G. Glaser,
Academician of the Russian Academy of Education, Moscow

About the Pythagorean theorem and how to prove it

The area of ​​a square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on its legs...

This is one of the most famous geometric theorems of antiquity, called the Pythagorean theorem. It is still known to almost everyone who has ever studied planimetry. It seems to me that if we want to let extraterrestrial civilizations know about the existence of intelligent life on Earth, then we should send an image of the Pythagorean figure into space. I think that if thinking beings can accept this information, they will understand without complex signal decoding that there is a fairly developed civilization on Earth.

The famous Greek philosopher and mathematician Pythagoras of Samos, after whom the theorem is named, lived about 2.5 thousand years ago. The biographical information about Pythagoras that has come down to us is fragmentary and far from reliable. Many legends are associated with his name. It is authentically known that Pythagoras traveled a lot in the countries of the East, visited Egypt and Babylon. In one of the Greek colonies of southern Italy, he founded the famous "Pythagorean school", which played an important role in the scientific and political life of ancient Greece. It is Pythagoras who is credited with proving the well-known geometric theorem. Based on the legends spread by famous mathematicians (Proclus, Plutarch, etc.), for a long time it was believed that this theorem was not known before Pythagoras, hence the name - the Pythagorean theorem.

However, there is no doubt that this theorem was known many years before Pythagoras. So, 1500 years before Pythagoras, the ancient Egyptians knew that a triangle with sides 3, 4 and 5 is rectangular, and used this property (i.e., the inverse theorem of Pythagoras) to construct right angles when planning land plots and structures buildings. And even today, rural builders and carpenters, laying the foundation of the hut, making its details, draw this triangle to get a right angle. The same thing was done thousands of years ago in the construction of magnificent temples in Egypt, Babylon, China, and probably in Mexico. In the oldest Chinese mathematical and astronomical work that has come down to us, Zhou-bi, written about 600 years before Pythagoras, among other proposals related to a right triangle, the Pythagorean theorem is also contained. Even earlier this theorem was known to the Hindus. Thus, Pythagoras did not discover this property of a right-angled triangle; he was probably the first to generalize and prove it, thereby transferring it from the field of practice to the field of science. We don't know how he did it. Some historians of mathematics assume that, nevertheless, Pythagoras's proof was not fundamental, but only a confirmation, a verification of this property on a number of particular types of triangles, starting with an isosceles right triangle, for which it obviously follows from Fig. one.

With Since ancient times, mathematicians have found more and more proofs of the Pythagorean theorem, more and more ideas for its proofs. More than one and a half hundred such proofs - more or less rigorous, more or less visual - are known, but the desire to increase their number has been preserved. I think that the independent "discovery" of the proofs of the Pythagorean theorem will be useful for modern schoolchildren.

Let us consider some examples of evidence that may suggest the direction of such searches.

Proof of Pythagoras

";A square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs. "; The simplest proof of the theorem is obtained in the simplest case of an isosceles right triangle. Probably, the theorem began with him. Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true. For example, for DABC: a square built on the hypotenuse AU, contains 4 initial triangles, and squares built on the legs by two. The theorem has been proven.

Proofs based on the use of the concept of equal area of ​​figures.

At the same time, we can consider evidence in which the square built on the hypotenuse of a given right-angled triangle is “composed” of the same figures as the squares built on the legs. We can also consider such proofs in which the permutation of the terms of the figures is used and a number of new ideas are taken into account.

On fig. 2 shows two equal squares. The length of the sides of each square is a + b. Each of the squares is divided into parts consisting of squares and right triangles. It is clear that if we subtract the quadruple area of ​​a right-angled triangle with legs a, b from the square area, then equal areas remain, i.e. c 2 \u003d a 2 + b 2. However, the ancient Hindus, to whom this reasoning belongs, usually did not write it down, but accompanied the drawing with only one word: “look!” It is quite possible that Pythagoras offered the same proof.

additive evidence.

These proofs are based on the decomposition of the squares built on the legs into figures, from which it is possible to add a square built on the hypotenuse.

Here: ABC is a right triangle with right angle C; CMN; CKMN; PO||MN; EF||MN.

Prove on your own the pairwise equality of the triangles obtained by splitting the squares built on the legs and the hypotenuse.

Prove the theorem using this partition.

 On the basis of al-Nairiziya's proof, another decomposition of squares into pairwise equal figures was made (Fig. 5, here ABC is a right triangle with right angle C).

 Another proof by the method of decomposing squares into equal parts, called the "wheel with blades", is shown in fig. 6. Here: ABC is a right triangle with right angle C; O - the center of a square built on a large leg; dashed lines passing through the point O are perpendicular or parallel to the hypotenuse.

 This decomposition of squares is interesting in that its pairwise equal quadrilaterals can be mapped onto each other by parallel translation. Many other proofs of the Pythagorean theorem can be offered using the decomposition of squares into figures.

Proofs by extension method.

The essence of this method is that equal figures are attached to the squares built on the legs and to the square built on the hypotenuse in such a way that equal figures are obtained.

The validity of the Pythagorean theorem follows from the equal size of the hexagons AEDFPB and ACBNMQ. Here CEP, line EP divides hexagon AEDFPB into two equal-area quadrangles, line CM divides hexagon ACBNMQ into two equal-area quadrangles; a 90° rotation of the plane around the center A maps quadrilateral AEPB to quadrilateral ACMQ.

On fig. 8 The Pythagorean figure is completed to a rectangle, the sides of which are parallel to the corresponding sides of the squares built on the legs. Let's break this rectangle into triangles and rectangles. First, we subtract all polygons 1, 2, 3, 4, 5, 6, 7, 8, 9 from the resulting rectangle, leaving a square built on the hypotenuse. Then, from the same rectangle, we subtract rectangles 5, 6, 7 and the shaded rectangles, we get squares built on the legs.

Now let us prove that the figures subtracted in the first case are equal in size to the figures subtracted in the second case.

KLOA = ACPF = ACED = a 2 ;

LGBO = CBMP = CBNQ = b 2 ;

AKGB = AKLO + LGBO = c 2 ;

hence c 2 = a 2 + b 2 .

OCLP=ACLF=ACED=b2;

CBML = CBNQ = a 2 ;

OBMP = ABMF = c 2 ;

OBMP = OCLP + CBML;

c 2 = a 2 + b 2 .

Algebraic method of proof.

	Rice. 12 illustrates the proof of the great Indian mathematician Bhaskari (the famous author of Lilavati, X 2nd century). The drawing was accompanied by only one word: LOOK! Among the proofs of the Pythagorean theorem by the algebraic method, the proof using similarity occupies the first place (perhaps the oldest).
	Let us present in a modern presentation one of such proofs, which belongs to Pythagoras.

H and fig. 13 ABC - rectangular, C - right angle, CMAB, b 1 - projection of leg b on the hypotenuse, a 1 - projection of leg a on the hypotenuse, h - height of the triangle drawn to the hypotenuse.

From the fact that ABC is similar to ACM it follows

b 2 \u003d cb 1; (one)

from the fact that ABC is similar to BCM it follows

a 2 = ca 1 . (2)

Adding equalities (1) and (2) term by term, we get a 2 + b 2 = cb 1 + ca 1 = c(b 1 + a 1) = c 2 .

If Pythagoras really offered such a proof, then he was also familiar with a number of important geometric theorems that modern historians of mathematics usually attribute to Euclid.

Möllmann's proof (Fig. 14). The area of ​​this right triangle, on the one hand, is equal on the other, where p is the semiperimeter of the triangle, r is the radius of the circle inscribed in it We have:

whence it follows that c 2 =a 2 +b 2 .

in the second

Equating these expressions, we obtain the Pythagorean theorem.

Combined method

Equality of triangles

c 2 = a 2 + b 2 . (3)

Comparing relations (3) and (4), we obtain that

c 1 2 = c 2 , or c 1 = c.

Thus, the triangles - given and constructed - are equal, since they have three correspondingly equal sides. The angle C 1 is right, so the angle C of this triangle is also right.

Ancient Indian evidence.

The mathematicians of ancient India noticed that to prove the Pythagorean theorem, it is enough to use the inside of the ancient Chinese drawing. In the treatise “Siddhanta Shiromani” (“Crown of Knowledge”) written on palm leaves by the largest Indian mathematician of the 20th century. Bha-skara placed a drawing (Fig. 4)

characteristic of Indian evidence l the word "look!". As you can see, right-angled triangles are stacked here with their hypotenuse outward and the square with 2 shifted to the "armchair of the bride" with 2 -b 2 . Note that special cases of the Pythagorean theorem (for example, the construction of a square whose area is twice as large fig.4 area of ​​this square) are found in the ancient Indian treatise "Sulva";

They solved a right triangle and squares built on its legs, or, in other words, figures made up of 16 identical isosceles right triangles and therefore fit into a square. That's a lily. a small fraction of the riches hidden in the pearl of ancient mathematics - the Pythagorean theorem.

Ancient Chinese evidence.

Mathematical treatises of ancient China have come down to us in the edition of the 2nd century. BC. The fact is that in 213 BC. Chinese emperor Shi Huang-di, seeking to eliminate the old traditions, ordered to burn all ancient books. In P c. BC. paper was invented in China and at the same time the reconstruction of ancient books began. The key to this proof is not difficult to find. Indeed, in the ancient Chinese drawing there are four equal right-angled triangles with legs a, b and hypotenuse with stacked G) so that their outer contour forms Fig. 2 a square with sides a + b, and the inner one is a square with side c, built on the hypotenuse (Fig. 2, b). If a square with side c is cut out and the remaining 4 shaded triangles are placed in two rectangles (Fig. 2, in), it is clear that the resulting void, on the one hand, is equal to With 2 , and on the other - with 2 +b 2 , those. c 2 \u003d  2 + b 2. The theorem has been proven. Note that with such a proof, the constructions inside the square on the hypotenuse, which we see in the ancient Chinese drawing (Fig. 2, a), are not used. Apparently, the ancient Chinese mathematicians had a different proof. Precisely if in a square with a side with two shaded triangles (Fig. 2, b) cut off and attach the hypotenuses to the other two hypotenuses (Fig. 2, G), it is easy to find that

The resulting figure, sometimes referred to as the "bride's chair", consists of two squares with sides a and b, those. c 2 == a 2 +b 2 .

H Figure 3 reproduces a drawing from the treatise "Zhou-bi ...". Here the Pythagorean theorem is considered for the Egyptian triangle with legs 3, 4 and hypotenuse 5 units. The square on the hypotenuse contains 25 cells, and the square inscribed in it on the larger leg contains 16. It is clear that the remaining part contains 9 cells. This will be the square on the smaller leg.

1

Shapovalova L.A. (station Egorlykskaya, MBOU ESOSH No. 11)

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This academic year, I got acquainted with an interesting theorem, known, as it turned out, from ancient times:

"The square built on the hypotenuse of a right triangle is equal to the sum of the squares built on the legs."

Usually the discovery of this statement is attributed to the ancient Greek philosopher and mathematician Pythagoras (VI century BC). But the study of ancient manuscripts showed that this statement was known long before the birth of Pythagoras.

I wondered why, in this case, it is associated with the name of Pythagoras.

Relevance of the topic: The Pythagorean theorem is of great importance: it is used in geometry literally at every step. I believe that the works of Pythagoras are still relevant, because wherever we look, everywhere we can see the fruits of his great ideas, embodied in various branches of modern life.

The purpose of my research was: to find out who Pythagoras was, and what relation he has to this theorem.

Studying the history of the theorem, I decided to find out:

Are there other proofs of this theorem?

What is the significance of this theorem in people's lives?

What role did Pythagoras play in the development of mathematics?

From the biography of Pythagoras

Pythagoras of Samos is a great Greek scientist. Its fame is associated with the name of the Pythagorean theorem. Although now we already know that this theorem was known in ancient Babylon 1200 years before Pythagoras, and in Egypt 2000 years before him a right-angled triangle with sides 3, 4, 5 was known, we still call it by the name of this ancient scientist.

Almost nothing is known for certain about the life of Pythagoras, but a large number of legends are associated with his name.

Pythagoras was born in 570 BC on the island of Samos.

Pythagoras had a handsome appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received for always speaking correctly and convincingly, like a Greek oracle. (Pythagoras - "persuasive speech").

In 550 BC, Pythagoras makes a decision and goes to Egypt. So, an unknown country and an unknown culture opens up before Pythagoras. Much amazed and surprised Pythagoras in this country, and after some observations of the life of the Egyptians, Pythagoras realized that the path to knowledge, protected by the caste of priests, lies through religion.

After eleven years of study in Egypt, Pythagoras goes to his homeland, where along the way he falls into Babylonian captivity. There he gets acquainted with the Babylonian science, which was more developed than the Egyptian. The Babylonians knew how to solve linear, quadratic and some types of cubic equations. Having escaped from captivity, he could not stay long in his homeland because of the atmosphere of violence and tyranny that reigned there. He decided to move to Croton (a Greek colony in northern Italy).

It is in Croton that the most glorious period in the life of Pythagoras begins. There he established something like a religious-ethical brotherhood or a secret monastic order, whose members were obliged to lead the so-called Pythagorean way of life.

Pythagoras and the Pythagoreans

Pythagoras organized in a Greek colony in the south of the Apennine Peninsula a religious and ethical brotherhood, such as a monastic order, which would later be called the Pythagorean Union. The members of the union had to adhere to certain principles: firstly, to strive for the beautiful and glorious, secondly, to be useful, and thirdly, to strive for high pleasure.

The system of moral and ethical rules, bequeathed by Pythagoras to his students, was compiled into a kind of moral code of the Pythagoreans "Golden Verses", which were very popular in the era of Antiquity, the Middle Ages and the Renaissance.

The Pythagorean system of studies consisted of three sections:

Teachings about numbers - arithmetic,

Teachings about figures - geometry,

Teachings about the structure of the universe - astronomy.

The education system laid down by Pythagoras lasted for many centuries.

The school of Pythagoras did much to give geometry the character of a science. The main feature of the Pythagorean method was the combination of geometry with arithmetic.

Pythagoras dealt a lot with proportions and progressions and, probably, with the similarity of figures, since he is credited with solving the problem: “Construct a third one, equal in size to one of the data and similar to the second, based on the given two figures.”

Pythagoras and his students introduced the concept of polygonal, friendly, perfect numbers and studied their properties. Arithmetic, as a practice of calculation, did not interest Pythagoras, and he proudly declared that he "put arithmetic above the interests of the merchant."

Members of the Pythagorean Union were residents of many cities in Greece.

The Pythagoreans also accepted women into their society. The Union flourished for more than twenty years, and then the persecution of its members began, many of the students were killed.

There were many different legends about the death of Pythagoras himself. But the teachings of Pythagoras and his disciples continued to live.

From the history of the creation of the Pythagorean theorem

It is currently known that this theorem was not discovered by Pythagoras. However, some believe that it was Pythagoras who first gave its full proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Elements. On the other hand, Proclus claims that the proof in the Elements is due to Euclid himself. As we can see, the history of mathematics has almost no reliable concrete data on the life of Pythagoras and his mathematical activity.

Let's start our historical review of the Pythagorean theorem with ancient China. Here the mathematical book of Chu-pei attracts special attention. This essay says this about the Pythagorean triangle with sides 3, 4 and 5:

"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."

It is very easy to reproduce their method of construction. Take a rope 12 m long and tie it to it along a colored strip at a distance of 3 m. from one end and 4 meters from the other. A right angle will be enclosed between sides 3 and 4 meters long.

Geometry among the Hindus was closely connected with the cult. It is highly probable that the hypotenuse squared theorem was already known in India around the 8th century BC. Along with purely ritual prescriptions, there are works of a geometrically theological nature. In these writings, dating back to the 4th or 5th century BC, we meet with the construction of a right angle using a triangle with sides 15, 36, 39.

In the Middle Ages, the Pythagorean theorem defined the limit, if not of the greatest possible, then at least of good mathematical knowledge. The characteristic drawing of the Pythagorean theorem, which is now sometimes turned by schoolchildren, for example, into a top hat dressed in a robe of a professor or a man, was often used in those days as a symbol of mathematics.

In conclusion, we present various formulations of the Pythagorean theorem translated from Greek, Latin and German.

Euclid's theorem reads (literal translation):

"In a right triangle, the square of the side spanning the right angle is equal to the squares on the sides that enclose the right angle."

As you can see, in different countries and different languages ​​there are different versions of the formulation of the familiar theorem. Created at different times and in different languages, they reflect the essence of one mathematical pattern, the proof of which also has several options.

Five Ways to Prove the Pythagorean Theorem

ancient chinese evidence

In an ancient Chinese drawing, four equal right-angled triangles with legs a, b and hypotenuse c are stacked so that their outer contour forms a square with side a + b, and the inner one forms a square with side c, built on the hypotenuse

a2 + 2ab + b2 = c2 + 2ab

Proof by J. Gardfield (1882)

Let us arrange two equal right-angled triangles so that the leg of one of them is a continuation of the other.

The area of ​​the trapezoid under consideration is found as the product of half the sum of the bases and the height

On the other hand, the area of ​​the trapezoid is equal to the sum of the areas of the resulting triangles:

Equating these expressions, we get:

The proof is simple

This proof is obtained in the simplest case of an isosceles right triangle.

Probably, the theorem began with him.

Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true.

For example, for the triangle ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two. The theorem has been proven.

Proof of the ancient Hindus

A square with a side (a + b), can be divided into parts either as in fig. 12. a, or as in fig. 12b. It is clear that parts 1, 2, 3, 4 are the same in both figures. And if equals are subtracted from equals (areas), then equals will remain, i.e. c2 = a2 + b2.

Euclid's proof

For two millennia, the most common was the proof of the Pythagorean theorem, invented by Euclid. It is placed in his famous book "Beginnings".

Euclid lowered the height BH from the vertex of the right angle to the hypotenuse and proved that its extension 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.

Application of the Pythagorean Theorem

The significance of the Pythagorean theorem lies in the fact that most of the theorems of geometry can be derived from it or with its help and many problems can be solved. In addition, the practical significance of the Pythagorean theorem and its inverse theorem is that they can be used to find the lengths of segments without measuring the segments themselves. This, as it were, opens the way from a straight line to a plane, from a plane to volumetric space and beyond. It is for this reason that the Pythagorean theorem is so important for humanity, which seeks to discover more dimensions and create technologies in these dimensions.

Conclusion

The Pythagorean theorem is so famous that it is difficult to imagine a person who has not heard about it. I learned that there are several ways to prove the Pythagorean theorem. I studied a number of historical and mathematical sources, including information on the Internet, and realized that the Pythagorean theorem is interesting not only for its history, but also because it occupies an important place in life and science. This is evidenced by the various interpretations of the text of this theorem given by me in this paper and the ways of its proofs.

So, the Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable in that in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly on the drawing. But no matter how much you look at a right triangle, you will never see that there is a simple relation between its sides: c2 = a2 + b2. Therefore, visualization is often used to prove it. The merit of Pythagoras was that he gave a full scientific proof of this theorem. The personality of the scientist himself, whose memory is not accidentally preserved by this theorem, is interesting. Pythagoras is a wonderful speaker, teacher and educator, the organizer of his school, focused on the harmony of music and numbers, goodness and justice, knowledge and a healthy lifestyle. He may well serve as an example for us, distant descendants.

Bibliographic link

Tumanova S.V. SEVERAL WAYS TO PROVE THE PYTHAGOREAN THEOREM // Start in science. - 2016. - No. 2. - P. 91-95;
URL: http://science-start.ru/ru/article/view?id=44 (date of access: 02/28/2020).

The text of the work is placed without images and formulas.
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Introduction

In the school course of geometry, using the Pythagorean theorem, only mathematical problems are solved. Unfortunately, the question of the practical application of the Pythagorean theorem is not considered.

In this regard, the purpose of my work was to find out the scope 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 of qualitative and quantitative research that allow solving problems put forward by practice.

I will consider examples of the practical application of the Pythagorean theorem. I will not try to give all examples of using the theorem - it would hardly be possible. The area of ​​application of the theorem is quite extensive and generally cannot be indicated with sufficient completeness.

Hypothesis:

Using the Pythagorean theorem, you can solve not only mathematical problems.

For this research work, the following goal is defined:

Find out the scope of the Pythagorean theorem.

Based on the above goal, the following tasks were identified:

Collect information on the practical application of the Pythagorean theorem in various sources and determine the areas of application of the theorem.

Learn some historical information about Pythagoras and his theorem.

Show the application of the theorem in solving historical problems.

Process the collected data on the topic.

I was engaged in the search and collection of information - I studied printed material, worked with material on the Internet, and processed the collected data.

Research methodology:

The study of theoretical material.

The study of research methods.

Practical implementation of the study.

Communicative (method of measurement, questioning).

Project type: information research. The work was done in my spare time.

About Pythagoras.

Pythagoras is an ancient Greek philosopher, mathematician, and astronomer. He substantiated many properties of geometric figures, developed the mathematical theory of numbers and their proportions. He made a significant contribution to the development of astronomy and acoustics. Author of "Golden Verses", founder of the Pythagorean school in Croton.

According to legend, Pythagoras was born around 580 BC. e. on the island of Samos in a wealthy merchant family. His mother, Pythasis, got her name in honor of the Pythia, the priestess of Apollo. The Pythia predicted to Mnesarchus and his wife the birth of a son, the son was also named after the Pythia. According to many ancient testimonies, the boy was fabulously handsome and soon showed his outstanding abilities. He received his first knowledge from his father Mnesarchus, a jeweler and gem carver, who dreamed that his son would continue his work. But life judged otherwise. The future philosopher showed great aptitude for the sciences. Among the teachers of Pythagoras were Pherekides of Syros and the elder Germodamant. The first instilled in the boy a love for science, and the second for music, painting and poetry. Subsequently, Pythagoras met the famous philosopher - mathematician Thales of Miletus and, on his advice, went to Egypt - the center of the then scientific and research activities. After living 22 years in Egypt and 12 years in Babylon, he returned to the island of Samos, then left it for unknown reasons and moved to the city of Croton, in southern Italy. Here he created the Pythagorean school (union), which studied various issues of philosophy and mathematics. At the age of about 60, Pythagoras married Theano, one of his students. They have three children, and they all become followers of their father. The historical conditions of that time are characterized by a broad movement of the demos against the power of the aristocrats. Fleeing from the waves of popular anger, Pythagoras and his students moved to the city of Tarentum. According to one version: Kilon, a rich and evil man, came to him, wanting to drunkenly join the brotherhood. Having been refused, Cylon began a fight with Pythagoras. During the fire, the students at their cost saved the life of the teacher. Pythagoras became homesick and soon committed suicide.

It should be noted that this is one of the variants of his biography. The exact dates of his birth and death have not been established, many facts of his life are contradictory. But one thing is clear: this man lived, and left to his descendants a great philosophical and mathematical heritage.

Pythagorean theorem.

The Pythagorean theorem is the most important statement of geometry. The theorem is formulated as follows: the area of ​​a square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on its legs.

The discovery of this statement is attributed to Pythagoras of Samos (XII century BC)

The study of Babylonian cuneiform tablets and ancient Chinese manuscripts (copies of even more ancient manuscripts) showed that the famous theorem was known long before Pythagoras, perhaps several millennia before him.

(But there is an assumption that Pythagoras gave her a full proof)

But there is another opinion: in the Pythagorean school it was a wonderful custom to attribute all the merits to Pythagoras and somewhat not appropriate the glory of the discoverers, except perhaps in a few cases.

(Iamblichus-Syriac Greek-speaking writer, author of the treatise "The Life of Pythagoras." (II century AD)

So the German historian of mathematics Kantor believes that the equality 3 2 + 4 2= 5 2 was

known to the Egyptians around 2300 BC. e. during the time of King Amenechmet (according to papyrus 6619 of the Berlin Museum). Some believe that Pythagoras gave the theorem a full proof, while others deny him this merit.

Some attribute to Pythagoras the proof given by Euclid in his Elements. On the other hand, Proclus (mathematician, 5th century) claims that the proof in the "Principles" belonged to Euclid himself, that is, the history of mathematics has almost not preserved reliable data on the mathematical activity of Pythagoras. In mathematics, perhaps, there is no other theorem that deserves all kinds of comparisons.

In some lists of the "Beginnings" of Euclid, this theorem was called the "nymph theorem" for the similarity of the drawing with a bee, butterfly ("butterfly theorem"), which in Greek was called a nymph. The Greeks also called this word some other goddesses, as well as young women and brides. The Arabic translator did not pay attention to the drawing and translated the word "nymph" as "bride". This is how the affectionate name "the bride's theorem" appeared. There is a legend that when Pythagoras of Samos proved his theorem, he thanked the gods by sacrificing 100 bulls. Hence another name - "theorem of a hundred bulls."

In English-speaking countries, it was called: "windmill", "peacock tail", "bride's chair", "donkey bridge" (if the student could not "cross" it, then he was a real "donkey")

In pre-revolutionary Russia, the drawing of the Pythagorean theorem for the case of an isosceles triangle was called "Pythagorean pants".

These "pants" appear when, on each side of a right triangle, build squares to the outside.

How many different proofs of the Pythagorean theorem are there?

Since the time of Pythagoras, more than 350 of them have appeared. The theorem was included in the Guinness Book of Records. If we analyze the proofs of the theorem, then they use few fundamentally different ideas.

Areas of application of the theorem.

It is widely used in solving geometric tasks.

It is with its help that you can geometrically find the values ​​of the square roots of integers:

To do this, we build a right triangle AOB (angle A is 90 °) with unit legs. Then its hypotenuse is √2. Then we build a single segment BC, BC is perpendicular to OB, the length of the hypotenuse OS=√3, etc.

(this method is found in Euclid and F. Kirensky).

Tasks in the course physics high school require knowledge of the Pythagorean theorem.

These are tasks related to the addition of velocities.

Pay attention to the slide: a task from a 9th grade physics textbook. In a practical sense, it can be formulated as follows: at what angle to the river flow should a boat carrying passengers between piers move in order to meet the schedule? (The piers are located on opposite banks of the river)

When a biathlete shoots at a target, he makes a "wind correction". If the wind blows from the right, and the athlete shoots in a straight line, then the bullet will go to the left. To hit the target, you need to move the sight to the right by the bullet displacement distance. Special tables have been compiled for them (based on the consequences of Comrade Pythagoras). The biathlete knows at what angle to shift the sight at a known wind speed.

Astronomy - also a wide area for application of the theorem path of the light beam. The figure shows the path of a light beam from A to B and back. The path of the beam is shown with a curved arrow for clarity, in fact, the light beam is straight.

What is the path of the beam? Light travels back and forth the same way. What is half the path the ray travels? If we mark the segment AB symbol l, half the time as t, and also denoting the speed of light by the letter c, then our equation will take the form

c*t=l

This is the product of the time spent on the speed!

Now let's try to look at the same phenomenon from another frame of reference, for example, from a spacecraft flying past a traveling beam with a speed v. With such an observation, the velocities of all bodies will change, and the stationary bodies will begin to move with a speed v in the opposite direction. Suppose the ship is moving to the left. Then the two points between which the bunny runs will move to the right with the same speed. Moreover, while the bunny runs its way, the starting point A shifts and the beam returns to a new point C.

Question: how much time will the point move (to turn into point C) while the light beam travels? More precisely: what is half of this offset equal to? If we denote half the travel time of the beam by the letter t", and half the distance AC letter d, then we get our equation in the form:

v * t" = d

letter v indicates the speed of the spacecraft.

Another question: what path will the ray of light travel in this case?(More precisely, what is half of this path? What is the distance to the unknown object?)

If we denote half the length of the path of light by the letter s, then we get the equation:

c * t" = s

Here c is the speed of light, and t" is the same time as discussed above.

Now consider the triangle ABC. It is an isosceles triangle whose height is l, which we introduced when considering the process from a fixed point of view. Since the movement is perpendicular l, then it could not affect her.

Triangle ABC composed of two halves - identical right-angled triangles, the hypotenuses of which AB and BC must be connected with the legs according to the Pythagorean theorem. One of the legs is d, which we just calculated, and the second leg is s, which the light passes through, and which we also calculated. We get the equation:

s 2 = l 2 +d 2

This is Pythagorean theorem!

Phenomenon stellar aberration, discovered in 1729, lies in the fact that all the stars in the celestial sphere describe ellipses. The semi-major axis of these ellipses is observed from Earth at an angle of 20.5 degrees. This angle is associated with the movement of the Earth around the Sun at a speed of 29.8 km per hour. In order to observe a star from a moving Earth, it is necessary to tilt the telescope tube forward along the movement of the star, since while the light travels the length of the telescope, the eyepiece moves forward along with the earth. The addition of the speeds of light and the Earth is done vectorially, using the so-called.

Pythagoras. U 2 \u003d C 2 + V 2

C is the speed of light

V-ground speed

telescope tube

At the end of the nineteenth century, various assumptions were made about the existence of inhabitants of Mars similar to humans, this was the result of the discoveries of the Italian astronomer Schiaparelli (he opened channels on Mars that were considered artificial for a long time). Naturally, the question of whether it is possible to communicate with these hypothetical creatures with the help of light signals caused a lively discussion. The Paris Academy of Sciences even established a prize of 100,000 francs for the first person to establish contact with some inhabitant of another celestial body; this award is still waiting for the lucky one. As a joke, although not completely unreasonable, it was decided to send a signal to the inhabitants of Mars in the form of the Pythagorean theorem.

It is not known how to do this; but it is obvious to everyone that the mathematical fact expressed by the Pythagorean theorem takes place everywhere, and therefore inhabitants of another world like us should understand such a signal.

mobile connection

Who in today's world does not use a cell phone? Each mobile subscriber is interested in its quality. And the quality, in turn, depends on the height of the antenna of the mobile operator. To calculate in what radius a transmission can be received, we use the Pythagorean theorem.

What is the maximum height of the mobile operator's antenna in order to receive a transmission within a radius of R=200 km? (Earth's radius is 6380 km.)

Decision:

Let be AB= x , BC=R=200 km , OC= r = 6380 km.

OB=OA+ABOB=r+x.

Using the Pythagorean theorem, we get Answer: 2.3 km.

When building houses and cottages, the question often arises about the length of the rafters for the roof, if the beams have already been made. For example: it is planned to build a gable roof in a house (sectional shape). What length should the rafters be if the beams are made AC=8 m., and AB=BF.

Decision:

Triangle ADC is isosceles AB=BC=4 m., BF=4 m. If we assume that FD=1.5 m., then:

A) From triangle DBC: DB=2.5 m.

B) From triangle ABF:

Window

In buildings Gothic and Romanesque style 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. The figure shows a simple example of such a window in the Gothic style. The method of constructing it is very simple: From the figure it is easy to find the centers of six arcs of circles, the radii of which are equal to

window width (b) for external arches

half width, (b/2) for internal arcs

There is still a complete circle touching the four arcs. Since it is enclosed between two concentric circles, its diameter is equal to the distance between these circles, i.e. b / 2 and, therefore, the radius is equal to b / 4. And then it becomes clear

the position of its center.

AT Romanesque architecture the motif shown in the figure is often found. If b still denotes the width of the window, then the radii of the semicircles will be equal to R = b / 2 and r = b / 4. The radius p of the inner circle can be calculated from the right triangle shown in fig. dotted line. The hypotenuse of this triangle, passing through the tangent point of the circles, is equal to b/4+p, one leg is equal to b/4, and the other is b/2-p. By the Pythagorean theorem we have:

(b/4+p) 2 =(b/4) 2 +(b/4-p) 2

b 2 /16+ bp / 2 + p 2 \u003d b 2 / 16 + b 2 / 4 - bp / 2 + p 2,

Dividing by b and bringing like terms, we get:

(3/2)p=b/4, p=b/6.

In the forest industry: for the needs of construction, logs are sawn into timber, while the main task is to get as little waste as possible. The smallest amount of waste will be when the beam has the largest volume. What should be in the section? As can be seen from the solution, the cross section must be square, and Pythagorean theorem and other considerations allow such a conclusion to be drawn.

Bar of the largest volume

Task

From a cylindrical log it is necessary to cut a rectangular beam of the largest volume. What shape should its cross section be (Fig. 23)?

Decision

If the sides of a rectangular section are x and y, then by the Pythagorean theorem

x 2 + y 2 \u003d d 2,

where d is the diameter of the log. The volume of the timber is greatest when its cross-sectional area is greatest, that is, when xy reaches its greatest value. But if xy is the largest, then the product x 2 y 2 will also be the largest. Since the sum x 2 + y 2 is unchanged, then, according to what was proved earlier, the product x 2 y 2 is the largest when

x 2 \u003d y 2 or x \u003d y.

So, the cross section of the beam should be square.

Transport tasks(the so-called optimization tasks; tasks, the solution of which allows answering the question: how to dispose of funds to achieve great benefits)

At first glance, nothing special: measure the height from floor to ceiling at several points, subtract a few centimeters so that the cabinet does not rest against the ceiling. Having done so, in the process of assembling furniture, difficulties may arise. After all, furniture makers assemble the frame by placing the cabinet in a horizontal position, and when the frame is assembled, they raise it to a vertical position. Consider the side wall of the cabinet. The height of the cabinet must be 10 cm less than the distance from the floor to the ceiling, provided that this distance does not exceed 2500 mm. And the depth of the cabinet is 700 mm. Why 10 cm, and not 5 cm or 7, and what does the Pythagorean theorem have to do with it?

So: side wall 2500-100=2400(mm) - the maximum height of the structure.

The side wall in the process of lifting the frame must pass freely both in height and diagonally. By the Pythagorean theorem

AC \u003d √ AB 2 + BC 2

AC= √ 2400 2 + 700 2 = 2500 (mm)

What happens if the cabinet height is reduced by 50mm?

AC= √ 2450 2 + 700 2 = 2548 (mm)

Diagonal 2548 mm. So, you can’t put a closet (you can ruin the ceiling).

Lightning rod.

It is known that a lightning rod protects all objects from lightning, the distance of which from its base does not exceed its doubled height. It is necessary to determine the optimal position of the lightning rod on a gable roof, providing its lowest available height.

According to the Pythagorean theorem h 2 ≥ a 2 +b 2 means h≥(a 2 +b 2) 1/2

Urgently at their summer cottage it is necessary to make a greenhouse for seedlings.

From the boards knocked down a square 1m1m. There are remnants of a film measuring 1.5m1.5m. At what height in the center of the square should the rail be fixed so that the film completely covers it?

1) Diagonal of the greenhouse d == 1.4; 0.7

2) Film diagonal d 1= 2,12 1,06

3) Rail height x= 0,7

Conclusion

As a result of the research, I found out some areas of application of the Pythagorean theorem. I have collected and processed a lot of material from literary sources and the Internet on this topic. I studied some historical information about Pythagoras and his theorem. Yes, indeed, using the Pythagorean theorem, you can solve not only mathematical problems. The Pythagorean theorem has found its application in construction and architecture, mobile communications, and literature.

Study and analysis of sources of information about the Pythagorean theorem

showed that:

a) the exclusive attention of mathematicians and mathematicians to the theorem is based on its simplicity, beauty and significance;

b) the Pythagorean theorem for many centuries serves as an impetus for interesting and important mathematical discoveries (Fermat's theorem, Einstein's theory of relativity);

in) the Pythagorean theorem - is the embodiment of the universal language of mathematics, valid throughout the world;

G) the scope of the theorem is quite extensive and generally cannot be indicated with sufficient completeness;

d) the secrets of the Pythagorean theorem continue to excite humanity and therefore each of us is given a chance to be involved in their disclosure.

Bibliography

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Alexander Danilovich Alexandrov (on his fiftieth birthday),

Alexandrov A.D., Werner A.L., Ryzhik V.I. Geometry, 10 - 11 cells. - M.: Enlightenment, 1992.

Atanasyan L.S. etc. Geometry, 10 - 11 cells. - M.: Enlightenment, 1992.

Vladimirov Yu.S. Space - time: explicit and hidden dimensions. - M.: "Nauka", 1989.

Voloshin A.V. Pythagoras. - M.: Enlightenment, 1993.

Newspaper "Mathematics", No. 21, 2006.

Newspaper "Mathematics", No. 28, 1995.

Geometry: Proc. For 7 - 11 cells. middle school / G.P. Bevz, V.G. Bevz, N.G. Vladimirova. - M.: Enlightenment, 1992.

Geometry: Textbook for 7 - 9 cells. general education Institutions/ L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - 6th ed. - M.: Enlightenment, 1996.

Glazer G.I. History of mathematics at school: IX - Xcl. A guide for teachers. - M.: Enlightenment, 1983.

Additional chapters to the school textbook 8th grade: Textbook for school students. and classes with deepening. study mathematics /L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - M .: Education, 1996.

Yelensky Sh. In the footsteps of Pythagoras. M., 1961.

Kiselev A.P., Rybkin N.A. Geometry: Planimetry: 7 - 9 cells: Textbook and problem book. - M.: Bustard, 1995.

Kline M. Mathematics. Search for Truth: Translation from English. / Ed. and foreword. IN AND. Arshinova, Yu.V. Sachkov. - M.: Mir, 1998.

Liturman V. The Pythagorean theorem. - M., 1960.

Mathematics: Handbook of schoolchildren and students / B. Frank and others; Translation from him. - 3rd ed., stereotype. - M.: Bustard, 2003.

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Perelman Ya. I. Entertaining mathematics. - M.: "Science", 1976.

Ponomareva T.D. Great scientists. - M .: LLC Astrel Publishing House, 2002.

Sveshnikova A. Journey into the history of mathematics. - M., 1995.

Semyonov E.E. We study geometry: Book. For students 6 - 8 cells. middle school - M.: Enlightenment, 1987.

Smyshlyaev V.K. About mathematics and mathematicians. - Mari book publishing house, 1977.

Tuchnin N.P. How to ask a question. - M.: Enlightenment, 1993.

Cherkasov O.Yu. Planimetry at the entrance exam. - M.: Moscow Lyceum, 1996.

Encyclopedic Dictionary of a Young Mathematician. Comp. A.P. Savin. - M.: Pedagogy, 1985.

Encyclopedia for children. T. 11. Mathematics. /Ch. Ed. M.D. Aksenova. - M.: Avanta +, 2001.

The Pythagorean theorem says:

In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:

a 2 + b 2 = c 2,

• a and b- legs forming a right angle.
• with is the hypotenuse of the triangle.

Formulas of the Pythagorean theorem

• a = \sqrt(c^(2) - b^(2))
• b = \sqrt (c^(2) - a^(2))
• c = \sqrt (a^(2) + b^(2))

Proof of the Pythagorean Theorem

The area of ​​a right triangle is calculated by the formula:

S = \frac(1)(2)ab

To calculate the area of ​​an arbitrary triangle, the area formula is:

• p- semiperimeter. p=\frac(1)(2)(a+b+c) ,
• r is the radius of the inscribed circle. For a rectangle r=\frac(1)(2)(a+b-c).

Then we equate the right sides of both formulas for the area of ​​a triangle:

\frac(1)(2) ab = \frac(1)(2)(a+b+c) \frac(1)(2)(a+b-c)

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

2 ab = \left((a+b)^(2) -c^(2) \right)

2ab = a^(2)+2ab+b^(2)-c^(2)

0=a^(2)+b^(2)-c^(2)

c^(2) = a^(2)+b^(2)

Inverse Pythagorean theorem:

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. That is, for any triple of positive numbers a, b and c, such that

a 2 + b 2 = c 2,

there is a right triangle with legs a and b and hypotenuse c.

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It was proved by the scientist mathematician and philosopher Pythagoras.

The meaning of the theorem in that it can be used to prove other theorems and solve problems.

Additional material:

In one thing, you 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 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.

In this case, it is necessary to complete another 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 picture. 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 was born: "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, 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 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 same 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.