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Pythagorean theorem


The Pythagorean theorem connects the three sides of a right-angled triangle with one formula, which is still used today. The theorem says that in a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: a 2 + b 2 = c 2 , where a and b are the legs of the triangle (sides intersecting at right angles), c is the hypotenuse of the triangle. The Pythagorean theorem is applicable in many cases, for example, using this theorem it is easy to find the distance between two points on the coordinate plane.

Lesson presentation

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The purpose of the lesson: the study of the Pythagorean theorem and its application.


  • To introduce students to the life of Pythagoras, his school.
  • Prove the theorem and show various methods of proof.
  • Show the application of the theorem in life
    (flash projects of students).
  • To develop logical thinking, independence and imagination of students.
  • Maintain interest in the subject.

Equipment and materials: multimedia projector, PC, textbook, handout, presentation for the lesson and flash projects of students.
The peculiarity of the lesson is that it is based on flash projects of students using PC.

“Yes, the path of knowledge is not smooth.
But we know from school years,
There are more puzzles than clues
And there is no limit to the search! ”

1. Pythagoras of Samos and the history of the proof of the theorem (5 min.) (Slides 5-9)

The famous Greek philosopher and mathematician Pythagoras of Samos, whose name is called the theorem, lived about 2.5 thousand years ago. The biographical information about Pythagoras that has reached us is fragmentary and far from reliable. Many legends are associated with his name.
It is reliably known that Pythagoras traveled a lot in the countries of the East, visited Egypt, India and Babylon, studied the ancient culture and achievements of science in different countries. Returning to his homeland, Pythagoras organized a circle of youth from representatives of the aristocracy, where they were accepted with great ceremonies after long trials.
As a result of the first lecture given, Pythagoras acquired 2,000 students who did not return home, and together with their wives and children formed a huge school. The Pythagorean theorem and the Pythagorean school admire humanity throughout history, they are dedicated to poems, songs, drawings, paintings. So the artist F.A. Bronnikov (1827-1902) painted the painting "Pythagorean Anthem to the Rising Sun"
A postage stamp was issued in Greece on the occasion of the renaming of the island of Samos into the island of Pythagoreion.
On the stamp is the inscription: “Pythagorean theorem. Ellas. 350 drams. "
This beautiful brand is almost the only one among many thousands of existing ones, which depicts a mathematical fact Pythagoras - This is not a name, but a nickname that the philosopher received for always speaking correctly and convincingly, like the Greek oracle. (Pythagoras - "convincing with a speech.")
There is a legend that states that, having proved his famous theorem, Pythagoras sacrificed a bull to the gods, and according to other sources, even 100 bulls.

2. Various formulations of the Pythagorean theorem translated from Greek, Latin and German (3 min.) (Slides 10-16)

In Euclid, this theorem states (literal translation):
"In a right-angled triangle, the square of the side stretched over a right angle is equal to the squares on the sides enclosing a right angle."

The Latin translation of the Arabic text by Annairitzi (about 900 BC), made by Gerhard of Cremona (early 12th century), translated into Russian reads:
"In any right-angled triangle, a square formed on a side stretched over a right angle is equal to the sum of two squares formed on two sides enclosing a right angle."
In Geometria Culmonensis (circa 1400), the theorem reads as follows:
Also, wird das vierecke Feld, gemessen an der langen Wand, so also gross ist als bei beide Vierecke, bei zwei werden gemessen von den zwei Wanden des deren, bei zwei gemeinde, tretten in dem rechten Winkel. Translated, this means:
"So, the area of ​​the square, measured along the long side, is as large as the two squares, which are measured on two sides of it, adjacent to the right angle."
The modern formulation of the Pythagorean theorem "In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs."

3. Proof of the Pythagorean theorem (5 min.) (Slides 17-20)


1. (a + b) 2 = 4S+ c 2
2. 4S= 4 · 1/2 ab = 2 ab
3. but 2 + 2ab + b 2 = 2ab + c 2
4. but 2 + b 2 = c 2

Middle Ages students considered the proof of the Pythagorean theorem very difficult and called it Dons asinorum - donkey bridge, or elefuga - the flight of the "wretched" since some “wretched” students who did not have serious mathematical training fled from geometry. Weak students who memorized theorems by heart, without understanding, and are called “donkeys” by this, were not able to overcome the Pythagorean theorem, which served as an irresistible bridge for them.

4. Examples of various methods of proving the theorem (5 min.) (Slides 21-29)

Since ancient times, mathematicians have found more and more proofs of the Pythagorean theorem, more and more new ideas for its proofs. Such evidence - more or less strict, more or less obvious - is known for more than one and a half hundred (according to other sources, more than five hundred), but the desire to increase their number has remained. Therefore, the Pythagorean theorem is listed in the Guinness Book of Records.

  • Ancient Chinese evidence.
  • Proof of Euclid.
  • Waldheim Proof
  • Hawkins proof.
  • Gutheil's proof.
  • Proof of Perigal.
  • Proof based on similarity theory.
  • Holes of Hippocrates.
  • Chinese evidence, 1670
  • Proof from the works of Bhaskara.
  • The proof is a model (video).

5. Examples of the application of the Pythagorean theorem in practice (18 min.) (Slides 30-31)

Pythagoras is remarkable in that it is simple, but not obvious. This combination of two conflicting principles and gives it a special attractive force, makes it beautiful. But, in addition, the Pythagorean theorem is of great practical importance: it is applied in geometry at every step. The Pythagorean theorem is one of the most important theorems of geometry. Most theorems can be deduced from or with its help. The theorem itself

  • in planimetry
  • in stereometry
  • in architecture
  • in construction
  • in physics
  • in astronomy
  • in literature

In planimetry:

1. Square with side but and diagonal d.

Consider the application of the Pythagorean theorem to find the diagonal of a square with side but.
By the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs, then d 2 = a 2 + a 2 from: d 2 = 2a 2 d = but 2

2. The diagonal d of the rectangle with sides a and b is calculated similarly to how the hypotenuse of a right triangle with legs is calculated a and b.
By the Pythagorean theorem: d 2 = a 2 + b 2

Consider an example of calculating the diagonal of a rectangle with sides of 5 cm and 12 cm.

3. Height h equilateral triangle with side but can be considered as a leg of a right triangle whose hypotenuse butand another cathet a/ 2. Thus, according to the Pythagorean theorem

but 2 = h 2 + (1/2a) 2
h 2 = a 2 – (1/2a) 2 h = 1/2a3

Consider an example of calculating the height length in an equilateral triangle with a side of 4 cm.

In stereometry:

Calculation of the diagonal length of a rectangular parallelepiped

In architecture:

In buildings of the Gothic and Romanesque style, the upper parts of the windows are divided by stone ribs, which not only play the role of an ornament, but also contribute to the strength of the windows. The figure shows a simple example of such a window in the Gothic style.
The way to build it is very simple: From the figure, it is easy to find the centers of six arcs of circles whose radii are equal to the width of the window (b) for external arcs and half width (b/ 2), for internal arcs. There remains a complete circle touching the four arcs. Since it is enclosed between two concentric circles, its diameter is equal to the distance between these circles, i.e. b/ 2 and therefore the radius is b/four. And then the position of its center becomes clear. In the considered example, the radii were without any difficulties. In other similar examples, calculations may be required; we show how the Pythagorean theorem is used in such problems.
In Romanesque architecture, the motif presented in the figure is often found. If a b still denotes the width of the window, then the radii of the semicircles will be equal to R = b/ 2 and r = b/four. Radius p the inner circle can be calculated from the right triangle shown in fig. dotted line. The hypotenuse of this triangle passing through the tangent point of the circles is b/4 + p, one leg is equal b/ 4 and the other b/2 – p.
By the Pythagorean theorem, we have: (b/4 + p) 2 = (b/4) 2 + (b/2 – p) 2
Having solved this equation, it is easy to find the radius of the inner circle R = b/6

In construction:

Perhaps someone will consider the application of the Pythagorean theorem to be purely theoretical. But this is not so. If, for example, we consider a triangular prism as the roof of a tower, then in our first question we are talking about how long it is necessary to make side ribs so that the prescribed roof height is maintained at a given area of ​​the attic. Note that the calculation of the roof area can be greatly simplified if you use one very simple rule, valid in all cases when all the roof slopes, no matter how many there are, have the same slope. It reads:
To find the surface area of ​​a gable roof, all slopes of which have an equal slope, you need to multiply the area of ​​the attic Sh to the length of the rafters and divide by half the width of the house.
For example, during the construction of any structure, distances, centers of gravity, placement of supports, beams, etc. are calculated. In general, the significance of the theorem, apart from the above, is that it is used in almost all modern technologies, and also opens up scope for creating and inventing new ones.

In physics:

Lightning rod, lightning rod, a device for the protection of buildings, industrial, transport, utilities, agricultural. and other structures from lightning strikes.
It is known that a lightning rod protects all objects from lightning whose distance from its base does not exceed its double height. It is necessary to determine the optimal position of the lightning rod on a gable roof, ensuring its lowest available height.

  • By the Pythagorean Theorem h 2 >a 2 + b 2 ,
  • mean h>a 2 + b 2

In astronomy:

At the end of the nineteenth century, various assumptions were made about the existence of the inhabitants of Mars like humans. This was the result of the discoveries of the Italian astronomer Schiaparelli (he discovered channels on Mars that had long been considered artificial) and others. Naturally, the question of whether light signals could be explained to these hypothetical creatures caused a lively discussion. The Paris Academy of Sciences even established a prize of 100,000 francs for the first to establish contact with some inhabitant of another celestial body, this award is still waiting for a lucky one. As a joke, although not entirely unfounded, it was decided to transmit to the inhabitants of Mars the Light signal in the form of the Pythagorean theorem.
It is not known how to do this, but it is obvious to everyone that the mathematical fact expressed by the Pythagorean theorem takes place everywhere and therefore inhabitants of another world like us should understand such a signal.

In literature:

Many, with the name Pythagoras, recall his theorem, but few people know that he was related not only to mathematics, but also to literature.
The great mathematician was also a great philosopher of his time.
Here are some of his statements:

  • When doing great, do not promise great.
  • No matter how short the words “yes” and “no” are, they still require the most serious reflection.
  • Do nothing shameful either in the presence of others or in secret.
  • Your first law should be self-respect
  • Do not close your eyes when you want to sleep without understanding all your actions over the past day.
  • Do not go on the dirt road.

Pythagorean theorem has long been widely used in various fields of science, technology and practical life.
Roman architect and engineer Vitruvius, a Greek moral writer o Plutarch, a Greek scientist of the 3rd century wrote about her in his works. Diogenes Laertius, mathematician of the 5th century Proclus and many others.
The legend that in honor of his discovery Pythagoras sacrificed a bull or, as others say, a hundred bulls, served as a reason for humor in the stories of writers and in poems of poets. So, for example, the German novelist A. Chamisso, who at the beginning of the XIX century. participated in a round-the-world trip on the Russian ship "Rurik", wrote the following verses:

Truth be eternal, how soon
A weak person knows her!
And now the Pythagorean theorem
True, as in his distant age.
The sacrifice was plentiful
To the gods from Pythagoras. One hundred bulls
He gave away for a spell and burning
Behind the light, a ray coming from the clouds.
Therefore always since
A little truth is born
The bulls roar, sensing her, after.
They cannot interfere with light.
And they can only, closing their eyes, tremble
For fear that Pythagoras instilled in them.

6. The legend of the death of Pythagoras

The sleepy silence of the night Metapont was cut through by a terrible scream. There was a heavy body falling to the ground, the clatter of running away legs, and everything fell silent. When the night guard arrived at the scene, in the flickering light of torches, everyone saw an old man spread out on the ground, and not far from him was a boy 12 with a face twisted with horror.
- Who is it? - the chief of the guard asked the boy
“This is Pythagoras,” he answered.
- Who is Pythagoras? Among the inhabitants of the city there is no citizen with that name.
- We recently arrived from Croton. My master had to hide from enemies, and left only at night. They tracked him down and killed him.
- How many were there?
- I did not have time to notice this in the dark. They threw me aside and pounced on him. The guard of the guard knelt down and put his hands on the elder’s chest.
“The end,” the chief said.
“The mind alone, as a wise guardian, should be entrusted with its life”

7. Summarizing the lesson. Test (3 min)

Students answer questions:

  • Now I found out that ...
  • Now I can…
  • I didn’t understand how ...
  • I didn’t know that ...
  • Now i know that

Test: (slides 32-34)

- To which triangles can the Pythagorean theorem be applied?

Choose the correct formulation of the Pythagorean theorem:

  1. In a triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
  2. In a right triangle, the square of the hypotenuse is the sum of the legs.
  3. In a right triangle, the hypotenuse is equal to the sum of the squares of the legs.
  4. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

- Is it true that in a right triangle any of the legs is smaller than the hypotenuse?

8. Homework (1 min.) (Slides 35-37)

  1. Learn the statement of the theorem
  2. To be able to prove the Pythagorean theorem
  3. Learn a poem


If a triangle is given to us
And besides, with a right angle,
That is the square of hypotenuse
We will always easily find:
We square the legs,
The sum of the degrees we find -
And in such a simple way
We will come to the result.


According to the historian of mathematics Moritz Kantor, in ancient Egypt during the time of King Amenemhat I (about the XXIII century BC), it was known about a rectangular triangle with sides 3, 4, 5 - it was used by the harpedonapts - “rope tensioners”. In the ancient Babylonian text dating back to the time of Hammurabi (XX century BC), an approximate calculation of the hypotenuse is given. According to van der Waerden, it is very likely that the general relation was known in Babylon already around the 18th century BC. e.

In the ancient Chinese book "Zhou bi xuan jing", attributed to the period of V — III centuries BC. e., there is a triangle with sides 3, 4 and 5, moreover, the image can be interpreted as a graphic justification of the relation of the theorem. In the Chinese collection of problems “Mathematics in nine books” (X — II centuries BC), a separate book is devoted to the application of the theorem.

It is generally accepted that the proof of the correlation was given by the ancient Greek philosopher Pythagoras (570-490 BC). There is evidence of Proclus (412-485 AD) that Pythagoras used algebraic methods to find the Pythagorean triples [⇨], but there were no direct references to the proof of his authorship for five centuries after the death of Pythagoras. However, when authors such as Plutarch and Cicero write about the Pythagorean theorem, it follows from the content that the authorship of Pythagoras is well-known and certain. There is a legend reported by Diogenes of Laertes, according to which Pythagoras allegedly celebrated the discovery of his theorem with a giant feast, having sung a hundred bulls for joy.

Around 400 BC e., according to Proclus, Plato gave a method for finding the Pythagorean triples combining algebra and geometry. Around 300 BC e. in the "Beginnings" of Euclid appeared the oldest axiomatic proof of the Pythagorean theorem.

The main formulation contains algebraic actions - in a right-angled triangle, the legs of which are a < displaystyle a> and b < displaystyle b>, and the length of the hypotenuse is c < displaystyle c>, the following relation holds:

An equivalent geometric formulation is also possible, resorting to the concept of the area of ​​a figure: in a right-angled triangle, the area of ​​a square constructed on the hypotenuse is equal to the sum of the squares of the squares built on the legs. In this form, the theorem is formulated in the Principles of Euclid.

At least 400 proofs of the Pythagorean theorem are recorded in the scientific literature, which is explained by both the fundamental value for geometry and the elementary nature of the result. Основные направления доказательств: алгебраическое использование соотношений элементов треугольника (таков, например, популярный метод подобия

  • [⇨] ), метод площадей
  • [⇨] , существуют также различные экзотические доказательства (например, с помощью дифференциальных уравнений).

    Доказательство Евклида Править

    The classic proof of Euclid is aimed at establishing the equality of the areas between the rectangles formed from the dissection of a square above the hypotenuse from a height from a right angle with squares above the legs.

    Thus, the proof establishes that the area of ​​the square above the hypotenuse composed of the rectangles A H J K < displaystyle AHJK> and B H J I < displaystyle BHJI> is equal to the sum of the squares over the legs.

    Through Squares of Similar Triangles

    The following evidence is based on the fact that the areas of such triangles are referred to as squares of the respective sides.

    area D B A area A B C = A B 2 B C 2. < displaystyle < frac << text<площадь>>

    In the same way, we obtain

    area D A C area A B C = A C 2 B C 2. < displaystyle < frac << text<площадь>>

    Similar geometric shapes on three sides. Edit

    An important geometric generalization of the Pythagorean theorem was given by Euclid in his Beginnings, moving from the squares of squares on the sides to the squares of arbitrary similar geometric figures: the sum of the squares of such figures constructed on catheti will be equal to the area of ​​a similar figure constructed on a hypotenuse.

    The main idea of ​​this generalization is that the area of ​​such a geometric figure is proportional to the square of any of its linear size and, in particular, to the square of the length of either side. Therefore, for such figures with areas A < displaystyle A>, B < displaystyle B> and C < displaystyle C>, built on legs with lengths a < displaystyle a> and b < displaystyle b> and hypotenuse c < displaystyle c> accordingly, the relation is:

    Pappe's Area Theorem Edit

    Pappe's area theorem, which allows for an arbitrary triangle and arbitrary parallelograms on its two sides to construct a parallelogram on the third side so that its area is equal to the sum of the areas of two given parallelograms, can also be considered as a generalization of the Pythagorean theorem: in the case where the original triangle Is rectangular, and squares are given as parallelograms on the legs, the square constructed on the hypotenuse turns out to satisfy the conditions of Papp's area theorem.

    Multidimensional Generalizations Edit

    A generalization of the Pythagorean theorem for three-dimensional Euclidean space is the de Gua theorem: if the tetrahedron has a right angle, then the square of the area of ​​the face lying opposite the right angle is equal to the sum of the squares of the areas of the other three faces. This conclusion can also be generalized as the “n-dimensional Pythagorean theorem” for higher-dimensional Euclidean spaces — for the faces of an orthogonal n < displaystyle n> -dimensional simplex with areas S 1, ..., S n < displaystyle S_ <1>, dots, S_> the orthogonal faces and the opposite face with an area of ​​S 0 < displaystyle S_ <0>> the relation is satisfied:

    Another multidimensional generalization arises from the problem of finding the squared length of the diagonal of a rectangular parallelepiped: to calculate it, the Pythagorean theorem must be applied twice, as a result, it will be the sum of the squared lengths of three adjacent sides of the parallelepiped. In the general case, the diagonal length of an n < displaystyle n> -dimensional rectangular box with adjacent sides with lengths a 1, ..., a n < displaystyle a_ <1>, dots, a_> is:

    as in the three-dimensional case, the result is a consequence of the consistent application of the Pythagorean theorem to right-angled triangles in perpendicular planes.

    A generalization of the Pythagorean theorem for infinite-dimensional space is the Parseval equality.

    Non-Euclidean geometry Edit

    The Pythagorean theorem is derived from the axioms of Euclidean geometry and is invalid for non-Euclidean geometry - the fulfillment of the Pythagorean theorem is equivalent to the Euclidean parallelism postulate.

    In non-Euclidean geometry, the relation between the sides of a right triangle will necessarily be in a form different from the Pythagorean theorem. For example, in spherical geometry, all three sides of a right triangle that limit the octant of the unit sphere are π / 2 < displaystyle pi / 2>, which contradicts the Pythagorean theorem.

    Moreover, the Pythagorean theorem is valid in hyperbolic and elliptic geometry, if the requirement of the rectangularity of the triangle is replaced by the condition that the sum of the two angles of the triangle should be equal to the third.