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 clichés and elementary truths - only in such conditions are all great discoveries born.

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

From the history of the issue

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

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

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

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

Proofs of the Pythagorean theorem

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, move them to opposite sides of the square with side c and attach the hypotenuses to the hypotenuses of the lilac triangles, you get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will see that the "bride's chair" is formed by two squares: small ones with a side b and big with a side a.

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

Proof 5

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

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

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

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

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

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

Let's write both ways to calculate the area of ​​​​a figure by putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify 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 has great importance in geometry. Pythagorean triples are used to solve many mathematical problems. The idea of ​​them can be useful to you in further education.

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

Pythagorean triples can be:

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

Even before our era, the ancient Egyptians were fascinated by the mania for the numbers of Pythagorean 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 finds in it wide application in tasks of different levels of complexity. For example, look at the Romanesque window:

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

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

Using the theorem, you can calculate the length of the rafters for a gable roof. Determine how high the mobile tower needs to be in order for the signal to reach a certain locality. And even stably install 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. Make sure on concrete examples that there is always room for creativity. 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.

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

When you first started learning about square roots and how to solve irrational equations (equalities containing an unknown under the root sign), you probably got the first idea of ​​\u200b\u200btheir practical use. The ability to extract Square root of numbers is also necessary for solving problems on the application of the Pythagorean theorem. This theorem relates the lengths of the sides of any right triangle.

Let the lengths of the legs of a right triangle (those two sides that converge at a right angle) be denoted by the letters and , and the length of the hypotenuse (the longest side of the triangle located opposite the right angle) will be denoted by the letter. Then the corresponding lengths are related by the following relation:

This equation allows you to find the length of a side of a right triangle in the case when the length of its other two sides is known. In addition, it allows you to determine whether the considered triangle is right-angled, provided that the lengths of all three sides are known in advance.

Solving problems using the Pythagorean theorem

To consolidate the material, we will solve the following problems for the application of the Pythagorean theorem.

So given:

1. The length of one of the legs is 48, the hypotenuse is 80.
2. The length of the leg is 84, the hypotenuse is 91.

Let's get to the solution:

a) Substituting the data into the equation above gives the following results:

48 2 + b 2 = 80 2

2304 + b 2 = 6400

b 2 = 4096

b= 64 or b = -64

Since the length of a side of a triangle cannot be expressed as a negative number, the second option is automatically discarded.

Answer to the first picture: b = 64.

b) The length of the leg of the second triangle is found in the same way:

84 2 + b 2 = 91 2

7056 + b 2 = 8281

b 2 = 1225

b= 35 or b = -35

As in the previous case, the negative solution is discarded.

Answer to the second picture: b = 35

We are given:

1. The lengths of the smaller sides of the triangle are 45 and 55, respectively, and the larger ones are 75.
2. The lengths of the smaller sides of the triangle are 28 and 45, respectively, and the larger ones are 53.

We solve the problem:

a) It is necessary to check whether the sum of the squares of the lengths of the smaller sides of a given triangle is equal to the square of the length of the larger one:

45 2 + 55 2 = 2025 + 3025 = 5050

Therefore, the first triangle is not a right triangle.

b) The same operation is performed:

28 2 + 45 2 = 784 + 2025 = 2809

Therefore, the second triangle is a right triangle.

First, find the length of the largest segment formed by points with coordinates (-2, -3) and (5, -2). To do this, we use the well-known formula for finding the distance between points in a rectangular coordinate system:

Similarly, we find the length of the segment enclosed between the points with coordinates (-2, -3) and (2, 1):

Finally, we determine the length of the segment between points with coordinates (2, 1) and (5, -2):

Since there is an equality:

then the corresponding triangle is a right triangle.

Thus, we can formulate the answer to the problem: since the sum of the squares of the sides with the shortest length is equal to the square of the side with the longest length, the points are the vertices of a right triangle.

The base (located strictly horizontally), the jamb (located strictly vertically) and the cable (stretched diagonally) form a right triangle, respectively, the Pythagorean theorem can be used to find the length of the cable:

Thus, the length of the cable will be approximately 3.6 meters.

Given: the distance from point R to point P (the leg of the triangle) is 24, from point R to point Q (hypotenuse) - 26.

So, we help Vitya solve the problem. Since the sides of the triangle shown in the figure are supposed to form a right triangle, you can use the Pythagorean theorem to find the length of the third side:

So, the width of the pond is 10 meters.

Sergey Valerievich

1

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

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3. Zenkevich I.G. "Aesthetics of the Mathematics Lesson". – M.: Enlightenment, 1981.

4. Litzman V. The Pythagorean theorem. - M., 1960.

5. Voloshinov A.V. "Pythagoras". - M., 1993.

6. Pichurin L.F. "Beyond the Pages of an Algebra Textbook". - M., 1990.

7. Zemlyakov A.N. "Geometry in the 10th grade." - M., 1986.

8. Newspaper "Mathematics" 17/1996.

9. Newspaper "Mathematics" 3/1997.

10. Antonov N.P., Vygodskii M.Ya., Nikitin V.V., Sankin A.I. "Collection of tasks for elementary mathematics". - M., 1963.

11. Dorofeev G.V., Potapov M.K., Rozov N.Kh. "Mathematics Handbook". - M., 1973.

12. Shchetnikov A.I. "The Pythagorean doctrine of number and magnitude". - Novosibirsk, 1997.

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15. URL: www.moypifagor.narod.ru/

16. URL: http://www.zaitseva-irina.ru/html/f1103454849.html.

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 has great value: 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 beautiful appearance, wore long beard and on his head a golden diadem. 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 the 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: “Based on the given two figures, construct a third, equal in size to one of the data and similar to the second.”

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 a historical overview of the Pythagorean theorem with ancient China. Here Special attention attracted by the mathematical book of Chu-pei. 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 mantle 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 we see, in different countries and different languages, there are different versions of the formulation of the familiar theorem. Created in different time 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. Besides, practical value the Pythagorean theorem and its inverse theorem is that with their help you can find the lengths of the 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 healthy lifestyle life. 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).

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Right triangle. Complete illustrated guide (2019)

RIGHT TRIANGLE. FIRST LEVEL.

In problems, a right angle is not at all necessary - the lower left one, so you need to learn how to recognize a right triangle in this form,

and in such

and in such

What is good about a right triangle? Well... first of all, there are special beautiful names for his sides.

Attention to the drawing!

Remember and do not confuse: legs - two, and the hypotenuse - only one(the only, unique and longest)!

Well, we discussed the names, now the most important thing: the Pythagorean theorem.

Pythagorean theorem.

This theorem is the key to solving many problems involving a right triangle. It was proved by Pythagoras in completely immemorial times, and since then it has brought many benefits to those who know it. And the best thing about her is that she is simple.

So, Pythagorean theorem:

Do you remember the joke: “Pythagorean pants are equal on all sides!”?

Let's draw these very Pythagorean pants and look at them.

Does it really look like shorts? Well, on which sides and where are they equal? Why and where did the joke come from? And this joke is connected precisely with the Pythagorean theorem, more precisely with the way Pythagoras himself formulated his theorem. And he formulated it like this:

"Sum area of ​​squares, built on the legs, is equal to square area built on the hypotenuse.

Doesn't it sound a little different, doesn't it? And so, when Pythagoras drew the statement of his theorem, just such a picture turned out.

In this picture, the sum of the areas of the small squares is equal to the area of ​​the large square. And so that the children better remember that the sum of the squares of the legs is equal to the square of the hypotenuse, someone witty invented this joke about Pythagorean pants.

Why are we now formulating the Pythagorean theorem

Did Pythagoras suffer and talk about squares?

You see, in ancient times there was no ... algebra! There were no signs and so on. There were no inscriptions. Can you imagine how terrible it was for the poor ancient students to memorize everything with words??! And we can be glad that we have a simple formulation of the Pythagorean theorem. Let's repeat it again to better remember:

Now it should be easy:

The square of the hypotenuse is equal to the sum of the squares of the legs.

Well, the most important theorem about a right triangle was discussed. If you are interested in how it is proved, read the next levels of theory, and now let's move on ... into the dark forest ... of trigonometry! To the terrible words sine, cosine, tangent and cotangent.

Sine, cosine, tangent, cotangent in a right triangle.

In fact, everything is not so scary at all. Of course, the "real" definition of sine, cosine, tangent and cotangent should be looked at in the article. But you really don't want to, do you? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

Why is it all about the corner? Where is the corner? In order to understand this, you need to know how statements 1 - 4 are written in words. Look, understand and remember!

1.
It actually sounds like this:

What about the angle? Is there a leg that is opposite the corner, that is, the opposite leg (for the corner)? Of course have! This is a cathet!

But what about the angle? Look closely. Which leg is adjacent to the corner? Of course, the cat. So, for the angle, the leg is adjacent, and

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

How to put it into words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the cathet? Adjacent to the corner. So what did we get?

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what the legs and hypotenuse are? If not, then look at the picture - refresh your knowledge

It is possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true. How would you prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cunningly we divided its sides into segments of lengths and!

Now let's connect the marked points

Here we, however, noted something else, but you yourself look at the picture and think about why.

What is the area of ​​the larger square? Correctly, . What about the smaller area? Of course, . The total area of ​​the four corners remains. Imagine that we took two of them and leaned against each other with hypotenuses. What happened? Two rectangles. So, the area of ​​"cuttings" is equal.

Let's put it all together now.

Let's transform:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite leg to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite leg to the adjacent leg.

The cotangent of an acute angle is equal to the ratio of the adjacent leg to the opposite leg.

And once again, all this in the form of a plate:

It is very comfortable!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! Here it is very important that the legs are "corresponding". For example, if it goes like this:

THEN THE TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both - opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Look at the topic “and pay attention to the fact that for the equality of “ordinary” triangles, you need the equality of their three elements: two sides and an angle between them, two angles and a side between them, or three sides. But for the equality of right-angled triangles, only two corresponding elements are enough. It's great, right?

Approximately the same situation with signs of similarity of right triangles.

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it happened that

1. - median:

Remember this fact! Helps a lot!

What is even more surprising is that the converse is also true.

What good can be gained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look closely. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But in a triangle there is only one point, the distances from which about all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCUM DEscribed. So what happened?

So let's start with this "besides...".

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What use can be drawn from this "triple" similarity.

Well, for example - two formulas for the height of a right triangle.

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get first formula "Height in a right triangle":

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

Both of these formulas must be remembered very well and the one that is more convenient to apply. Let's write them down again.

Pythagorean theorem:

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

Signs of equality of right triangles:

• on two legs:
• along the leg and hypotenuse: or
• along the leg and the adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one sharp corner: or
• from the proportionality of the two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent one:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

• through the catheters:

An animated proof of the Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It is believed that it was proved by the Greek mathematician Pythagoras, after whom it is named (there are other versions, in particular, an alternative opinion that this theorem is in general view was formulated by the Pythagorean mathematician Hippasus).
The theorem says:

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

Denoting the length of the hypotenuse of the triangle c, and the lengths of the legs as a and b, we get the following formula:

Thus, the Pythagorean theorem establishes a relation that allows you to determine the side of a right triangle, knowing the lengths of the other two. The Pythagorean theorem is a special case of the cosine theorem, which determines the relationship between the sides of an arbitrary triangle.
The converse statement is also proved (also called the inverse Pythagorean theorem):

For any three positive numbers a, b and c such that a ? +b? = c ?, there is a right triangle with legs a and b and hypotenuse c.

Visual evidence for the triangle (3, 4, 5) from Chu Pei 500-200 BC. The history of the theorem can be divided into four parts: knowledge about Pythagorean numbers, knowledge about the ratio of the sides in a right triangle, knowledge about the ratio adjacent corners and proof of the theorem.
Megalithic structures around 2500 BC in Egypt and Northern Europe, contain right-angled triangles with integer sides. Barthel Leendert van der Waerden conjectured that in those days the Pythagorean numbers were found algebraically.
Written between 2000 and 1876 BC papyrus from the Middle Kingdom of Egypt Berlin 6619 contains a problem whose solution is the Pythagorean numbers.
During the reign of Hammurabi the Great, a Vibylonian tablet Plimpton 322, written between 1790 and 1750 BC contains many entries closely related to Pythagorean numbers.
In the Budhayana sutras, which are dated according to different versions to the eighth or second centuries BC. in India, contains Pythagorean numbers derived algebraically, a formulation of the Pythagorean theorem, and a geometric proof for an isosceles right triangle.
The Sutras of Apastamba (circa 600 BC) contain a numerical proof of the Pythagorean theorem using area calculations. Van der Waerden believes that it was based on the traditions of its predecessors. According to Albert Burko, this is the original proof of the theorem and he suggests that Pythagoras visited Arakoni and copied it.
Pythagoras, whose years of life are usually indicated 569 - 475 BC. uses algebraic methods calculation of Pythagorean numbers, according to Proklov's comments on Euclid. Proclus, however, lived between 410 and 485 AD. According to Thomas Giese, there is no indication of authorship of the theorem for five centuries after Pythagoras. However, when authors such as Plutarch or Cicero attribute the theorem to Pythagoras, they do so as if the authorship is widely known and certain.
Around 400 BC According to Proclus, Plato gave a method for calculating Pythagorean numbers, combining algebra and geometry. Around 300 BC, in Beginnings Euclid, we have the oldest axiomatic proof that has survived to this day.
Written sometime between 500 B.C. and 200 BC, the Chinese mathematical book "Chu Pei" (? ? ? ?), gives a visual proof of the Pythagorean theorem, which in China is called the gugu theorem (????), for a triangle with sides (3, 4, 5). During the reign of the Han Dynasty, from 202 BC. before 220 AD Pythagorean numbers appear in the book "Nine Sections of the Mathematical Art" along with a mention of right triangles.
The use of the theorem is first documented in China, where it is known as the gugu theorem (????) and in India, where it is known as Baskar's theorem.
Many are debating whether the Pythagorean theorem was discovered once or repeatedly. Boyer (1991) believes that the knowledge found in the Shulba Sutra may be of Mesopotamian origin.
Algebraic proof
Squares are formed from four right triangles. More than a hundred proofs of the Pythagorean theorem are known. Here the evidence is based on the existence theorem for the area of ​​a figure:

Place four identical right triangles as shown in the figure.
Quadrilateral with sides c is a square, since the sum of two acute angles is , and the straightened angle is .
The area of ​​the whole figure is equal, on the one hand, to the area of ​​a square with side "a + b", and on the other, to the sum of the areas of four triangles and the inner square.

Which is what needs to be proven.
By the similarity of triangles
Use of similar triangles. Let ABC is a right triangle in which the angle C straight, as shown in the picture. Let's draw a height from a point c, and call H point of intersection with a side AB. Triangle formed ACH like a triangle abc, since they are both rectangular (by definition of height) and they share an angle A, obviously the third angle will be the same in these triangles as well. Similarly mirkuyuyuchy, triangle CBH also similar to triangle ABC. From the similarity of triangles: If

This can be written as

If we add these two equalities, we get

HB + c times AH = c times (HB + AH) = c ^ 2, ! Src = "http://upload.wikimedia.org/math/7/0/9/70922f59b11b561621c245e11be0b61b.png" />

In other words, the Pythagorean theorem:

Euclid's proof
Proof of Euclid in the Euclidean "Principles", the Pythagorean theorem proved by the method of parallelograms. Let A, B, C vertices of a right triangle, with a right angle A. Drop a perpendicular from a point A to the side opposite the hypotenuse in a square built on the hypotenuse. The line divides the square into two rectangles, each of which has the same area as the squares built on the legs. The main idea in the proof is that the upper squares turn into parallelograms of the same area, and then go back and turn into rectangles in the lower square and again with the same area.

Let's draw segments CF and AD, we get triangles BCF and BDA.
corners CAB and BAG- straight; points C, A and G are collinear. Same way B, A and H.
corners CBD and FBA- both are straight, then the angle ABD equal to the angle fbc, since both are the sum of a right angle and an angle ABC.
Triangle ABD and FBC level on two sides and the angle between them.
Because the dots A, K and L– collinear, the area of ​​the rectangle BDLK is equal to two areas of the triangle ABD (BDLK) = BAGF = AB2)
Similarly, we get CKLE = ACIH = AC 2
On one side the area CBDE equal to the sum of the areas of the rectangles BDLK and CKLE, on the other hand, the area of ​​the square BC2, or AB 2 + AC 2 = BC 2.

Using Differentials
The use of differentials. The Pythagorean theorem can be arrived at by studying how the increment of a side affects the length of the hypotenuse as shown in the figure on the right and applying a little calculation.
As a result of the growth of the side a, from similar triangles for infinitesimal increments

Integrating we get

If a a= 0 then c = b, so the "constant" is b 2. Then

As can be seen, the squares are due to the proportion between increments and sides, while the sum is the result of the independent contribution of the increments of the sides, not evident from the geometric evidence. In these equations da and dc are, respectively, infinitesimal increments of the sides a and c. But instead of them we use? a and? c, then the limit of the ratio if they tend to zero is da / dc, derivative, and is also equal to c / a, the ratio of the lengths of the sides of the triangles, as a result we obtain a differential equation.
In the case of an orthogonal system of vectors, an equality takes place, which is also called the Pythagorean theorem:

If - These are the projections of a vector onto the coordinate axes, then this formula coincides with the Euclidean distance and means that the length of the vector is equal to the square root of the sum of the squares of its components.
The analogue of this equality in the case of an infinite system of vectors is called Parseval's equality.