Pythagorean theorem: history, proof, examples of practical application. Ancient theorems. History of the Pythagorean Theorem What Pythagoras proved
Prividentsev Vladislav, Farafonova Ekaterina
Students' project work for a mathematical conference
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BOU TR OO "Trosnyanskaya Secondary School"
Student mathematical conference dedicated to the great mathematician Pythagoras
(as part of Mathematics Week at School)
History of Pythagorean Theorem
(project)
Prepared
9th grade students
Farafonova Ekaterina and Prividentsev Vladislav
Teacher Bilyk T.V.
January – 2016
Goals:
- 1.Expand your knowledge of the history of mathematics.
- 2. Get acquainted with biographical facts from the life of Pythagoras related to the theorem.
- 3. Study the history of the Pythagorean theorem through myths and legends of antiquity.
- 4. Consider the application of the Pythagorean theorem in solving problems from various branches of geometry.
Plan.
1. Introduction
2. From the history of the theorem
3. Poems about Pythagoras
4. Summary
5. Conclusion
Introduction.
The Pythagorean theorem has long been widely used in various fields of science, technology and practical life. The Roman architect and engineer Vitruvius, the Greek moralist writer Plutarch, and the Greek scientist lll century wrote about it in their works. Diogenes Laertius, 5th century mathematician 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 the poems of poets.
The poet Heinrich Heine (1797-1856), known for his anti-religious views and caustic ridicule of superstitions, in one of his works ridicules the “doctrine” of transmigration of souls as follows:
"Who knows! Who knows! The soul of Pythagoras settled, perhaps, on a poor candidate who was unable to prove Pythagoras’s theorems and therefore failed the exam, while in his examiners live the souls of those very bulls that Pythagoras once sacrificed to the immortal gods, delighted by the discovery of his theorem.” The history of the Pythagorean theorem begins long before Pythagoras. Numerous different proofs of the Pythagorean Theorem have been given over the centuries.
From the history of the theorem
Let's start our historical review with ancient China. Here the mathematical book Chu-pei attracts special attention. This work says this about the Pythagorean triangle with sides 3, 4 and 5: “If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3 and the height is 4.” In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.
- Cantor (the leading German historian of mathematics) believes that equality 32 + 42 = 52 was already known to the Egyptians still around 2300 BC. e., during the time of the king Amenemhet I (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonaptes, or “rope pullers,” built right angles using right-angled 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 enclosed 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 carpenter's workshop.
- A little more is known about the Pythagorean theorem Babylonians . In one text related to time Hammurabi , i.e. by 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. Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other, on a critical study of Greek sources, Van der Waerden (Dutch mathematician) came to 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.” Hindu geometry , like the Egyptians and Babylonians, was closely associated with the 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.
- In the first Russian translation of Euclidean Elements, made by F. I. Petrushevsky, the Pythagorean 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."It is now known that this theorem was not discovered by Pythagoras. However, some believe that Pythagoras was the first to give its full proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Elements. On the other hand, Proclus claims that the proof in the Elements belongs to Euclid himself. As we see, the history of mathematics has preserved almost no reliable data about the life of Pythagoras and his mathematical activities. But the legend even tells us the immediate circumstances that accompanied the discovery of the theorem. They say that in honor of this discovery, Pythagoras sacrificed 100 bulls.
- For a long time it was believed that this theorem was not known before Pythagoras and was therefore called the “Pythagorean theorem.” This name has survived to this day. However, it has now been established that this most important theorem is found in Babylonian texts written 1200 years before Pythagoras.
- The fact that a triangle with sides 3, 4 and 5 is a rectangle was known 2000 BC. the Egyptians, who probably used this ratio to construct right angles when constructing buildings. In China, the proposal for the square of the hypotenuse was known at least 500 years before Pythagoras. This theorem was also known in Ancient India; This is evidenced by the sentences contained in the Sutras.
Pythagoras made many important discoveries, but the theorem he proved, which now bears his name, brought the greatest fame to the scientist. Indeed, in modern textbooks the theorem is formulated as follows: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.” - How to write the Pythagorean theorem for a right triangle ABC with legs a, b and hypotenuse c.
a 2 + b 2 = c 2
It is believed that in the time of Pythagoras the theorem sounded differently: “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.” Really, With 2 – area of the square built on the hypotenuse, a 2 and b 2 – areas of squares built on legs.
It is likely that the fact stated in the Pythagorean theorem was first established for isosceles right triangles. A square built on the hypotenuse contains four triangles. And on each side there is a square containing two triangles. From Figure 9 it is clear that the area of the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs.
Poems about Pythagoras.
German novelist A. Chamisso, who at the beginning of the 10th century. He took part in a trip around the world on the Russian ship “Rurik” and wrote the following poems:
The truth will remain eternal, as soon as
A weak person will know it!
And now the Pythagorean theorem
True, like his distant century.
The sacrifice was abundant
To the gods from Pythagoras. One Hundred Bulls
He gave it up to be slaughtered and burned
Behind the light is a ray that came from the clouds.
Therefore, ever since then,
The truth is just being born,
The bulls roar, sensing her, following her.
They are unable to stop the light,
Or they can only close their eyes and tremble
From the fear that Pythagoras instilled in them
To summarize:
If we are given a triangle
And, moreover, with a right angle,
That is the square of the hypotenuse
We can always easily find:
We square the legs,
We find the sum of powers
And in such a simple way
We will come to the result.
A test in geometry is approaching, and during tests and exams there are sometimes cases when students, having pulled out a ticket, remember the formulation of the theorem, but forget where to start the proof. To prevent this from happening to you, I suggest a drawing - a reference signal. I think he will remain in your memory for a long time.
Ivan Tsarevich cut off the dragon's head, and two new ones grew from him. In mathematical language this means: spent in Δ ABC height CD , and two new right triangles were formed ADC and BDC.
Conclusion.
After studying the material constructed, we can conclude that the Pythagorean theorem is one of the most important theorems of geometry because with its help you can prove many other theorems and solve many problems.
Pythagoras and the Pythagorean school played a major role in improving methods for solving scientific problems: the need for rigorous proof was firmly established in mathematics, which gave it the significance of a special science.
Introduction
It is difficult to find a person who does not associate the name of Pythagoras with his theorem. Perhaps even those who have said goodbye to mathematics forever in their lives retain memories of “Pythagorean pants” - a square on the hypotenuse, equal in size to two squares on the sides.
The reason for the popularity of the Pythagorean theorem is triune: it
simplicity - beauty - significance. Indeed, the Pythagorean theorem is simple, but not obvious. This is a combination of two contradictory
began to give her a special attractive force, makes her beautiful.
In addition, the Pythagorean theorem is of great importance: 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.) testifies to the gigantic number of its specific implementations .
In modern textbooks, the theorem is formulated as follows: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.”
In the time of Pythagoras, it sounded like this: “Prove that a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs” or “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.”
Goals and objectives
The main goal of the work was to showthe importance of the Pythagorean theorem in the development of science and technology of manycountries and peoples of the world, as well as in the most simple and interestingform to teach the content of the theorem.
The main method used in the work isis a method of organizing and processing data.
Involving information technology, diversifyingzili material with various colorful illustrations.
"GOLDEN VERSES" OF PYTHAGORUS
Be fair both in your words and in your actions... Pythagoras (c. 570-c. 500 BC)
Ancient Greek philosopher and mathematiciandeveloped with his teaching about cosmic harmony andtransmigration of souls. Tradition credits Pythagoras with proving the theorem that bears his name. Much inPlato's teachings go back to Pythagoras and his successors tel.
There are no written documents left about Pythagoras of Samos, the son of Mnesarchus, and from later evidence it is difficult to reconstruct the true picture of his life and achievements.(Electronic encyclopedia:StarWorld) It is known that Pythagoras left his native island of Samos in the Aegean Sea at the shoregov of Asia Minor in protest against the tyranny of the ruler and already in adulthoodage (according to legend, 40 years old) appeared in the Greek city of Crotone in southern Italy. Pythagoras and his followers - the Pythagoreans - formed a secret alliance that played a significant role in the life of the Greek colonies in ItaLii. The Pythagoreans recognized each other by a star-shaped pentagon - a pentagram. But Pythagoras had to retire to Metapontum, where hedied. Later in the second halfVBC e., his order was destroyed.
The teachings of Pythagoras were greatly influenced by philosophy and religiongia of the East. He traveled a lot in the countries of the East: he was inEgypt and Babylon. There Pythagoras also met Eastern mathematics tikoy.
The Pythagoreans believed that secrets were hidden in numerical patterns.on the world. The world of numbers lived a special life for the Pythagorean; numbers hadits own special life meaning. Numbers equal to the sum of their divisors were perceived as perfect (6, 28, 496, 8128); friendlynamed pairs of numbers, each of which was equal to the sum of the other's divisorsgogo (for example, 220 and 284). Pythagoras was the first to divide numbers into even andodd, prime and composite, introduced the concept of figured numbers. In hisThe school examined in detail Pythagorean triplets of natural numbers, in which the square of one was equal to the sum of the squares of the other two (Fermat’s last theorem).
Pythagoras is credited with saying: “Everything is a number.” To the numbers(and he meant only natural numbers) he wanted to bring the whole world together, andmathematics in particular. But in the school of Pythagoras itself a discovery was made that violated this harmony. It has been proven that the root of 2 is notis a rational number, i.e. it cannot be expressed in terms of natural numbers numbers.
Naturally, Pythagoras’ geometry was subordinated to arithmetic.This was clearly manifested in the theorem that bears his name and later becamethe basis for the application of numerical methods of geometry. (Later, Euclid again brought geometry to the forefront, subordinating algebra to it.) Apparently, the Pythagoreans knew the correct solids: tetrahedron, cube and dodecahedron.
Pythagoras is credited with the systematic introduction of proofs into geometry, the creation of planimetry of rectilinear figures, the doctrine of bii.
The name of Pythagoras is associated with the doctrine of arithmetic, geometric and harmonic proportions.
It should be noted that Pythagoras considered the Earth to be a ball movingaround the sun. When inXVIcentury the church began to be fiercely persecutedIf we take the teaching of Copernicus, this teaching was stubbornly called Pythagorean.(Encyclopedic Dictionary of a Young Mathematician: E-68. A. P. Savin.- M.: Pedagogy, 1989, p. 28.)
Some fundamental concepts undoubtedly belongto Pythagoras himself. The first one- the idea of space as mathematicsa tically ordered whole. Pythagoras came to him after discovering that the fundamental harmonic intervals, i.e., octave, perfect fifth and perfect fourth, arise when the lengths of vibrating strings are related as 2:1, 3:2 and 4:3 (legend has it that the discovery was made whenPythagoras passed by a forge: anvils with different massesgenerated the corresponding sound ratios upon impact). UsmotRevealing an analogy between the orderliness in music, expressed by the relationships discovered by it, and the orderliness of the material world, Pythagorascame to the conclusion that it is permeated with mathematical relationshipsthe whole space. An attempt to apply the mathematical discoveries of Pythagoras to speculative physical constructions led to interesting consequences.results. Thus, it was assumed that each planet during its revolutionaround the Earth it emits as it passes through the clear upper air, or "ether",tone of a certain pitch. The pitch of the sound changes depending on the speedspeed of the planet's movement, the speed depends on the distance to the Earth. PlumWhen celestial sounds come together, they form what is called the “harmony of the spheres,” or “music of the spheres,” references to which are frequent in European literature.
The early Pythagoreans believed that the Earth was flat and in the centerspace. Later they began to believe that the Earth has a spherical shape and, together with other planets (which they included the Sun), is shapedrevolves around the center of space, i.e., the “hearth”.
In antiquity, Pythagoras was best known as a preachersecluded lifestyle. Central to his teaching was the ideatalk about reincarnation (transmigration of souls), which, of course, presupposes the ability of the soul to survive the death of the body, and therefore its immortality. Since in a new incarnation the soul can move into the body of an animal, Pythagoras was opposed to killing animals, eating their meat, and even stated that one should not deal with those who slaughter animals or butcher their carcasses. As far as one can judge from the writings of Empedocles, who shared the religious views of Pythagoras, the shedding of blood was considered here as an original sin, for which the soul is expelled into the mortal world, where it wanders, being imprisoned in one body or another. The soul passionately desires liberation, but out of ignorance it invariably repeats the sinful act.
Can save the soul from an endless series of reincarnationscleansing The simplest cleansing consists in observing certainprohibitions (for example, abstaining from intoxication or drinkingeating beans) and rules of behavior (for example, honoring elders, obeying the law and not being angry).
The Pythagoreans highly valued friendship, and according to their concepts, all the property of friends should be common. A select few were offered the highest form of purification - philosophy, that is, love of wisdom, and therefore the desire for it (this word, according to Cicero, was first used by Pythagoras, who called himself not a sage, but a lover of wisdom). By means of these means the soul comes into contact with the principles of cosmic order and becomes in tune with them, it is freed from its attachment to the body, its lawless and disordered desires. Mathematics is one of the components of religionPythagoreans, who taught that God laid the number at the basis of the worldorder.
Influence of the Pythagorean Brotherhood in the first halfVV. BC e. Notincreased continuously. But his desire to give power to the “best” came into conflict with the rise of democratic sentiment in the Greek cities of southern Italy, and soon after 450 BC. e. there was an outbreak in Crotonea rebellion against the Pythagoreans that resulted in the murder and expulsion of many, if not all, members of the brotherhood. However, still inIVV. BC e. pythagoThe Reichs enjoyed influence in southern Italy, and in Tarentum, where Plato’s friend Archytas lived, it remained even longer. However, much more important for the history of philosophy was the creation of Pythagorean centers in Greece itself,for example in Thebes, in the second halfVV. BC e. Hence the Pythagoreanideas penetrated to Athens, where, according to Plato's dialoguePhaedo,they were adopted by Socrates and turned into a broad ideological movement,started by Plato and his student Aristotle.
In subsequent centuries, the figure of Pythagoras himself was surrounded
many legends: he was considered the reincarnated god Apollo,
it was believed that he had a golden thigh and was capable of teaching in
the same time in two places. Early Christian Church Fathers answer
whether Pythagoras has a place of honor between Moses and Plato. Also inXVIV[
there were frequent references to the authority of Pythagoras in matters not only of science |.:
but also magic.(Electronic encyclopedia:StarWorld.).
Behind the legend is the truth
The discovery of the Pythagorean theorem is surrounded by a halo of beautiful legendsProclus, commenting on the last sentenceIbooks "Elements" by Euclid,writes: “If you listen to those who like to repeat ancient legends, thenwe have to say that this theorem goes back to Pythagoras; they saythat he sacrificed a bull in honor of this.” This legend has firmly grown togetherwith the Pythagorean theorem and after 2000 years continued to cause hot clicks. Thus, the optimist Mikhailo Lomonosov wrote: “Pythagoras for the invention of one geometricAccording to the rule of Zeus, he sacrificed a hundred oxen.But if for those found in modern times fromwitty mathematicians rules according to his superstitiousjealousy to act, then barelyif there were so many in the whole worldcattle have been found."
But the ironic Heinrich Heine saw the development of the same situation somewhat differently : « Who knows ! Who knows ! Maybe , the soul of Pythus the mountain moved into the poor candidate , who could not prove the Pythagorean theorem and failed from - for this in exams , while in his examiners dwell the souls of those bulls , which Pythagoras , delighted by the discovery of his theorem , sacrificed to the immortal gods ».
History of the discovery of the theorem
The discovery of the Pythagorean theorem is usually attributed to the ancient Greek philosopher and mathematician Pythagoras (VIV. BC e.). But a study of Babylonian cuneiform tables and ancient Chinese manuscripts (copies of even more ancient manuscripts) showed that this statement was known long before Pythagoras, perhaps millennia before him. The merit of Pythagoras was that he discovered the proof of this theorem.
Let's start our historical review with ancient China. There is a special note heremania is attracted by the mathematical book Chu-pei. This work talks about the Pythagorean triangle with sides 3, 4 and 5:“If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3 and the height is 4.”
In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.
Also, the Pythagorean theorem was discovered in the ancient Chinese treatise “Zhou-bi suan jin” (“Mathematical treatiseabout the gnomon"), the time of creation of which is unknown exactly, but where it is stated that inXVV. BC e. the Chinese knew the properties of the Egyptian triangle, and inXVIV. BC e. - and the general form of the theorem.
Cantor (the greatest German historian of mathematics) believes that the equality 3 2 + 4 2 = 5 2 was already known to the Egyptians around 2300 BC. e. during the time of King AmenemhetI(according to papyrus 6619 of the Berlin Museum).
According to Cantor, harpedonaptes, or “rope pullers,” built right angles when
using right triangles with sides 3, 4 and 5.
It is very easy to reproduce their methodconstruction. Let's take a rope 12 m long and tie a colored strip to it at a distance3 m from one end and 4 m from the other. Right anglewill be enclosed between sides 3 and 4 m long. It could be objected to the Harpedonaptes that their method of construction becomes redundant 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 carpenter's workshop.Somewhat more is known aboutPythagorean theorem among the Babylonians.In one text dating back to the timeMeni Hammurabi, i.e. by 2000BC e., an approximate calculation of the hypotenuse is given directlycoal triangle. From herewe can conclude that in Dvurawho knew how to do calculationswith right trianglesmi, at least in somecases. Based on onesides, at today's levelknowledge about Egyptian and Babylonianmathematics, and on the other - in criticismlogical study of Greek sources, Van der Waerden (DutchRussian 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. The computational recipe is in their hands you, based on vague ideas, have turned into precise new science."
The geometry of the Hindus, like that of the Egyptians and Babylonians, was closelyassociated with a cult. It is very likely that the square theorem is hypotenuse was known in India already aboutXVIIIcentury BC e., alsoit was also known in ancient Indian geometrictheological treatiseVII- Vcenturies BC e. "Sulva Sutra" ("Rulesropes").
But despite all this evidence, the name of Pythagoras is sofirmly fused with the Pythagorean theorem, which is simply impossible nowone can imagine that this phrase will fall apart. Same fromalso refers to the legend of the spell of Pythagoras' bulls. And it's unlikelyneed to be dissected with a historical-mathematical scalpeldeep ancient legends.
Methods to prove the theorem
Proof of the Pythagorean Theorem by Middle Ages Studentsconsidered it very difficult and called itDons asinorum - donkey bridge, orelefuga - flight of the “poor”, as some “poor” students who did not have serious mathematical training fledwhether from geometry. Weak students who have memorized theoremswithout understanding and therefore nicknamed “donkeys”, were unableability to overcome the Pythagorean theorem, which seemed to serve themsurmountable bridge. Because of the drawings accompanying the theoremPythagoras, students also called it a “windmill”, withthey wrote poems like “Pythagorean trousers are equal on all sides” and drew cartoons.
A). The simplest proof
Probably the fact stated in the Pythagorean theorem was a dreamchala is set for isosceles rectangles. Just look at the mosaic of black and light triangles,to verify the validity of the theorem for triangleska ABC : a square built on the hypotenuse contains four triangles, and on each side a square is built containingtwo triangles (Fig. 1, 2).
Proofs based on the use of the concept of equal size of figures.
In this case, we can consider evidence in which quadrath built on the hypotenuse of a given rectangular trianglesquare, “made up” of the same figures as squares built on the sides. The following evidence can also be consideredva, in which the permutation of summand figures anda number of new ideas are taken into account.
In Fig. 3 shows two equal squares. Length of sides eachequal to the squarea + b. Each of the squares is divided into parts,consisting of squares and right triangles. It is clear that if you subtract quadruple the area of a right triangle with legs from the area of a squarea, b, then they will remain equal have mercy, i.e. With 2 = a 2 + b 2 . However, the ancient Hindus, who belonged tothis reasoning lies, usually they did not write it down, but accompanied itdrawing with just one word: “Look!” It is quite possible that shePythagoras also offered some proof.
b). Evidence by the method of completion.
The essence of this method is that to the squares, constructon the legs, and to a square built on the hypotenuse, withconnect equal figures so that they are equalnew figures.
In Fig. 4 shows a regular Pythagorow figure right triangleABCwith squares built on its sides. Attached to this figure are threesquares 1 and 2, equal to the original straightcoal triangle.
The validity of the Pythagorean theorem follows from the equal size of hexagonsAEDFPB And ACBNMQ. Here is a direct EP delit hexagonAEDFPBinto two equal quadrilaterals, line CM divides the hexagonACBNMQinto two equal quadrangles; rotating the plane 90° around center A maps the quadrilateral AERB onto a quadrilateralACMQ.
(This proof was first given by Leonard before da Vinci.)
Pythagorean figure completedto a rectangle whose sides are parallelaligned with the corresponding sides of the quadracoms built on legs. Let's divide this rectangle into triangles and straightsquares. From the resulting rectangleFirst, we subtract all the polygons 1, 2, 3, 4, 5, 6, 7, 8, 9, leaving a square built on the hypotenuse. Then from the same rectangle we subtract rectangles 5, 6, 7 and shaded straightsquares, we get squares built on the legs.
Now let us prove that the figures subtracted in the first case areare equal in size to the figures subtracted in the second case.
This illustrates the proof,given by Nassir-ed-Din (1594). Here: P.L.- straight;
KLOA = ACPF = ACED = a 2 ;
LGBO= SVMR = CBNQ = b 2 ;
AKGB = AKLO + LGBO= c 2 ;
hence with 2 = a 2 + b 2 .
Rice. 7 illustrates the proof,given by Hoffmann (1821). HereThe Pythagorean figure is constructed in such a way thatsquares lie on one side of a lineAB. Here: 
OCLP = ACLF = ACED = b 2 ;
CBML=CBNQ= A 2 ;
OVMR =ABMF= With 2 ;
OVMR = OCLP + CBML;
Hence c 2 = a 2 + b.
This illustrates another more original evidence offeredHoffman. Here: triangleABC with straight wash angle C; line segmentB.F.perpendicularNE and equal to it, segmentBEperpendicularAB and equal to it, segmentAD perpendicular ren AC and equal to it; pointsF, WITH, D belongs reap one straight line; quadrilateralsADFBand ACVE are equal in size, sinceABF= ESV; trianglesADF And ACEs are equal in size;
subtract from both equal quadranglesnicks have a common triangleABC, we get ½ a* a + ½ b* b – ½ c* c
V). Algebraic method of proof.
The figure illustrates the proof of the great Indian mathematician Bhaskari (the famous author of Li-lavati,XIIV.). The drawing was accompanied by only one word: LOOK! Among the proofs of the Pythagorean theorem by the algebraic method, first place (perhaps the oldest) fortakes evidence using subtext bee.
Historians believe that Bhaskara was born sting area with 2 square built onhypotenuse, as the sum of the areas of four triangles 4(ab/2) and the area of a square with a side equal to the difference of the legs.
Let us present in a modern presentation one of these proofs:bodies belonging to Pythagoras.
I "
In Fig. 10 ABC - rectangular, C - right angle, ( C.M.L AB) b - leg projection b to the hypotenuse, A - leg projectionA on the hypotenuse, h - altitude of the triangle drawn to hypotenuse. From the fact that ABC is similar to AFM, it followsb 2 = cb; (1) from the fact that ABC is similar to VSM, it follows that 2 = CA (2) Adding equalities (1) and (2) term by term, we obtain a 2 + b 2 = cb + ca = = c (b + a) = c 2 .
If Pythagoras actually offered such a proof,then he was familiar with a number of important geometric theorems,which modern historians of mathematics usually attribute Euclid.
Proof of Möhl- manna. Area given right trianglenika, on the one hand, is equal to 0,5 a* b, on the other 0.5* p*g, where p - semiperimeter of a triangler - radius inscribed in it is approx.circumference (r = 0.5 - (a + b - c)).We have: 0.5*a*b - 0.5*p*g - 0.5 (a + b + c) * 0.5-(a + b - c), from where it follows that c 2 = a 2 + b 2 .
d) Garfield's proof.
In Figure 12 there are three straighttriangles form a trapezoid. That's why.the area of this figure is possible.\ find using the area formuladi rectangular trapezoid,or as the sum of areasthree triangles. In the laneIn this case, this area is equal toby 0.5 (a + b) (a + b), in second rum - 0.5* a* b+ 0.5*a* b+ 0.5*s 2
Equating these expressions, we obtain the Pythagorean theorem.
There are many proofs of the Pythagorean theorem, carried outusing both each of the described methods and using a combinationtion of various methods. Concluding the review of examples of various docksstatements, here are some more drawings illustrating the eight waysbov, to which there are references in Euclid’s “Elements” (Fig. 13 - 20).In these drawings the Pythagorean figure is depicted as a solid lineher, and additional constructions - dotted.
As mentioned above, the ancient Egyptians for more than 2000 yearsago, they practically used the properties of a triangle with sides 3, 4, 5 to construct a right angle, that is, they actually used the theorem inverse to the Pythagorean theorem. Let us present a proof of this theorem based on the criterion for the equality of triangles (i.e., one that can be introduced very early into schoolnew practice). So let the sides of the triangleABC (Fig. 21) related to 2 = a 2 + b 2 . (3)
Let us prove that this triangle is right-angled.
Let's construct a right triangleA B C on two sides, whose lengths are equal to the lengthsA And b legs of a given triangle. Let the length of the hypotenuse of the constructed triangle be on c . Since the constructed triangle is right-angled, then by theoryin the Pythagorean rheme we havec = a + b (4)
Comparing relations (3) and (4), we obtain thatWith= with or c = c Thus, the triangles - the given one and the one constructed - are equal, since they have three respectively equal sides. Angle Cis straight, therefore angle C of this triangle is also right.
Additive evidence.
These proofs are based on the decomposition of squares built on the sides into figures from which a quad can be formedrath built on the hypotenuse.
Einstein's proof ( rice. 23) based on decompositiona square built on the hypotenuse into 8 triangles.
Here: ABC- rectangular triangle with right angle C;COMN; SK MN; P.O.|| MN; E.F.|| MN.
Prove it yourselfequality of triangles, halfcalculated by dividing the squares according tobuilt on legs and hypotenuse.
b) Based on the proof of al-Nayriziyah, another decomposition of squares into pairwise equal figures was carried out (hereABC - right triangle with right angle C).
This proof is also called “hinged” becausethat here only two parts, equal to the original triangle, change their position, and they are, as it were, attached to the restfigure on hinges around which they rotate (Fig. 25).
c) Another proof by the method of decomposing squares intoequal parts, called a "wheel with blades", is shown in rice. 26. Here: ABC - right triangle with right angle scrap S, O - the center of a square built on a large side; dotted lines passing through a pointABOUT, perpendicular orparallel to the hypotenuse.
This decomposition of squares is interesting because its pairwise equal quadrilaterals can be mapped onto each other by parallel translation.
"Pythagorean trousers" (Euclid's proof).
For two thousand yearschanged the proof inventedEuclid, which is placed in histhe famous "Principles". Euclid opus cal height VN from the vertex of a right triangle to the hypotenuse and proved that its continuation divides the square constructed on the hypotenuse into two rectangles whose areas are equal
areas of the corresponding squares built on the sides. Euclid's proof in comparison with the ancient Chinese or ancient Indian looks likeoverly complicated. For this reasonhe was often called “stilted” and “contrived.” But this opinionsuperficial. The drawing used to prove the theorem is jokingly called “Pythagorean pants.” Duringfor a long time it was considered one of the symbols of mathematical science.
Ancient Chinese evidence.
Mathematical treatises of Ancient China have reached us in editionsIIV. BC e. The fact is that in 213 BC. e. chinese emperor
Shi Huangdi, trying to eliminate previous traditions, ordered all ancient books to be burned. InIIV. BC e. Paper was invented in China and at the same time the restoration beganancient books. This is how “Mathematics in Nine Books” arose -the main surviving mathematical and astronomical works ny.
In the 9th book of "Mathematics" there is a drawingwho proves the Pythagorean theorem.The key to this proof is not difficult to find (Fig. 27).
In fact, in ancient Chinesethe same four equal rectangular trianglessquare with legsa, c and hypotenuse With laid so that their outer contour isthere is a square with a sidea + b, and internal - a square with side c, built on the hypotenuse (Fig. 28).
If a square with sideWith cut and the remaining 4 shaded trianglesplaced in two rectangles, it is clear that the resulting void, on the one hand,
equal to With, and on the other
a + b 2 , i.e. With 2 = a 2 + b
The theorem has been proven.
Note that with such a proof
Constructions inside the square on the hypotenwe see
dim in the ancient Chinese drawing are not used (Fig. 30). Apparently, ancient Chinese mathematicians had something different beforeproof, namely: if squared with
sideWith
two shaded trianglescut off the nick and attach the hypotenuses totwo other hypotenuses, then it is easy to findconfirm that the resulting figure, which
sometimes called the "bride's chair", withconsists of two squares with sidesA
Andb,
i.e. with 2
=
A
2
+ b
2
.
The figure reproduces blackfrom the treatise “Zhou-bi...”. HerePythagorean theorem considered forEgyptian triangle with legs3, 4 and hypotenuse 5 units of measurement.The square on the hypotenuse contains 25cells, and the square inscribed in it on the larger side is 16. It is clear that the remaining part contains 9 cells. This andthere will be a square on the smaller side.
The Pythagorean theorem states:
In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:
a 2 + b 2 = c 2,
- a And b– legs forming a right angle.
- With– hypotenuse of the triangle.
Formulas of the Pythagorean theorem
- a = \sqrt(c^(2) - b^(2))
- b = \sqrt (c^(2) - a^(2))
- c = \sqrt (a^(2) + b^(2))
Proof of the Pythagorean Theorem
The area of a right triangle is calculated by the formula:
S = \frac(1)(2) ab
To calculate the area of an arbitrary triangle, the area formula is:
- p– semi-perimeter. p=\frac(1)(2)(a+b+c) ,
- r– radius of the inscribed circle. For a rectangle r=\frac(1)(2)(a+b-c).
Then we equate the right sides of both formulas for the area of the triangle:
\frac(1)(2) ab = \frac(1)(2)(a+b+c) \frac(1)(2)(a+b-c)
2 ab = (a+b+c) (a+b-c)
2 ab = \left((a+b)^(2) -c^(2) \right)
2 ab = a^(2)+2ab+b^(2)-c^(2)
0=a^(2)+b^(2)-c^(2)
c^(2) = a^(2)+b^(2)
Converse Pythagorean theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. That is, for any triple of positive numbers a, b And c, such that
a 2 + b 2 = c 2,
there is a right triangle with legs a And b and hypotenuse c.
Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It was proven by the learned mathematician and philosopher Pythagoras.
The meaning of the theorem The point is that it can be used to prove other theorems and solve problems.
Additional material:
One thing you can be one hundred percent sure of is that when asked what the square of the hypotenuse is, any adult will boldly answer: “The sum of the squares of the legs.” This theorem is firmly ingrained in the minds of every educated person, but you just need to ask someone to prove it, and difficulties can arise. Therefore, let's remember and consider different ways to prove the Pythagorean theorem.
Brief biography
The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who brought it into the world is not so popular. This can be fixed. Therefore, before exploring the different ways to prove Pythagoras’ theorem, you need to briefly get to know his personality.
Pythagoras - philosopher, mathematician, thinker originally from Today it is very difficult to distinguish his biography from the legends that have developed in memory of this great man. But as follows from the works of his followers, Pythagoras of Samos was born on the island of Samos. His father was an ordinary stone cutter, but his mother came from a noble family.
Judging by the legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, the born boy was supposed to bring a lot of benefit and good to humanity. Which is exactly what he did.
Birth of the theorem
In his youth, Pythagoras moved to Egypt to meet famous Egyptian sages there. After meeting with them, he was allowed to study, where he learned all the great achievements of Egyptian philosophy, mathematics and medicine.
It was probably in Egypt that Pythagoras was inspired by the majesty and beauty of the pyramids and created his great theory. This may shock readers, but modern historians believe that Pythagoras did not prove his theory. But he only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.
Be that as it may, today not one method of proving this theorem is known, but several at once. Today we can only guess how exactly the ancient Greeks performed their calculations, so here we will look at different ways to prove the Pythagorean theorem.
Pythagorean theorem
Before you begin any calculations, you need to figure out what theory you want to prove. The Pythagorean theorem goes like this: “In a triangle in which one of the angles is 90°, the sum of the squares of the legs is equal to the square of the hypotenuse.”
There are a total of 15 different ways to prove the Pythagorean theorem. This is a fairly large number, so we will pay attention to the most popular of them.
Method one
First, let's define what we've been given. These data will also apply to other methods of proving the Pythagorean theorem, so it is worth immediately remembering all the available notations.
Suppose we are given a right triangle with legs a, b and a hypotenuse equal to c. The first method of proof is based on the fact that you need to draw a square from a right triangle.
To do this, you need to add a segment equal to leg b to leg length a, and vice versa. This should result in two equal sides of the square. All that remains is to draw two parallel lines, and the square is ready.
Inside the resulting figure, you need to draw another square with a side equal to the hypotenuse of the original triangle. To do this, from the vertices ас and св you need to draw two parallel segments equal to с. Thus, we get three sides of the square, one of which is the hypotenuse of the original right triangle. All that remains is to draw the fourth segment.
Based on the resulting figure, we can conclude that the area of the outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, there are four right triangles. The area of each is 0.5av.
Therefore, the area is equal to: 4 * 0.5ab + c 2 = 2av + c 2
Hence (a+c) 2 =2ab+c 2
And, therefore, c 2 =a 2 +b 2
The theorem has been proven.
Method two: similar triangles
This formula for proving the Pythagorean theorem was derived based on a statement from the section of geometry about similar triangles. It states that the leg of a right triangle is the average proportional to its hypotenuse and the segment of the hypotenuse emanating from the vertex of the 90° angle.
The initial data remains the same, so let's start right away with the proof. Let us draw a segment CD perpendicular to side AB. Based on the above statement, the sides of the triangles are equal:
AC=√AB*AD, SV=√AB*DV.
To answer the question of how to prove the Pythagorean theorem, the proof must be completed by squaring both inequalities.
AC 2 = AB * AD and CB 2 = AB * DV
Now we need to add up the resulting inequalities.
AC 2 + CB 2 = AB * (AD * DV), where AD + DV = AB
It turns out that:
AC 2 + CB 2 =AB*AB
And therefore:
AC 2 + CB 2 = AB 2
The proof of the Pythagorean theorem and various methods for solving it require a versatile approach to this problem. However, this option is one of the simplest.
Another calculation method
Descriptions of different methods of proving the Pythagorean theorem may not mean anything until you start practicing on your own. Many techniques involve not only mathematical calculations, but also the construction of new figures from the original triangle.
In this case, it is necessary to complete another right triangle VSD from the side BC. Thus, now there are two triangles with a common leg BC.
Knowing that the areas of similar figures have a ratio as the squares of their similar linear dimensions, then:
S avs * c 2 - S avd * in 2 = S avd * a 2 - S vsd * a 2
S avs *(from 2 - to 2) = a 2 *(S avd -S vsd)
from 2 - to 2 =a 2
c 2 =a 2 +b 2
Since out of the various methods of proving the Pythagorean theorem for grade 8, this option is hardly suitable, you can use the following method.
The easiest way to prove the Pythagorean Theorem. Reviews
According to historians, this method was first used to prove the theorem in ancient Greece. It is the simplest, as it does not require absolutely any calculations. If you draw the picture correctly, then the proof of the statement that a 2 + b 2 = c 2 will be clearly visible.
The conditions for this method will be slightly different from the previous one. To prove the theorem, assume that right triangle ABC is isosceles.
We take the hypotenuse AC as the side of the square and draw its three sides. In addition, it is necessary to draw two diagonal lines in the resulting square. So that inside it you get four isosceles triangles.
You also need to draw a square to the legs AB and CB and draw one diagonal straight line in each of them. We draw the first line from vertex A, the second from C.
Now you need to carefully look at the resulting drawing. Since on the hypotenuse AC there are four triangles equal to the original one, and on the sides there are two, this indicates the veracity of this theorem.
By the way, thanks to this method of proving the Pythagorean theorem, the famous phrase was born: “Pythagorean pants are equal in all directions.”
Proof by J. Garfield
James Garfield is the twentieth President of the United States of America. In addition to making his mark on history as the ruler of the United States, he was also a gifted autodidact.
At the beginning of his career, he was an ordinary teacher in a public school, but soon became the director of one of the higher educational institutions. The desire for self-development allowed him to propose a new theory for proving the Pythagorean theorem. The theorem and an example of its solution are as follows.
First you need to draw two right triangles on a piece of paper so that the leg of one of them is a continuation of the second. The vertices of these triangles need to be connected to ultimately form a trapezoid.
As you know, the area of a trapezoid is equal to the product of half the sum of its bases and its height.
S=a+b/2 * (a+b)
If we consider the resulting trapezoid as a figure consisting of three triangles, then its area can be found as follows:
S=av/2 *2 + s 2 /2
Now we need to equalize the two original expressions
2ab/2 + c/2=(a+b) 2 /2
c 2 =a 2 +b 2
More than one volume of textbooks could be written about the Pythagorean theorem and methods of proving it. But is there any point in it when this knowledge cannot be applied in practice?
Practical application of the Pythagorean theorem
Unfortunately, modern school curricula provide for the use of this theorem only in geometric problems. Graduates will soon leave school without knowing how they can apply their knowledge and skills in practice.
In fact, anyone can use the Pythagorean theorem in their daily life. And not only in professional activities, but also in ordinary household chores. Let's consider several cases when the Pythagorean theorem and methods of proving it may be extremely necessary.
Relationship between the theorem and astronomy
It would seem how stars and triangles on paper can be connected. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.
For example, consider the movement of a light beam in space. It is known that light moves in both directions at the same speed. Let's call the trajectory AB along which the light ray moves l. And let's call half the time it takes light to get from point A to point B t. And the speed of the beam - c. It turns out that: c*t=l
If you look at this same ray from another plane, for example, from a space liner that moves with speed v, then when observing bodies in this way, their speed will change. In this case, even stationary elements will begin to move with speed v in the opposite direction.
Let's say the comic liner is sailing to the right. Then points A and B, between which the beam rushes, will begin to move to the left. Moreover, when the beam moves from point A to point B, point A has time to move and, accordingly, the light will already arrive at a new point C. To find half the distance by which point A has moved, you need to multiply the speed of the liner by half the travel time of the beam (t ").
And to find how far a ray of light could travel during this time, you need to mark half the path with a new letter s and get the following expression:
If we imagine that points of light C and B, as well as the space liner, are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two right triangles. Therefore, thanks to the Pythagorean theorem, you can find the distance that a ray of light could travel.
This example, of course, is not the most successful, since only a few can be lucky enough to try it in practice. Therefore, let's consider more mundane applications of this theorem.
Mobile signal transmission range
Modern life can no longer be imagined without the existence of smartphones. But how much use would they be if they couldn’t connect subscribers via mobile communications?!
The quality of mobile communications directly depends on the height at which the mobile operator’s antenna is located. In order to calculate how far from a mobile tower a phone can receive a signal, you can apply the Pythagorean theorem.
Let's say you need to find the approximate height of a stationary tower so that it can distribute a signal within a radius of 200 kilometers.
AB (tower height) = x;
BC (signal transmission radius) = 200 km;
OS (radius of the globe) = 6380 km;
OB=OA+ABOB=r+x
Applying the Pythagorean theorem, we find out that the minimum height of the tower should be 2.3 kilometers.
Pythagorean theorem in everyday life
Oddly enough, the Pythagorean theorem can be useful even in everyday matters, such as determining the height of a wardrobe, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements using a tape measure. But many people wonder why certain problems arise during the assembly process if all measurements were taken more than accurately.
The fact is that the wardrobe is assembled in a horizontal position and only then raised and installed against the wall. Therefore, during the process of lifting the structure, the side of the cabinet must move freely both along the height and diagonally of the room.
Let's assume there is a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 2600 mm. An experienced furniture maker will say that the height of the cabinet should be 126 mm less than the height of the room. But why exactly 126 mm? Let's look at an example.
With ideal cabinet dimensions, let’s check the operation of the Pythagorean theorem:
AC =√AB 2 +√BC 2
AC=√2474 2 +800 2 =2600 mm - everything fits.
Let's say the height of the cabinet is not 2474 mm, but 2505 mm. Then:
AC=√2505 2 +√800 2 =2629 mm.
Therefore, this cabinet is not suitable for installation in this room. Because lifting it into a vertical position can cause damage to its body.
Perhaps, having considered different ways of proving the Pythagorean theorem by different scientists, we can conclude that it is more than true. Now you can use the information received in your daily life and be completely confident that all calculations will be not only useful, but also correct.
It would not be associated with the Pythagorean theorem. Even those who are far from mathematics in their lives continue to retain memories of the “Pythagorean pants” - a square on the hypotenuse, equal in size to two squares on the sides. The reason for the popularity of the Pythagorean theorem is clear: it is simplicity - beauty - significance. Indeed, the Pythagorean theorem is simple, but not obvious. The contradiction of the two principles gives her a special attractive force and makes her beautiful. But, in addition, the Pythagorean theorem is of great importance. It is used in geometry literally at every step. There are about five hundred different proofs of this theorem, which indicates a gigantic number of its specific implementations.
Historical studies date the birth of Pythagoras to approximately 580 BC. The happy father Mnesarchus surrounds the boy with care. He had the opportunity to give his son a good upbringing and education.
The future great mathematician and philosopher already in childhood showed great abilities for science. From his first teacher Hermodamas, Pythagoras learned the basics of music and painting. To exercise his memory, Hermodamas forced him to learn songs from the Odyssey and the Iliad. The first teacher instilled in young Pythagoras a love of nature and its secrets.
Several years have passed, and on the advice of his teacher, Pythagoras decides to continue his education in Egypt. With the help of his teacher, Pythagoras manages to leave the island of Samos. But it’s still a long way from Egypt. He lives on the island of Lesbos with his relative Zoil. There Pythagoras meets the philosopher Pherecydes, a friend of Thales of Miletus. From Pherecydes, Pythagoras studied astrology, the prediction of eclipses, the secrets of numbers, medicine and other sciences required for that time.
Then in Miletus he listens to lectures by Thales and his younger colleague and student Anaximander, an outstanding geographer and astronomer. Pythagoras acquired a lot of important knowledge during his stay at the Milesian school.
Before Egypt, he stops for some time in Phenicia, where, according to legend, he studies with the famous Sidonian priests.
According to ancient legends, while in captivity in Babylon, Pythagoras met with Persian magicians, became familiar with eastern astrology and mysticism, and became acquainted with the teachings of the Chaldean sages. The Chaldeans introduced Pythagoras to the knowledge accumulated by the eastern peoples over many centuries: astronomy and astrology, medicine and arithmetic.
Pythagoras spent twelve years in Babylonian captivity until he was freed by the Persian king Darius Hystaspes, who had heard about the famous Greek. Pythagoras is already sixty, he decides to return to his homeland in order to introduce his people to the accumulated knowledge.
Since Pythagoras left Greece, great changes have occurred there. The best minds, fleeing the Persian yoke, moved to Southern Italy, which was then called Magna Graecia, and founded the colony cities of Syracuse, Agrigentum, and Croton there. This is where Pythagoras decided to create his own philosophical school.
Quite quickly it gains great popularity among residents. Pythagoras skillfully uses the knowledge gained from traveling around the world. Over time, the scientist stops performing in churches and on the streets. Already in his home, Pythagoras taught medicine, the principles of political activity, astronomy, mathematics, music, ethics and much more. Outstanding political and statesmen, historians, mathematicians and astronomers came from his school. He was not only a teacher, but also a researcher. His students also became researchers. Pythagoras developed the theory of music and acoustics, creating the famous “Pythagorean scale” and conducting fundamental experiments on the study of musical tones: he expressed the relationships he found in the language of mathematics. The School of Pythagoras first suggested the sphericity of the Earth. The idea that the movement of celestial bodies obeys certain mathematical relationships, the ideas of “harmony of the world” and “music of the spheres,” which later led to a revolution in astronomy, first appeared precisely in the School of Pythagoras.
The scientist also did a lot in geometry. Proclus assessed the Greek scientist’s contribution to geometry in the following way: “Pythagoras transformed geometry, giving it the form of a free science, considering its principles in a purely abstract way and exploring the theorems from an immaterial, intellectual point of view. It was he who found the theory of irrational quantities and the design of cosmic bodies.”
In the school of Pythagoras, geometry was for the first time formalized into an independent scientific discipline. It was Pythagoras and his students who were the first to study geometry systematically - as a theoretical doctrine about the properties of abstract geometric figures, and not as a collection of applied recipes for land surveying.
The most important scientific merit of Pythagoras is considered to be the systematic introduction of proof into mathematics, and, above all, into geometry. Strictly speaking, only from this moment does mathematics begin to exist as a science, and not as a collection of ancient Egyptian and ancient Babylonian practical recipes. With the birth of mathematics, science in general was born, for “no human research can be called true science if it has not gone through mathematical proof” (Leonardo da Vinci).
So, the merit of Pythagoras was that he, apparently, was the first to come to the following thought: in geometry, firstly, abstract ideal objects should be considered, and, secondly, the properties of these ideal objects should not be established with using measurements on a finite number of objects, but using reasoning that is valid for an infinite number of objects. This chain of reasoning, which, using the laws of logic, reduces non-obvious statements to known or obvious truths, is a mathematical proof.
The discovery of the theorem by Pythagoras is surrounded by an aura of beautiful legends. Proclus, commenting on the last sentence of Book 1 of the Elements, writes: “If you listen to those who like to repeat ancient legends, you will have to say that this theorem goes back to Pythagoras; they say that in honor of this discovery he sacrificed a bull.” However, more generous storytellers turned one bull into one hecatomb, and this is already a whole hundred. And although Cicero noted that any shedding of blood was alien to the charter of the Pythagorean order, this legend firmly merged with the Pythagorean theorem and, two thousand years later, continued to evoke ardent responses.
