Interesting ways to prove the Pythagorean theorem presentation. Presentation on the topic "methods of proving the Pythagorean theorem"

Lesson plan Organizational moment Organizational point Repetition Repetition Message about the life of Pythagoras of Samos Message about the life of Pythagoras of Samos Historical information about the Pythagorean theorem Historical information about the Pythagorean theorem Work on the theorem Work on the theorem Solving problems using the theorem Solving problems using the theorem Summing up the lesson Summing up the lesson Homework Homework

Pythagoras of Samos Pythagoras was born in 580 BC. in Ancient Greece on the island of Samos, which is located in the Aegean Sea off the coast of Asia Minor, which is why he is called Pythagoras of Samos. In the family of a stone carver who found fame rather than wealth. Even as a child, he showed extraordinary abilities, and when he grew up, the young man’s restless imagination became cramped on the small island.

Pythagoras Pythagoras moved to the city of Miletus and became a student of Thales, who at that time was in his eighties. The wise scientist advised the young man to go to Egypt, where he himself once studied science. An unknown country opened up before Pythagoras. He was struck by the fact that in his native Greece the gods were in the form of people, and the Egyptian gods were in the form of half-humans - half-animals. Knowledge was concentrated in temples, access to which was limited.

Pythagoras took years to deeply study Egyptian culture before he was allowed to become acquainted with the centuries-old achievements of Egyptian science. When Pythagoras comprehended the science of the Egyptian priests, he went home to create his own school there. The priests, who did not want to spread their knowledge beyond the temples, did not want to let him go. With great difficulty he managed to overcome this obstacle.

However, on the way home, Pythagoras was captured and ended up in Babylon. The Babylonians valued smart people, so he found his place among the Babylonian sages. Babylonian science was more developed than Egyptian science. The most striking successes were in algebra. Pythagoras The Babylonians invented and used the positional number system when counting, and were able to solve linear, quadratic and some types of cubic equations. Pythagoras lived in Babylon for about ten years and returned to his homeland at the age of forty. But he did not remain on the island of Samos for long. As a sign of protest against the tyrant Polycrates, who then ruled the island, he settled in one of the Greek colonies of Southern Italy in the city of Crotone.

There Pythagoras organized a secret league of youth from representatives of the aristocracy. They were accepted into this union with great ceremonies after long trials. Each entrant renounced his property and swore an oath to keep the founder's teachings secret. The Pythagoreans, as they were later called, studied mathematics, philosophy, and natural sciences. There was a decree at school according to which the authorship of all mathematical works was attributed to the teacher. The Pythagorean alliance was secret. The emblem or identification mark of the union was a pentagram - a five-pointed star. The pentagram was assigned the ability to protect a person from evil spirits.

The Pythagoreans made many important discoveries in arithmetic and geometry. It is also known that in addition to the spiritual and moral development of Pythagoras’s students, he was concerned about their physical development. Not only did he himself participate in the Olympic Games and twice won fist fights, but he also trained a galaxy of great Olympians. Pythagoras The scientist devoted about forty years to the school he created and, according to one version, at the age of eighty Pythagoras was killed in a street fight in time of popular uprising. After his death, the students surrounded the name of their teacher with many legends.

In Babylonian texts it is found 1200 years before Pythagoras. Apparently, he was the first to find its proof. In this regard, the following entry was made: “... when he discovered that in a right triangle the hypotenuse corresponds to the legs, he sacrificed a bull made of wheat dough.” 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.

Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Given: Δ ABC, C = 90° Prove: Proof: D Considering cos B, we obtain: Adding (1) and (2), we obtain: Considering cos B, we obtain: Let us lower the height SD from the vertex of the right angle

Slide 2

a2+b2=c2 c a b P

Slide 3

Pythagoras did not discover this property of a right triangle; he was probably the first to generalize and prove it, thereby transferring it from the field of practice to the field of science. We don't know how he did it. It is assumed that Pythagoras’s proof was not fundamental, but only a confirmation, a test of this property on a number of particular types of triangles, starting with an isosceles right triangle, for which it obviously follows from Fig. 1.

Slide 4

Slide 5

Proofs based on the use of the concept of equal size of figures.

Slide 6

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 equal areas will remain, i.e. c2 = a2 + b2. However, the ancient Hindus, to whom this reasoning belongs, usually did not write it down, but accompanied the drawing with only one word: “look!” It is quite possible that Pythagoras offered the same proof.

Slide 7

Additive evidence. These proofs are based on the decomposition of squares built on the legs into figures from which one can add a square built on the hypotenuse. Einstein's proof (Fig. 3) is based on the decomposition of a square built on the hypotenuse into 8 triangles.

Slide 8

In Fig. 4 shows the proof of the Pythagorean theorem using the partition of al-Nayriziyah, the medieval Baghdad commentator on Euclid’s Elements. In this partition, the square built on the hypotenuse is divided into 3 triangles and 2 quadrilaterals. Here: ABC is a right triangle with right angle C; DE = BF. Prove the theorem using this partition. D E

Slide 9

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

Slide 10

The validity of the Pythagorean theorem follows from the equal size of the hexagons AEDFPB and ACBNMQ. F

Slide 11

In Fig. 13 ABC – rectangular, C – right angle, CM AB, b1 – projection of leg b onto the hypotenuse, a1 – projection of leg a onto the hypotenuse, h – altitude of the triangle drawn to the hypotenuse. Since ABC is similar to ACM, it follows that b2 = c*b1; (1) from the fact that ABC is similar to BCM, it follows that a2 = c*a1. (2) Adding equalities (1) and (2) term by term, we obtain a2 + b2 = c*b1 + c*a1 = c*(b1 + a1) = c2. b

Slide 12

In Figure 15, three right triangles form a trapezoid. Therefore, the area of ​​this figure can be found using the formula for the area of ​​a rectangular trapezoid, or as the sum of the areas of three triangles. Garfield's proof.

Slide 13

Biography of Pythagoras. The great scientist Pythagoras was born around 570 BC. on the island of Samos. Pythagoras's father was Mnesarchus, a gem cutter. The name of Pythagoras' mother is not known. According to many ancient testimonies, the boy who was born was fabulously handsome, and soon showed his extraordinary abilities. Among the teachers of young Pythagoras were the elder Hermodamantus and Pherecydes of Syros. Young Pythagoras spent whole days at the feet of the elder Hermo, listening to the melody of the cithara and the hexameters of Homer. Pythagoras retained his passion for the music and poetry of the great Homer throughout his life. And, being a recognized sage, surrounded by a crowd of disciples, Pythagoras began the day by singing one of Homer’s songs. Pherecydes was a philosopher and was considered the founder of the Italian school of philosophy. But be that as it may, the restless imagination of young Pythagoras very soon became cramped in little Samos, and he went to Miletus, where he met another scientist - Thales. Thales advises him to go to Egypt for knowledge, which Pythagoras did. In 548 BC. Pythagoras arrived in Naucratis, a Samian colony, where there was someone to find shelter and food.

Slide 14

Having studied the language and religion of the Egyptians, he leaves for Memphis. Despite the pharaoh's letter of recommendation, the cunning priests were in no hurry to reveal their secrets to Pythagoras, offering him difficult tests. But, driven by a thirst for knowledge, Pythagoras overcame them all, although according to excavations, the Egyptian priests could not teach him much, because at that time, Egyptian geometry was a purely applied science (satisfying the need of that time for counting and measuring land). Therefore, having learned everything that the priests gave him, he ran away from them and moved to his homeland in Hellas. However, having completed part of the journey, Pythagoras decided on an overland journey, during which he was captured by Cambyses, the king of Babylon, who was heading home. There is no need to dramatize the life of Pythagoras in Babylon, because... the great ruler Cyrus was tolerant of all captives. Babylonian mathematics was undoubtedly more developed (an example of this is the positional system of calculus) than Egyptian, and Pythagoras had a lot to learn. But in 530 BC. Cyrus went on a campaign against the tribes in Central Asia. And, taking advantage of the commotion in the city, Pythagoras fled to his homeland.

Slide 15

And on Samos at that time the tyrant Polycrates reigned. Of course, Pythagoras was not satisfied with the life of a court slave, and he retired to caves in the vicinity of Samos. After several months of claims from Polycrates, Pythagoras moved to Croton. In Croton, Pythagoras established something like a religious-ethical brotherhood or a secret monastic order (“Pythagoreans”), whose members pledged to lead the so-called Pythagorean way of life. It was at the same time a religious union, a political club, and a scientific society. It must be said that some of the principles preached by Pythagoras are worthy of imitation even now. ...20 years have passed. The fame of the brotherhood spread throughout the world. One day, Cylon, a rich but evil man, comes to Pythagoras, wanting to join the brotherhood while drunk. Having received a refusal, Cylon begins to fight Pythagoras, taking advantage of the arson of his house. During the fire, the Pythagoreans saved the life of their teacher at the cost of their own, after which Pythagoras became sad and soon committed suicide.

View all slides

History of the theorem. Ancient China Let's start our historical review with ancient China. Here the mathematical book Chu-pei attracts special attention. This essay talks about the Pythagorean triangle with sides 3, 4 and 5: Let's start with ancient China. Here the mathematical book Chu-pei attracts special attention. This work says this about a 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.” In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara. In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.

Somewhat more is known about the Pythagorean theorem among the Babylonians. In one text dating back to the time of Hammurabi, that is, to 2000 BC. e., an approximate calculation of the hypotenuse of a right triangle is given. From this we can conclude that in Mesopotamia they were able to perform calculations with right triangles, at least in some cases. Geometry among the Hindus, like among the Egyptians and Babylonians, was closely connected with cult. It is very likely that the theorem on the square of the hypotenuse was already known in India around the 18th century BC. e. Ancient India

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 Amenemhat I (according to papyrus 6619 of the Berlin Museum) According to Cantor, the harpedonaptes, or “rope pullers,” built right angles using right triangles with sides 3, 4 and 5. Their method of construction can be very easily reproduced. Let's take a rope 12 meters long and tie a colored stripe to it at a distance of 3 meters from one end and 4 meters from the other. The right angle will be enclosed between sides 3 and 4 meters long.

Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other hand, on a critical study of Greek sources, Van der Waerden (Dutch mathematician) made the following conclusion: “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 turned into an exact science."

The great scientist Pythagoras was born around 570 BC. on the island of Samos. Pythagoras's father was Mnesarchus, a gem cutter. The name of Pythagoras' mother is unknown. According to many ancient testimonies, the boy who was born was fabulously handsome, and soon showed his extraordinary abilities. Pythagoras retained his passion for the music and poetry of the great Homer throughout his life. Soon, the restless imagination of young Pythagoras became cramped in little Samos, and he went to Miletus, where he met another scientist, Thales. Then he goes on a journey and is captured by the Babylonian king Cyrus. In 530 BC. Cyrus went on a campaign against the tribes in Central Asia. And, taking advantage of the commotion in the city, Pythagoras fled to his homeland.

And on Samos at that time the tyrant Polycrates reigned. After several months of claims from Polycrates, Pythagoras moved to Croton. In Croton, Pythagoras established something like a religious-ethical brotherhood or a secret monastic order (“Pythagoreans”), whose members pledged to lead the so-called Pythagorean way of life.... 20 years passed. The fame of the brotherhood spread throughout the world. One day, Cylon, a rich but evil man, comes to Pythagoras, wanting to join the brotherhood while drunk. Having received a refusal, Cylon begins to fight Pythagoras, taking advantage of the arson of his house. During the fire, the Pythagoreans saved the life of their teacher at the cost of their own, after which Pythagoras became sad and soon committed suicide.

Pythagorean theorem. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Other formulations of the theorem. Euclid's theorem states (literal translation): "In a right triangle, the square of the side spanning the right angle is equal to the squares of the sides enclosing the right angle." In Geometria Culmonensis (c. 1400), the translation of the theorem reads: “The area of ​​a square, then, measured along its long side, is as great as that of two squares measured along its two sides adjacent to a right angle.”

The simplest proof. The simplest proof of the theorem is obtained in the simplest 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.

Proof by subtraction method. Let's look at another proof using the subtraction method. Let us enclose the familiar drawing of the Pythagorean theorem in a rectangular frame, the directions of the sides of which coincide with the directions of the legs of the triangle. Let's continue some of the segments of the figure as indicated in the figure, while the rectangle breaks up into several triangles, rectangles and squares. Let's first remove several parts from the rectangle so that only the square built on the hypotenuse remains. These parts are as follows: 1. triangles 1, 2, 3, 4; 2. rectangle 5; 3. rectangle 6 and square 8; 4. rectangle 7 and square 9;

Then we throw out the parts from the rectangle so that only the squares built on the sides remain. These parts will be: 1. rectangles 6 and 7; 2. rectangle 5; 3. rectangle 1 (shaded); 4. rectangle 2 (shaded); All we have to do is show that the taken away parts are equal in size. This is easy to see due to the arrangement of the figures. From the figure it is clear that: 1. rectangle 5 is equal in size to itself; 2. four triangles 1,2,3,4 are equal in size to two rectangles 6 and 7; 3. rectangle 6 and square 8, taken together, are equal in size to rectangle 1 (shaded);; 4. rectangle 7 together with square 9 are equal in size to rectangle 2 (shaded); The theorem is proven

Einstein's proof Points E, C and F lie on the same line; this follows from simple calculations of the degree measure of the angle ECF (it is unfolded). CD is drawn perpendicular to EF. The left and right sides of the square built on the hypotenuse are extended upward until they intersect with EF; side EA is extended until it intersects with CD. Accordingly, equal triangles are equally numbered.

In fact, triangles ABD and BFC are equal in two sides and the angle between them: FB = AB, BC = BD, and the angles between them are equal as obtuse angles with mutually perpendicular sides. S ABD = 0.5 S BJLD, since triangle ABD and rectangle BJLD have a common base BD and a common height LD. Similarly S FBC=0.5 S ABFH (BF-common base, AB-common height). Hence, taking into account that S ABD= S FBC, we have S BJLD= S ABFH. Similarly, if you draw the segment AE using the equality of triangles ВСК and ACE, you will prove that S JCEL = S ACKG. So, S ABFH+ S ACKG= S BJLD+ S JCEL= S BCED, which is what needed to be proven. This proof was given by Euclid in his Elements. According to Proclus (Byzantium), it was invented by Euclid himself. Euclid's proof is given in sentence 47 of the first book of the Elements. On the hypotenuse and legs of the right triangle ABC, the corresponding squares are constructed and it is proved that the rectangle BJLD is equal to the square ABFH, and the rectangle JCEL is equal to the square AGKC. Then the sum of the areas of the squares on the legs will be equal to the area of ​​the square on the hypotenuse.

The second mystery is the precisely unknown number of proofs of the famous theorem of Pythagoras of Samos. It was for this reason that I decided to conduct a sociological survey, which showed that most people of the older generation agree with the existence of 250 proofs, although I know from additional sources that there are more than 350 proofs of this theorem, which is why it even got into the Guinness Book of Records! But, of course, relatively few fundamentally different ideas are used in these proofs.

The third secret is that the Pythagorean theorem is a symbol of mathematics today. The fourth secret - the Pythagorean theorem - provides us with a wealth of material for generalization - the most important type of mental activity, the basis of theoretical thinking, which many scientists are fluent in. Here we can add that from the Pythagorean theorem one can move on to other theorems.

The fifth secret is that some researchers attribute to Pythagoras the proof that Euclid gave in the first book of his Elements. On the other hand, Proclus (5th century mathematician) argued that the proof in the Elements belonged to Euclid himself. But still, today the method of proof of Pythagoras remains unknown.

The sixth secret is the legend about Pythagoras himself, the man who first proved this theorem. There is a legend that when Pythagoras of Samos proved his theorem, he thanked the gods by sacrificing 100 bulls. There were also legends about the scientist’s hypnotic abilities: as if he could change the flight direction of birds with just his glance. They also said that this amazing man was simultaneously seen in different cities, between which there were several days of travel. And that he supposedly owned a “wheel of fortune”, by rotating which he not only predicted the future, but also intervened, if necessary, in the course of events.

Description of the presentation by individual slides:

1 slide

Slide description:

Teacher of the Lyceum at KazGASA Auelbekova G.U. "The Pythagorean theorem and various ways of proving it." 2016

2 slide

Slide description:

OBJECTIVE: The main objective is to look at the different ways to prove the Pythagorean Theorem. Show what significance the Pythagorean theorem has in the development of science and technology, in mathematics in general.

3 slide

Slide description:

From the biography of Pythagoras The most that the population now knows about this respected ancient Greek fits into one phrase: “Pythagoras’ trousers are equal on all sides.” The authors of this tease are clearly separated by centuries from Pythagoras, otherwise they would not have dared to tease. Because Pythagoras is not at all the square of the hypotenuse, equal to the sum of the squares of the legs. This is a famous philosopher. Pythagoras lived in the sixth century BC, had a beautiful appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received because he always spoke correctly and convincingly, like a Greek oracle. (Pythagoras - “persuasive by speech.”) With his speeches he acquired 2,000 students, who, together with their families, formed a school-state, where the laws and rules of Pythagoras were in effect. He was the first to give a name to his line of work. The word “philosopher”, like the word “cosmos”, came to us from Pythagoras. There is a lot of cosmic in his philosophy. He argued that to understand God, man and nature, one must study algebra with geometry, music and astronomy. By the way, it is the Pythagorean system of knowledge that is called “mathematics” in Greek. As for the notorious triangle with its hypotenuse and legs, this, according to the great Greek, is more than a geometric figure. This is the “key” to all encrypted phenomena of our life. Everything in nature, said Pythagoras, is divided into three parts. Therefore, before solving any problem, it must be represented in the form of a triangular diagram. "See the triangle - and the problem is two-thirds solved."

4 slide

Slide description:

Now there are three formulations of the Pythagorean theorem: 1. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. 2. 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 the legs. 3. A square built on the hypotenuse of a right triangle is equivalent to squares built on the legs. Converse Pythagorean theorem: For every triple of positive numbers a, b and c such that a2 + b2 = c2, there exists a right triangle with legs a and b and hypotenuse c. you

5 slide

Slide description:

From the history of the theorem From the history of the theorem 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 we can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. This is confirmed by about 500 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. .

6 slide

Slide description:

Formulations Statements of the theorem translated from Greek, Latin and German In Euclid, this theorem states (literal translation): “In a right triangle, the square of the side spanning the right angle is equal to the squares on the sides enclosing the right angle.” The Latin translation of the Arabic text Annairitsi (about 900 BC), made by Gerhard of Clemons (early 12th century), translated into Russian reads: “In every right triangle, the square formed on the side stretched over the right angle is equal to the sum of two squares formed on two sides enclosing a right angle." In Geometria Culmonensis (c. 1400), the translation of the theorem reads: “The area of ​​a square, then, measured along its long side, is as great as that of two squares measured along its two sides adjacent to a right angle.” In the first Russian translation of Euclidean Elements, made by F. I. Petrushevsky, Pythagoras’ theorem is stated as follows: “In right triangles, the square of the side opposite the right angle is equal to the sum of the squares of the sides containing the right angle.”

7 slide

Slide description:

The construction used for the proof is as follows: for a right triangle with a right angle, squares above the legs and and a square above the hypotenuse, an altitude and a ray extending it are constructed, dividing the square above the hypotenuse into two rectangles and. The proof is aimed at establishing the equality of the areas of the rectangle with the square above the leg, the equality of the areas of the second rectangle constituting the square with the hypotenuse and the rectangle above the other leg in a similar way. The equality of the areas of the rectangle is established through the congruence of the triangles and, the area of ​​each of which is equal to half the area of ​​the squares and, accordingly, in connection with the following property: the area of ​​the triangle is equal to half the area of ​​the rectangle if the figures have a common side, and the height of the triangle to the common side is the other side of the rectangle . The congruence of triangles follows from the equality of two sides (sides of squares) and the angle between them (composed of a right angle and an angle at. Thus, the proof establishes that the area of ​​a square above the hypotenuse, composed of rectangles and, is equal to the sum of the areas of the squares above the legs. SIMPLE PROOF

8 slide

Slide description:

AJ is the height lowered to the hypotenuse. Let us prove that its continuation divides the square built on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding Squares built on the sides. Let us prove that the rectangle BJLD is equal in size to the square ABFH. Triangle ABD=BFC (on two sides and the angle between them BF=AB; BC=BD; angle FBC=angle ABD).

Slide 9

Slide description:

S triangle ABD=1/2 S rectangle BJLD, because Triangle ABD and Rectangle BJLD have a common base BD and a common height LD. SIMILARLY, S triangle FBC=1/2 S rectangle ABFH(BF-common base, AB-common height). Hence, taking into account that S of triangle ABD =S of triangle FBC, we have: S BJLD=S ABFH. SIMILARLY, using the equality of triangles BCK and ACE, it is proved that S JCEL=S ACKG. S ABFH+S ACKJ=S BJLD+ S JCEL=S BCED. Triangle S=1/2AB x BD=1/2LD x BD=1/2 S BJLD The theorem is proven. A L B D

10 slide

Slide description:

Proof of the Indian mathematician Bhaskari a in c in a - in in in c Bhaskari's method is as follows: express the area of ​​the square built on the hypotenuse (c ²) as the sum of the areas of the triangles (4S = 4· 0.5 a b) and the area of ​​the square (a – c) ². That is, it turns out that c ² = 4 · 0.5 a b + (a – c) ² c ² = 2 a b + a ² - 2 a b + b ² c ² = a ² + b ² The theorem is proven.

11 slide

Slide description:

Waldheim's proof a b c a b c Waldheim uses the fact that the area of ​​a right triangle is equal to half the product of its legs, and the area of ​​a trapezoid is equal to the product of half the sum of its parallel bases and its height. Now, to prove the theorem, it is enough only to express the area of ​​the trapezoid in two ways S trapezoid = 0.5(a + b) (a + b) = 0.5 (a + b) ² S trapezoid = 0.5 a b + 0, 5 a b + 0.5 c ² Equating the right sides, we get 0.5 (a + b) ² = 0.5 a b + 0.5 a b + 0.5 c ² (a + b) ² = a b + а в + с ² а ² + 2 а в + в ² = 2 а в + с ² с ² = а ² + в ² The theorem is proven

12 slide

Slide description:

Hawkins' proof A B C A1 B1 a c D c a c 1. Let us rotate the rectangular ∆ABC (with right angle C) around the center at point C by 90º so that it takes the position A1 B1 C, as shown in the figure. 2. Let’s continue the hypotenuse B1 A1 beyond point A1 until it intersects with line AB at point D. The segment B1 D will be of height ∆B1AB (since ∟B1DA = 90º). 3. Consider the quadrilateral A1AB1B. On the one hand, SА1АВ1В = SАА1 + SСВВ1 =0.5в · в + 0.5а · а=0.5(а² + в²) On the other hand, SA1АВ1В = SA1ВВ1 + SАА1В1 = 0.5 s · ВД + 0.5 s · AD = = 0.5 · s ·(AD + VD) = 0.5 · s² Equating the resulting expressions, we obtain 0.5 (a² + b²) = 0.5 c² a² + b² = c² The theorem is proven.

Slide 13

Slide description:

Geometric proof. (Hoffmann's method) Construct triangle ABC with right angle C. Construct BF=CB, BFCB Construct BE=AB, BEAB Construct AD=AC, ADAC Points F, C, D belong to the same line.

Slide 14

Slide description:

As we see, the quadrilaterals ADFB and ACBE are equal in size, because ABF=ECB. Triangles ADF and ACE are equal in size. Let us subtract the triangle ABC they share from both equal quadrilaterals, and we obtain: 1/2a2+1/2b 2=1/2c 2 Accordingly: a2+ b 2 =c 2 The theorem is proven.

15 slide

Slide description:

Algebraic proof (Möhlmann's method) The area of ​​a given rectangle on one side is 0.5ab, on the other 0.5pr, where p is the semi-perimeter of the triangle, r is the radius of the inscribed circle (r=0.5(a+b-c)). A C

16 slide

Slide description:

We have: 0.5ab=0.5pr=0.5(a+b+c)*0.5(a+b-c) It follows that c2= a2+b2 The theorem is proven. A C

Slide 17

Slide description:

The meaning of the Pythagorean theorem The Pythagorean theorem is rightfully one of the main theorems of mathematics. The significance of this theorem is that with its help one can derive most of the theorems in geometry. Its value in the modern world is also great, since the Pythagorean theorem is used in many branches of human activity. For example, it is used in the placement of lightning rods on the roofs of buildings, in the production of windows of certain architectural styles, and even in calculating the height of antennas of mobile operators. And this is not the entire list of practical applications of this theorem. This is why it is very important to know the Pythagorean theorem and understand its meaning.

18 slide

Slide description:

Pythagorean theorem in literature. Pythagoras is not only a great mathematician, but also a great thinker of his time. Let's get acquainted with some of his philosophical statements...

Slide 19

Slide description:

1. Thought is above all else between people on earth. 2. Do not sit on a grain measure (i.e., do not live idly). 3. When leaving, do not look back (i.e., before death, do not cling to life). 4. Don’t walk down the beaten path (that is, follow not the opinions of the crowd, but the opinions of the few who understand). 5. Don’t keep swallows in your house (i.e., don’t receive guests who are talkative or unrestrained in their language). 6. 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). 7. Do not wear images in the ring (that is, do not flaunt in front of people how you judge and think about the gods).

Chernov Maxim

A project on geometry, designed in the form of a presentation on the topic "Pythagorean Theorem and various methods of proving it"

Download:

Preview:

To use presentation previews, create a Google account and log in to it: https://accounts.google.com

Slide captions:

The Pythagorean theorem and various methods of proving it Completed by: Chernov Maxim 8A

Project goal: To present the Pythagorean theorem and present different ways of proving it.

History The ancient Chinese book Zhou Bi Xuan Jing talks about a Pythagorean triangle with sides 3, 4 and 5. The same book offers a drawing that coincides with one of the drawings of the Hindu geometry of Bashara. Moritz Cantor (the leading German historian of mathematics) believes that the equality 3² + 4² = 5² was already known to the Egyptians around 2300 BC, during the time of King Amenemhat I (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. Their method of construction can be very easily reproduced. Let's take a rope 12 m long and tie a colored strip to it at a distance of 3 m from one end and 4 meters from the other. The right angle will be between sides 3 and 4 meters long. It could be objected to the Harpedonaptians that their method of construction becomes superfluous if one uses, for example, a wooden square, which is used by all carpenters. Indeed, Egyptian drawings are known in which such a tool is found, for example, drawings depicting a carpentry workshop. Somewhat more is known about the Pythagorean theorem among the Babylonians. One text dating back to the time of Hammurabi, that is, 2000 BC, gives an approximate calculation of the hypotenuse of an isosceles right triangle. From this we can conclude that in Mesopotamia they were able to perform calculations with right triangles, at least in some cases. Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other hand, on a critical study of Greek sources, Van der Varden (Dutch mathematician) concluded that there was a high probability that the theorem on the square of the hypotenuse was known in Babylon already around the 18th century BC. e. According to Proclus's commentary on Euclid, Pythagoras (whose years are generally considered to be 570-490 BC) used algebraic methods to find Pythagorean triplets. However, Proclus wrote between 410 and 485. n. e. Thomas Little Heath believed that there is no explicit reference, dating back to the 5 centuries after the death of Pythagoras, that Pythagoras was the author of the theorem. However, when authors such as Plutarch and Cicero write about the Pythagorean theorem, they write as if the authorship of Pythagoras was widely known and undoubted. “Whether this formula belongs to Pythagoras personally ..., but we can confidently assume that it belongs to the period of Pythagorean mathematics." According to legend, Pythagoras celebrated the discovery of his theorem with a gigantic feast, slaughtering a hundred bulls to celebrate. Around 400 BC. BC, according to Proclus, Plato gave a method for finding Pythagorean triplets, combining algebra and geometry. Around 300 BC. e. the oldest axiomatic proof of the Pythagorean theorem appeared in Euclid's Elements.

Formulations: Geometric formulation: Initially, the theorem was formulated as follows: 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. Algebraic formulation: 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. That is, denoting the length of the hypotenuse of the triangle by, and the lengths of the legs by a and b: a2+b2=c2 Both formulations of the theorem are equivalent, but the second formulation is more elementary, it does not require the concept of area. That is, the second statement can be verified without knowing anything about the area and by measuring only the lengths of the sides of a right triangle.

Proofs Currently, 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. Such diversity can only be explained by the fundamental significance of the theorem for geometry. Of course, conceptually all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs (for example, using differential equations).

Through similar triangles The following proof of the algebraic formulation is the simplest of the proofs constructed directly from the axioms. In particular, it does not use the concept of area of ​​a figure. Let ABC be a right triangle with right angle C. Let's draw the height from C and denote its base by H. Triangle ACH is similar to triangle ABC in two angles. Likewise, triangle CBH is similar to ABC. By introducing the notation we get What is equivalent Adding, we get or, which is what we needed to prove

Proofs using the area method The following proofs, despite their apparent simplicity, are not so simple at all. All of them use the properties of area, the proof of which is more complex than the proof of the Pythagorean theorem itself. Proof through equicomplementarity Let us arrange four equal right triangles as shown in Figure 1. A quadrilateral with sides c is a square, since the sum of two acute angles is 90°, and a straight angle is 180 °. The area of ​​the entire figure is equal, on the one hand, to the area of ​​a square with side (a + b), and on the other hand, to the sum of the areas of the four triangles and the area of ​​the inner square. Q.E.D. .

Euclid's proof The idea of ​​Euclid's proof is as follows: let's try to prove that half the area of ​​the square built on the hypotenuse is equal to the sum of the half areas of the squares built on the legs, and then the areas of the large and two small squares are equal. Let's look at the drawing on the left. On it we constructed squares on the sides of a right triangle and drew a ray s from the vertex of the right angle C perpendicular to the hypotenuse AB, it cuts the square ABIK, built on the hypotenuse, into two rectangles - BHJI and HAKJ, respectively. It turns out that the areas of these rectangles are exactly equal to the areas of the squares built on the corresponding legs. Let's try to prove that the area of ​​the square DECA is equal to the area of ​​the rectangle AHJK. To do this, we will use an auxiliary observation: The area of ​​a triangle with the same height and base as the given rectangle is equal to half the area of ​​the given rectangle. This is a consequence of defining the area of ​​a triangle as half the product of the base and the height. From this observation it follows that the area of ​​triangle ACK is equal to the area of ​​triangle AHK (not shown in the figure), which in turn is equal to half the area of ​​rectangle AHJK. Let us now prove that the area of ​​triangle ACK is also equal to half the area of ​​square DECA. The only thing that needs to be done for this is to prove the equality of triangles ACK and BDA (since the area of ​​triangle BDA is equal to half the area of ​​the square according to the above property). This equality is obvious: the triangles are equal on both sides and the angle between them. Namely - AB=AK, AD=AC - the equality of the angles CAK and BAD is easy to prove by the method of motion: we rotate the triangle CAK 90° counterclockwise, then it is obvious that the corresponding sides of the two triangles in question will coincide (due to the fact that the angle at the vertex of the square is 90°). The reasoning for the equality of the areas of the square BCFG and the rectangle BHJI is completely similar. Thus, we proved that the area of ​​a square built on the hypotenuse is composed of the areas of squares built on the legs. The idea behind this proof is further illustrated by the animation above. This proof is also called “Pythagorean pants.”

Leonardo da Vinci's proof The main elements of the proof are symmetry and movement. Let's consider the drawing, as can be seen from the symmetry, the segment cuts the square into two identical parts (since the triangles are equal in construction). Using a 90-degree counterclockwise rotation around the point, we see the equality of the shaded figures and. Now it is clear that the area of ​​the figure we have shaded is equal to the sum of half the areas of the small squares (built on the legs) and the area of ​​the original triangle. On the other hand, it is equal to half the area of ​​the large square (built on the hypotenuse) plus the area of ​​the original triangle. Thus, half the sum of the areas of small squares is equal to half the area of ​​the large square, and therefore the sum of the areas of squares built on the legs is equal to the area of ​​the square built on the hypotenuse.

The meaning of the Pythagorean theorem 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.

Thank you for your attention!