection of the 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. We know that light travels in both directions at the same speed. We call the trajectory AB along which the light ray moves 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 beam from another plane, for example, from a space liner that moves at a speed v, then with such an observation of the bodies, their speed will change. In this case, even stationary elements will move with a 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 rushes, 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 that 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 far a ray of light could travel during this time, you need to designate half the path of the new beech 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 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 most successful, since only a few can be lucky enough to try it out in practice. Therefore, we consider more mundane applications of this theorem.

Mobile signal transmission range

Modern life can no longer be imagined without the existence of smartphones. But how much would they be of use if they could not connect subscribers via mobile communications?!

The quality of mobile communications directly depends on the height at which the antenna of the mobile operator is located. In order to calculate how far from a mobile tower a phone can receive a signal, 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;

BC (radius of signal transmission) = 200 km;

OS (radius of the globe) = 6380 km;

OB=OA+ABOB=r+x

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

Pythagorean theorem in everyday life

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

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

Suppose there is a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 2600 mm. An experienced furniture maker will say 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, let's check the operation of the Pythagorean theorem:

AC \u003d √AB 2 + √BC 2

AC \u003d √ 2474 2 +800 2 \u003d 2600 mm - everything converges.

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

AC \u003d √2505 2 + √800 2 \u003d 2629 mm.

Therefore, this cabinet is not suitable for installation in this room. Since when lifting it to a vertical position, damage to its body can be caused.

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.