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Presentation on the topic: Pythagorean pants are equal in all directions. What are "Pythagorean pants" for?

What are “Pythagorean pants” needed for? The work was completed by 8th grade students

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... Or The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.

This is one of the most famous geometric theorems of antiquity, called the Pythagorean theorem. Almost everyone who has ever studied planimetry knows it even now. The reason for such popularity of the Pythagorean theorem is its simplicity, beauty, and significance. The Pythagorean theorem is simple, but not obvious. This combination of two contradictory principles gives her a special attractive force and makes her beautiful. It is used in geometry literally at every step, and the fact that there are about 500 different proofs of this theorem (geometric, algebraic, mechanical, etc.) indicates its wide application.

The theorem almost everywhere bears the name of Pythagoras, but at present everyone agrees that it was not discovered by Pythagoras. However, some believe that he was the first to give a full proof of it, while others deny him this merit. This theorem was known many years before Pythagoras. Thus, 1500 years before Pythagoras, the ancient Egyptians knew that a triangle with sides 3, 4 and 5 is rectangular, and used this property to construct right angles when planning land plots and building structures.

The proof of the theorem was considered very difficult in the circles of students of the Middle Ages and was called the “donkey’s bridge” or “the flight of the wretched,” and the theorem itself was called “ windmill" or the "Bride's Theorem." Students even drew cartoons and composed poems like this: Pythagorean pants Equal in all directions.

Proof based on the use of the concept of equal size of figures. The figure 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 quadruple the area of ​​a right triangle with legs a, b from the area of ​​the square, then we will be left with equal areas, i.e. 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.

Proof offered by a school textbook. CD is the height of triangle ABC. AC = √ AD*AB AC 2 = AD*AB Similarly, BC 2 = BD*AB Considering that AD + BD = AB, we obtain AC 2 + BC 2 = AD*AB+ BD*AB = (AD+BD)*AB = AB 2 A C B D

Problem No. 1 Two planes took off from the airfield at the same time: one to the west, the other to the south. After two hours, the distance between them was 2000 km. Find the speeds of the planes if the speed of one was 75% of the speed of the other. Solution: According to the Pythagorean theorem: 4x2+(0.75x*2)2=20002 6.25x2=20002 2.5x=2000 x=800 0.75x=0.75*800=600. Answer: 800 km/h; 600 km/h.

Problem No. 2. What should a young mathematician do in order to reliably obtain a right angle? Solution: You can use the Pythagorean theorem and construct a triangle, giving its sides such a length that the triangle turns out to be rectangular. The easiest way to do this is to take strips of length 3, 4 and 5 of any randomly selected equal segments.

Problem No. 3. Find the resultant of three forces of 200 N each, if the angle between the first and second forces and between the second and third forces is 60°. Solution: The modulus of the sum of the first pair of forces is equal to: F1+22=F12+F22+2*F1*F2cosα where α is the angle between vectors F1 and F2, i.e. F1+2=200√ 3 N. As is clear from symmetry considerations, vector F1+2 is directed along the bisector of angle α, therefore the angle between it and the third force is equal to: β=60°+60°/2=90°. Now let’s find the resultant of the three forces: R2=(F3+F1+2) R=400 N. Answer: R=400 N.

Task No. 4. A lightning rod protects from lightning all objects whose distance from its base does not exceed its double height. Determine the optimal position of the lightning rod on a gable roof, ensuring its lowest accessible height. Solution: According to the Pythagorean theorem, h2≥ a2+b2, which means h≥(a2+b2)1/2. Answer: h≥(a2+b2)1/2.

Famous Pythagorean theorem - “in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs”- everyone knows it from school.

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

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

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

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

Let's take a right triangle with vertices X, Y And Z, Where Z- a right angle and drop the perpendicular from right angle Z to the hypotenuse. Here W- the point at which the altitude intersects the hypotenuse.

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

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

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

This is not difficult to prove.

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

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

But, let’s return to our “Pythagorean pants”...

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

Of course, the area of ​​the triangle WITH equal to the sum of the areas of the triangles A And B.

Those. A+ B= WITH

Now let’s divide the figure at the top (“Pythagorean Pants”) into three house figures:

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

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

Thus we have A/a² = B/b² = C/c² .

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

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

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

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

Or a² + b² = c²

And this is it proof of the Pythagorean theorem!

“Pythagorean pants are equal on all sides.
To prove this, we need to film it and show it.”

This poem is known to everyone high school, ever since we studied the famous Pythagorean theorem in geometry class: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. Although Pythagoras himself never wore pants - in those days the Greeks did not wear them. Who is Pythagoras?
Pythagoras of Samos from lat. Pythagoras, Pythian broadcaster (570-490 BC) - ancient Greek philosopher, mathematician and mystic, creator of the religious and philosophical school of the Pythagoreans.
Among the contradictory teachings of his teachers, Pythagoras sought a living connection, a synthesis of a single great whole. He set himself a goal - to find the path leading to the light of truth, that is, to experience life in unity. For this purpose, Pythagoras visited the entire ancient world. He believed that he should expand his already broad horizons by studying all religions, doctrines and cults. He lived among the rabbis and learned much about the secret traditions of Moses, the lawgiver of Israel. Then he visited Egypt, where he was initiated into the Mysteries of Adonis, and, having managed to cross the Euphrates Valley, he stayed for a long time with the Chaldeans to learn their secret wisdom. Pythagoras visited Asia and Africa, including Hindustan and Babylon. In Babylon he studied the knowledge of magicians.
The merit of the Pythagoreans was the promotion of ideas about the quantitative laws of the development of the world, which contributed to the development of mathematical, physical, astronomical and geographical knowledge. The basis of things is Number, Pythagoras taught, to know the world means to know the numbers that control it. By studying numbers, the Pythagoreans developed numerical relationships and found them in all areas human activity. Pythagoras taught in secret and did not leave behind written works. Pythagoras gave great importance number. His philosophical views are largely determined by mathematical concepts. He said: “Everything is a number”, “all things are numbers”, thus highlighting one side in the understanding of the world, namely, its measurability in numerical expression. Pythagoras believed that number controls all things, including moral and spiritual qualities. He taught (according to Aristotle): “Justice... is a number multiplied by itself.” He believed that in every object, in addition to its changeable states, there is an unchangeable being, a certain unchangeable substance. This is the number. Hence the main idea of ​​Pythagoreanism: number is the basis of everything that exists. The Pythagoreans saw in number and in mathematical relationships the explanation hidden meaning phenomena, laws of nature. According to Pythagoras, the objects of thought are more real than objects sensory knowledge, since numbers have a timeless nature, i.e. eternal. They are a kind of reality that stands above the reality of things. Pythagoras says that all properties of an object can be destroyed or changed, except for one numerical property. This property is Unit. Unity is the existence of things, indestructible and indecomposable, unchangeable. Break any object into the smallest particles - each particle will be one. Arguing that numerical being is the only unchanging being, Pythagoras came to the conclusion that all objects are copies of numbers.
Unit is an absolute number. Unit has eternity. The unit does not need to be in any relation to anything else. It exists on its own. Two is only a relation of one to one. All numbers are only
numerical relations of the Unit, its modifications. And all forms of being are only certain sides of infinity, and therefore Units. The original One contains all numbers, therefore, contains the elements of the whole world. Objects are real manifestations of abstract existence. Pythagoras was the first to designate the cosmos with all the things in it as an order that is established by number. This order is accessible to the mind and is recognized by it, which allows you to see the world in a completely new way.
The process of cognition of the world, according to Pythagoras, is the process of cognition of the numbers that control it. After Pythagoras, the cosmos began to be viewed as ordered by the number of the universe.
Pythagoras taught that the human soul is immortal. He came up with the idea of ​​the transmigration of souls. He believed that everything that happens in the world is repeated again and again after certain periods of time, and the souls of the dead, after some time, inhabit others. The soul, as a number, represents the Unit, i.e. the soul is essentially perfect. But every perfection, insofar as it comes into motion, turns into imperfection, although it strives to regain its former perfect state. Pythagoras called deviation from Unity imperfection; therefore Two was considered a cursed number. The soul in man is in a state of comparative imperfection. It consists of three elements: reason, intelligence, passion. But if animals also have intelligence and passions, then only man is endowed with reason (reason). Any of these three sides in a person can prevail, and then the person becomes predominantly either reasonable, or sane, or sensual. Accordingly, he turns out to be either a philosopher, or an ordinary person, or an animal.
However, let's get back to the numbers. Yes, indeed, numbers are an abstract manifestation of the basic philosophical law of the Universe - the Unity of Opposites.
Note. Abstraction serves as the basis for the processes of generalization and concept formation. She - necessary condition categorization. It forms generalized images of reality, which make it possible to identify connections and relationships of objects that are significant for a certain activity.
Unity of Opposites of the Universe consist of Form and Content, Form is a quantitative category, and Content is a qualitative category. Naturally, numbers express quantitative and qualitative categories in abstraction. Hence, addition (subtraction) of numbers is a quantitative component of the abstraction of Forms, and multiplication (division) is a qualitative component of the abstraction of Contents. The numbers of abstraction of Form and Content are in an inextricable connection of the Unity of Opposites.
Let's try to produce mathematical operations, over numbers, establishing an inextricable connection between Form and Content.

So let's look at the number series.
1,2,3,4,5,6,7,8,9. 1+2= 3 (3) 4+5=9 (9)… (6) 7+8=15 -1+5=6 (9). Next 10 – (1+0) + 11 (1+1) = (1+2= 3) - 12 –(1+2=3) (3) 13-(1+3= 4) + 14 –(1 +4=5) = (4+5= 9) (9) …15 –(1+5=6) (6) … 16- (1+6=7) + 17 – (1+7 =8) ( 7+8=15) – (1+5= 6) … (18) – (1+8=9) (9). 19 – (1+9= 10) (1) -20 – (2+0=2) (1+2=3) 21 –(2+1=3) (3) – 22- (2+2= 4 ) 23-(2+3=5) (4+5=9) (9) 24- (2+4=6) 25 – (2+5=7) 26 – (2+6= 8) – 7+ 8= 15 (1+5=6) (6) Etc.
From here we observe a cyclic transformation of Forms, which corresponds to the cycle of Contents - 1st cycle - 3-9-6 - 6-9-3 2nd cycle - 3-9- 6 -6-9-3, etc.
6
9 9
3

The cycles reflect the inversion of the torus of the Universe, where the Opposites of the abstraction numbers of Form and Content are 3 and 6, where 3 determines Compression, and 6 - Stretching. The compromise for their interaction is the number 9.
Next 1,2,3,4,5,6,7,8,9. 1x2=2 (3) 4x5=20 (2+0=2) (6) 7x8=56 (5+6=11 1+1= 2) (9), etc.
The cycle looks like this 2-(3)-2-(6)- 2- (9)… where 2 is the constituent element of the cycle 3-6-9.
Below is the multiplication table:
2x1=2
2x2=4
(2+4=6)
2x3=6
2x4=8
2x5=10
(8+1+0 = 9)
2x6=12
(1+2=3)
2x7=14
2x8=16
(1+4+1+6=12;1+2=3)
2x9=18
(1+8=9)
Cycle -6.6- 9- 3.3 – 9.
3x1=3
3x2=6
3x3=9
3x4=12 (1+2=3)
3x5=15 (1+5=6)
3x6=18 (1+8=9)
3x7=21 (2+1=3)
3x8=24 (2+4=6)
3x9=27 (2+7=9)
Cycle 3-6-9; 3-6-9; 3-6-9.
4x1=4
4x2=8 (4+8=12 1+2=3)
4x3=12 (1+2=3)
4x4=16
4x5=20 (1+6+2+0= 9)
4x6=24 (2+4=6)
4x7=28
4x8= 32 (2+8+3+2= 15 1+5=6)
4x9=36 (3+6=9)
Cycle 3.3 – 9 - 6.6 - 9.
5x1=5
5x2=10 (5+1+0=6)
5x3=15 (1+5=6)
5x4=20
5x5=25 (2+0+2+5=9)
5x6=30 (3+0=3)
5x7=35
5x8=40 (3+5+4+0= 12 1+2=3)
5x9=45 (4+5=9)
Cycle -6.6 – 9 - 3.3- 9.
6x1= 6
6x2=12 (1+2=3)
6x3=18 (1+8=9)
6x4=24 (2+4=6)
6x5=30 (3+0=3)
6x6=36 (3+6=9)
6x7=42 (4+2=6)
6x8=48 (4+8=12 1+2=3)
6x9=54 (5+4=9)
Cycle – 3-9-6; 3-9-6; 3-9.
7x1=7
7x2=14 (7+1+4= 12 1+2=3)
7x3=21 (2+1=3)
7x4=28
7x5=35 (2+8+3+5=18 1+8=9)
7x6=42 (4+2=6)
7x7=49
7x8=56 (4+9+5+6=24 2+4=6)
7x9=63 (6+3=9)
Cycle – 3.3 – 9 – 6.6 – 9.
8x1= 8
8x2=16 (8+1+6= 15 1+5=6.
8x3=24 (2+4=6)
8x4=32
8x5=40 (3+2+4+0 =9)
8x6=48 (4+8=12 1+2=3)
8x7=56
8x8=64 (5+6+6+4= 21 2+1=3)
8x9=72 (7+2=9)
Cycle -6.6 – 9 – 3.3 – 9.
9x1=9
9x2= 18 (1+8=9)
9x3= 27 (2+7=9)
9x4=36 (3+6=9)
9x5=45 (4+5= 9)
9x6=54 (5+4=9)
9x7=63 (6+3=9)
9x8=72 (7+2=9)
9x9=81 (8+1=9).
The cycle is 9-9-9-9-9-9-9-9-9.

Numbers of the qualitative category of Content – ​​3-6-9, indicate the nucleus of an atom with different quantities neutrons, and quantitative categories indicate the number of electrons in an atom. Chemical elements are nuclei whose masses are multiples of 9, and multiples of 3 and 6 are isotopes.
Note. Isotope (from the Greek “equal”, “identical” and “place”) - varieties of atoms and nuclei of the same chemical element with different numbers of neutrons in the nucleus. A chemical element is a collection of atoms with identical nuclear charges. Isotopes are varieties of atoms of a chemical element with equal charge nuclei, but with different mass numbers.

All real objects are made of atoms, and atoms are determined by numbers.
Therefore, it is natural that Pythagoras was convinced that numbers are real objects, and not simple symbols. A number is a certain state of material objects, the essence of a thing. And Pythagoras was right about this.

The potential for creativity is usually attributed to the humanities, leaving the natural science to analysis, a practical approach and the dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity you won’t go far in the “queen of all sciences” - people have known 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 what we know today as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be exciting. And that this adventure is suitable not only for nerds with 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 were 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. What is known is 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 triangle are found in Egyptian sources from the time of Pharaoh Amenemhat 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. This is confirmed by about 367 different pieces of evidence that exist today. In this, no other theorem can compete with it. Among the famous authors of proofs we can recall Leonardo da Vinci and the twentieth US President 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 are somehow 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 consider those proofs of the famous theorem that are based on this science.

Evidence 1

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

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 construct a square consisting of four triangles equal to the original ABC. And on sides AB and BC a square is built, each of which contains two similar triangles.

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

Evidence 2

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

Construct a right triangle with sides a, b and c(Fig. 1). Then construct 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 triangles similar to those in Figure 1. The result is two squares: 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 checked by calculating the area 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 triangles inscribed in the square from the area of ​​a large square with a side (a+b).

Writing all this down, we have: a 2 +b 2 =(a+b) 2 – 2ab. Open the brackets, carry out all the necessary algebraic calculations and get that a 2 +b 2 = a 2 +b 2. In this case, the area inscribed in Fig. 3. square can also be calculated using the traditional formula S=c 2. Those. a 2 +b 2 =c 2– you have proven the Pythagorean theorem.

Evidence 3

The ancient Indian proof itself was 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 observation skills of students and followers: “Look!”

But we will analyze this proof in more detail:

Inside the square, build four right triangles as indicated in the drawing. Let us denote the side of the large square, also known as the hypotenuse, 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 formula for the area of ​​a square S=c 2 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 areas of all four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options for calculating the area of ​​a square to make sure that they give the same result. And this 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 receive the formula of the Pythagorean theorem c 2 =a 2 +b 2. The theorem has been proven.

Proof 4

This curious ancient Chinese proof was called the “Bride’s Chair” - because of the chair-like figure that results from all the constructions:

It uses the drawing that we have already seen in Fig. 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 rectangular 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 will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will make sure 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 c 2 =a 2 +b 2.

Evidence 5

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

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

To do this, continue the leg AC and construct a segment CD, which is equal to the leg AB. Lower the perpendicular AD line segment ED. Segments ED And AC are equal. Connect the dots E And IN, and E And WITH and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tried: 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 up 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=SE– this will allow us to simplify the recording and not overload it. So, S ABED =2*1/2(AB*AC)+1/2ВС 2.

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

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

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proven 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, you can 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 at all 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. Understanding them may be useful to you in further education.

So what are Pythagorean triplets? That's what they call it integers, collected in threes, the sum of the squares of two of which is equal to the third number in the square.

Pythagorean triples can be:

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

Even before our era, the ancient Egyptians were fascinated by the mania for numbers of Pythagorean triplets: in problems they considered a right 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 rectangular by default.

Examples of Pythagorean triplets: (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 is used not only in mathematics, but also in architecture and construction, astronomy and even literature.

First, about construction: the Pythagorean theorem is widely used in problems of various levels of complexity. For example, look at a Romanesque window:

Let us denote the width of the window as b, then the radius of the major semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed through 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 is just useful to calculate R. To do this, we use a right 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 represents the 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 =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 terms by b, we present 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 gable roof. Determine how tall the tower is mobile communications the signal needs to reach a certain settlement. And even install steadily christmas tree on the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

In literature, the Pythagorean theorem has inspired writers since antiquity and continues to do so in our time. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired to write a sonnet:

The light of truth will not dissipate soon,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
It will not cause doubt or controversy.

The wisest when it touches your gaze
Light of truth, thank the gods;
And a hundred bulls, slaughtered, lie -
A return gift from the lucky Pythagoras.

Since then the bulls have been roaring desperately:
Forever alarmed the bull tribe
Event mentioned here.

It seems to them that the time is about to come,
And they will be sacrificed again
Some great theorem.

(translation by Viktor Toporov)

And in the twentieth century, the Soviet writer Evgeny Veltistov, in his book “The Adventures of Electronics,” devoted an entire chapter to proofs of the Pythagorean theorem. And another half chapter to the story about the two-dimensional world that could exist if the Pythagorean theorem became a fundamental law and even a religion for a single world. Living there would be much easier, 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 mathematics teacher Taratar, says: “The main thing in mathematics is the movement of thought, new ideas.” It is precisely this creative flight of thought that gives rise to the Pythagorean theorem - it is not for nothing that it has so many varied proofs. It helps you go beyond the boundaries of the familiar and look at familiar things in a new way.

Conclusion

This article is designed to help you look beyond 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 and other interesting 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 qualify for higher scores in mathematics lessons - information on the subject from additional sources is always highly appreciated.

Secondly, we wanted to help you feel how interesting mathematics is. Make sure 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 independently explore and make 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 useful in your studies? Write to us what you think about the Pythagorean theorem and this article - we will be happy to discuss all this with you.

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

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

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Objectives: Study the biography of Pythagoras. Explore the history and proof of the theorem. Find out how the theorem is used in art. Find historical problems in which the Pythagorean theorem is used. Get acquainted with the attitude of children of different times to this theorem. Create a project.

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Progress of the research Biography of Pythagoras. Commandments and aphorisms of Pythagoras. Pythagorean theorem. History of the theorem. Why are “Pythagorean pants equal in all directions”? Various proofs of the Pythagorean theorem by other scientists. Application of the Pythagorean theorem. Survey. Conclusion.

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

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Commandments and aphorisms of Pythagoras Thought is above all else between people on earth. Do not sit on the grain measure (i.e., do not live idly). When leaving, do not look back (i.e., before death, do not cling to life). Don’t walk down the beaten path (that is, follow not the opinions of the crowd, but the opinions of the few who understand). Don’t keep swallows in your house (i.e., don’t receive guests who are talkative or unrestrained in their language). Be with those who shoulder the burden, do not be with those who dump the burden (i.e., encourage people not to idleness, but to virtue, to work). In the field of life, like a sower, walk with an even and constant step. The true fatherland is where there are good morals. Do not be a member of a learned society: the wisest, when they form a society, become commoners. Honor sacred numbers, weight and measure, like children of graceful equality. Measure your desires, weigh your thoughts, count your words. Do not be surprised at anything: the gods were surprised.

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

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

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

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

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

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Theorem in Ancient Egypt Cantor (the greatest German historian of mathematics) believes that the equality 3² + 4² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhet (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonaptes, or "rope pullers", built right angles using right triangles with sides of 3, 4 and 5.

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

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

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The attitude of ancient children to the proof of the Pythagorean theorem was considered very difficult by students of the Middle Ages. Weak students who memorized the theorems without understanding them, and were therefore nicknamed “donkeys,” were unable to overcome the Pythagorean theorem, which served as an insurmountable bridge for them. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill,” composed poems like “Pythagorean pants are equal on all sides,” and drew cartoons.

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Proof of the theorem The simplest proof of the theorem is obtained in the case of an isosceles right triangle. In fact, it is enough just to look at the mosaic of isosceles right triangles to be convinced of the validity of the theorem. For example, for triangle ABC: the square built on the hypotenuse AC contains 4 original triangles, and the squares built on the sides contain two.

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“Bride's Chair” In the figure, the squares built on the legs are placed in steps, one next to the other. This figure, which appears in evidence dating back to no later than the 9th century AD. e., the Hindus called it the “bride’s chair.”

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Application of the Pythagorean theorem Currently, it is generally recognized that the success of the development of many areas of science and technology depends on the development of various areas of mathematics. An important condition increasing production efficiency is the widespread introduction of mathematical methods into technology and National economy, which involves the creation of new, effective methods quality and quantitative research, which allow solving problems posed by practice.

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

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