The Pythagorean theorem is direct. Different Ways to Prove the Pythagorean Theorem

The potential for creativity is usually attributed to the humanities, leaving the natural scientific analysis, practical approach and dry language of formulas and numbers. 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 cliches 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

School textbooks mostly 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 right-angled, 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 evidence was 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 With 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 the right-hand side of the notation: 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 is of 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 triplets: 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 is widely used in it in problems 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 rafters for a 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 the story about the 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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Ways to prove the Pythagorean theorem.

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

"The 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 "bride-lo chair" 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 catheters 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 reliably known about the life of Pythagoras, but it is associated with his name a large number of legends.

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 "placed 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 obtained 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: 21.02.2019).

Those who are interested in the history of the Pythagorean theorem, which is studied in the school curriculum, will also be curious about such a fact as the publication in 1940 of a book with three hundred and seventy proofs of this seemingly simple theorem. But it intrigued the minds of many mathematicians and philosophers of different eras. In the Guinness Book of Records, it is recorded as a theorem with the maximum number of proofs.

History of the Pythagorean theorem

Associated with the name of Pythagoras, the theorem was known long before the birth of the great philosopher. So, in Egypt, during the construction of structures, the ratio of the sides of a right-angled triangle was taken into account five thousand years ago. The Babylonian texts mention the same ratio of the sides of a right triangle 1200 years before the birth of Pythagoras.

The question arises why then the story says - the emergence of the Pythagorean theorem belongs to him? There can be only one answer - he proved the ratio of the sides in the triangle. He did what those who simply used the aspect ratio and the hypotenuse, established by experience, did not do centuries ago.

From the life of Pythagoras

The future great scientist, mathematician, philosopher was born on the island of Samos in 570 BC. Historical documents preserved information about the father of Pythagoras, who was a gem carver, but there is no information about his mother. They said about the born boy that he was an outstanding child who showed a passion for music and poetry from childhood. Historians attribute Hermodamant and Pherekides of Syros to the teachers of young Pythagoras. The first introduced the boy into the world of the Muses, and the second, being a philosopher and founder of the Italian school of philosophy, directed the young man's gaze to the logos.

At the age of 22 (548 BC), Pythagoras went to Naucratis to study the language and religion of the Egyptians. Further, his path lay in Memphis, where, thanks to the priests, having passed through their ingenious tests, he comprehended Egyptian geometry, which, perhaps, prompted the inquisitive young man to prove the Pythagorean theorem. History will later ascribe this name to the theorem.

Captured by the king of Babylon

On his way home to Hellas, Pythagoras is captured by the king of Babylon. But being in captivity benefited the inquisitive mind of the novice mathematician, he had a lot to learn. Indeed, in those years, mathematics in Babylon was more developed than in Egypt. He spent twelve years studying mathematics, geometry and magic. And, perhaps, it was the Babylonian geometry that was involved in the proof of the ratio of the sides of the triangle and the history of the discovery of the theorem. Pythagoras had enough knowledge and time for this. But that this happened in Babylon, there is no documentary confirmation or refutation of this.

In 530 BC Pythagoras flees from captivity to his homeland, where he lives at the court of the tyrant Polycrates in the status of a semi-slave. Such a life does not suit Pythagoras, and he retires to the caves of Samos, and then goes to the south of Italy, where the Greek colony of Croton was located at that time.

Secret monastic order

On the basis of this colony, Pythagoras organized a secret monastic order, which was a religious union and a scientific society at the same time. This society had its charter, which spoke about the observance of a special way of life.

Pythagoras argued that in order to understand God, a person must know such sciences as algebra and geometry, know astronomy and understand music. Research work was reduced to the knowledge of the mystical side of numbers and philosophy. It should be noted that the principles preached at that time by Pythagoras make sense in imitation at the present time.

Many of the discoveries made by the disciples of Pythagoras were attributed to him. Nevertheless, in short, the history of the creation of the Pythagorean theorem by ancient historians and biographers of that time is directly associated with the name of this philosopher, thinker and mathematician.

The teachings of Pythagoras

Perhaps the historians were inspired by the statement of the great Greek that the proverbial triangle with its legs and hypotenuse encoded all the phenomena of our life. And this triangle is the "key" to solving all the problems that arise. The great philosopher said that one should see a triangle, then we can assume that the problem is two-thirds solved.

Pythagoras told about his teaching only to his students orally, without making any notes, keeping it secret. Unfortunately, the teachings of the greatest philosopher have not survived to this day. Some of it has leaked out, but it is impossible to say how much is true and how much is false in what has become known. Even with the history of the Pythagorean theorem, not everything is certain. Historians of mathematics doubt the authorship of Pythagoras, in their opinion, the theorem was used many centuries before his birth.

Pythagorean theorem

It may seem strange, but there are no historical facts of the proof of the theorem by Pythagoras himself - neither in the archives, nor in any other sources. In the modern version, it is believed that it belongs to none other than Euclid himself.

There is evidence of one of the greatest historians of mathematics, Moritz Kantor, who discovered on a papyrus stored in the Berlin Museum, written by the Egyptians around 2300 BC. e. equality, which read: 3² + 4² = 5².

Briefly from the history of the Pythagorean theorem

The formulation of the theorem from the Euclidean "Beginnings" in translation sounds the same as in the modern interpretation. There is nothing new in its reading: the square of the side opposite the right angle is equal to the sum of the squares of the sides adjacent to the right angle. The fact that the ancient civilizations of India and China used the theorem is confirmed by the treatise Zhou Bi Suan Jin. It contains information about the Egyptian triangle, which describes the aspect ratio as 3:4:5.

No less interesting is another Chinese mathematical book "Chu-pei", which also mentions the Pythagorean triangle with an explanation and drawings that coincide with the drawings of the Hindu geometry of Baskhara. About the triangle itself, the book says that if a right angle can be decomposed into its component parts, then the line that connects the ends of the sides will be equal to five, if the base is three, and the height is four.

The Indian treatise "Sulva Sutra", dating back to about the 7th-5th centuries BC. e., tells about the construction of a right angle using the Egyptian triangle.

Proof of the theorem

In the Middle Ages, students considered proving a theorem too difficult. Weak students learned theorems by heart, without understanding the meaning of the proof. In this regard, they received the nickname "donkeys", because the Pythagorean theorem was an insurmountable obstacle for them, like a bridge for a donkey. In the Middle Ages, students came up with a playful verse on the subject of this theorem.

To prove the Pythagorean theorem in the easiest way, you should simply measure its sides, without using the concept of areas in the proof. The length of the side opposite the right angle is c, and the a and b adjacent to it, as a result we get the equation: a 2 + b 2 \u003d c 2. This statement, as mentioned above, is verified by measuring the lengths of the sides of a right triangle.

If we start the proof of the theorem by considering the area of ​​the rectangles built on the sides of the triangle, we can determine the area of ​​the entire figure. It will be equal to the area of ​​a square with a side (a + b), and on the other hand, the sum of the areas of four triangles and the inner square.

(a + b) 2 = 4 x ab/2 + c 2 ;

a 2 + 2ab + b 2 ;

c 2 = a 2 + b 2 , which was to be proved.

The practical significance of the Pythagorean theorem is that it can be used to find the lengths of segments without measuring them. During the construction of structures, distances, placement of supports and beams are calculated, centers of gravity are determined. The Pythagorean theorem is also applied in all modern technologies. They did not forget about the theorem when creating movies in 3D-6D dimensions, where, in addition to the usual 3 values: height, length, width, time, smell and taste are taken into account. How are tastes and smells related to the theorem, you ask? Everything is very simple - when showing a film, you need to calculate where and what smells and tastes to direct in the auditorium.

It's only the beginning. Boundless scope for discovering and creating new technologies awaits inquisitive minds.

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 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 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 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, 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 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.