Different ways to prove the Pythagorean theorem. Different ways to prove the Pythagorean theorem: examples, description and reviews The first Pythagorean theorem

Those who are interested in the history of the Pythagorean theorem, which is studied in the school curriculum, will also be curious about such a fact as the publication in 1940 of a book with three hundred and seventy proofs of this seemingly simple theorem. But it intrigued the minds of many mathematicians and philosophers of different eras. In the Guinness Book of Records it is recorded as the theorem with the maximum number of proofs.

History of Pythagorean Theorem

Associated with the name of Pythagoras, the theorem was known long before the birth of the great philosopher. Thus, in Egypt, during the construction of structures, the aspect ratio of a right triangle was taken into account five thousand years ago. Babylonian texts mention the same aspect ratio of a right triangle 1200 years before the birth of Pythagoras.

The question arises, why then does history say that the origin of the Pythagorean theorem belongs to him? There can be only one answer - he proved the ratio of sides in a triangle. He did what those who simply used the aspect ratio and hypotenuse established by experience did not do centuries ago.

From the life of Pythagoras

The future great scientist, mathematician, philosopher was born on the island of Samos in 570 BC. Historical documents have preserved information about Pythagoras’s father, who was a precious stone carver, but there is no information about his mother. They said about the boy who was born that he was an extraordinary child who showed a passion for music and poetry from childhood. Historians include Hermodamas and Pherecydes of Syros as the teachers of young Pythagoras. The first introduced the boy into the world of the muses, and the second, being a philosopher and founder of the Italian school of philosophy, directed the young man’s gaze to the logos.

At the age of 22 (548 BC), Pythagoras went to Naucratis to study the language and religion of the Egyptians. Next, his path lay in Memphis, where, thanks to the priests, having gone through their ingenious tests, he comprehended Egyptian geometry, which, perhaps, prompted the inquisitive young man to prove the Pythagorean theorem. History will later assign this name to the theorem.

Captivity of the King of Babylon

On his way home to Hellas, Pythagoras is captured by the king of Babylon. But being in captivity benefited the inquisitive mind of the aspiring mathematician; he had a lot to learn. Indeed, in those years mathematics in Babylon was more developed than in Egypt. He spent twelve years studying mathematics, geometry and magic. And, perhaps, it was Babylonian geometry that was involved in the proof of the ratio of the sides of a triangle and the history of the discovery of the theorem. Pythagoras had enough knowledge and time for this. But there is no documentary confirmation or refutation that this happened in Babylon.

In 530 BC. Pythagoras escapes from captivity to his homeland, where he lives at the court of the tyrant Polycrates in the status of a semi-slave. Pythagoras is not satisfied with such a life, and he retires to the caves of Samos, and then goes to the south of Italy, where at that time the Greek colony of Croton was located.

Secret monastic order

On the basis of this colony, Pythagoras organized a secret monastic order, which was a religious union and a scientific society at the same time. This society had its own charter, which spoke about observing a special way of life.

Pythagoras argued that in order to understand God, a person must know such sciences as algebra and geometry, know astronomy and understand music. Research work boiled down to knowledge of the mystical side of numbers and philosophy. It should be noted that the principles preached by Pythagoras at that time make sense to be imitated at the present time.

Many of the discoveries made by Pythagoras' students were attributed to him. However, in short, the history of the creation of the Pythagorean theorem by ancient historians and biographers of that time is directly associated with the name of this philosopher, thinker and mathematician.

Teachings of Pythagoras

Perhaps the idea of the connection between the theorem and the name of Pythagoras was prompted by the statement of the great Greek that all the phenomena of our life are encrypted in the notorious triangle with its legs and hypotenuse. And this triangle is the “key” to solving all emerging problems. The great philosopher said that you should see the triangle, then you can consider that the problem is two-thirds solved.

Pythagoras spoke about his teaching only to his students orally, without making any notes, keeping it secret. Unfortunately, the teachings of the greatest philosopher have not survived to this day. Something leaked out of it, but it is impossible to say how much is true and how much is false in what became known. Even with the history of the Pythagorean theorem, not everything is certain. Historians of mathematics doubt the authorship of Pythagoras; in their opinion, the theorem was used many centuries before his birth.

Pythagorean theorem

It may seem strange, but there are no historical facts proving the theorem by Pythagoras himself - neither in the archives nor in any other sources. In the modern version it is believed that it belongs to none other than Euclid himself.

There is evidence from one of the greatest historians of mathematics, Moritz Cantor, who discovered on a papyrus stored in the Berlin Museum, written down by the Egyptians around 2300 BC. e. equality, which read: 3² + 4² = 5².

Brief history of the Pythagorean theorem

The formulation of the theorem from Euclidean “Principles”, in translation, sounds the same as in the modern interpretation. There is nothing new in her reading: the square of the side opposite the right angle is equal to the sum of the squares of the sides adjacent to the right angle. The fact that the ancient civilizations of India and China used the theorem is confirmed by the treatise “Zhou - bi suan jin”. It contains information about the Egyptian triangle, which describes the aspect ratio as 3:4:5.

No less interesting is another Chinese mathematical book, “Chu Pei,” which also mentions the Pythagorean triangle with explanations and drawings that coincide with the drawings of Hindu geometry by Bashara. About the triangle itself, the book says that if a right angle can be decomposed into its component parts, then the line that connects the ends of the sides will be equal to five if the base is equal to three and the height is equal to four.

Indian treatise "Sulva Sutra", dating back to approximately the 7th-5th centuries BC. e., talks about constructing a right angle using the Egyptian triangle.

Proof of the theorem

In the Middle Ages, students considered proving a theorem too difficult. Weak students learned theorems by heart, without understanding the meaning of the proof. In this regard, they received the nickname “donkeys”, because the Pythagorean theorem was an insurmountable obstacle for them, like a bridge for a donkey. In the Middle Ages, students came up with a humorous verse on the subject of this theorem.

To prove the Pythagorean theorem in the easiest way, you should simply measure its sides, without using the concept of areas in the proof. The length of the side opposite the right angle is c, and a and b adjacent to it, as a result we obtain the equation: a 2 + b 2 = c 2. This statement, as mentioned above, is verified by measuring the lengths of the sides of a right triangle.

If we begin the proof of the theorem by considering the area of the rectangles built on the sides of the triangle, we can determine the area of the entire figure. It will be equal to the area of a square with side (a+b), and on the other hand, the sum of the areas of four triangles and the inner square.

(a + b) 2 = 4 x ab/2 + c 2 ;

a 2 + 2ab + b 2 ;

c 2 = a 2 + b 2 , which is what needed to be proven.

The practical significance of the Pythagorean theorem is that it can be used to find the lengths of segments without measuring them. During the construction of structures, distances, placement of supports and beams are calculated, and centers of gravity are determined. The Pythagorean theorem is also applied in all modern technologies. They didn’t forget about the theorem when creating movies in 3D-6D dimensions, where in addition to the three dimensions we are used to: height, length, width, time, smell and taste are taken into account. How are tastes and smells related to the theorem, you ask? Everything is very simple - when showing a film, you need to calculate where and what smells and tastes to direct in the auditorium.

It's only the beginning. Unlimited scope for discovering and creating new technologies awaits inquisitive minds.

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relation

between the sides of a right triangle.

It is believed that it was proven by the Greek mathematician Pythagoras, after whom it was named.

Geometric formulation of the Pythagorean theorem.

The theorem was originally 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 legs.

Algebraic formulation of the Pythagorean theorem.

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

That is, denoting the length of the hypotenuse of the triangle by c, and the lengths of the legs through a And b:

Both formulations Pythagorean theorem are equivalent, but the second formulation is more elementary, it does not

requires 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.

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

right triangle.

Or, in other words:

For every triple of positive numbers a, b And c, such that

there is a right triangle with legs a And b and hypotenuse c.

Pythagorean theorem for an isosceles triangle.

Pythagorean theorem for an equilateral triangle.

Proofs of the Pythagorean theorem.

Currently, 367 proofs of this theorem have been recorded in the scientific literature. Probably theorem

Pythagoras 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:

proof area method, axiomatic And exotic evidence(For example,

by using differential equations).

1. Proof of the Pythagorean theorem using 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 there is a right triangle with a right angle C. Let's draw the height from C and denote

its foundation through H.

Triangle ACH similar to a triangle AB C at two corners. Likewise, triangle CBH similar ABC.

By introducing the notation:

we get:

which corresponds to -

Folded a 2 and b 2, we get:

or , which is what needed to be proven.

2. Proof of the Pythagorean theorem using the area method.

The proofs below, despite their apparent simplicity, are not so simple at all. All of them

use properties of area, the proofs of which are more complex than the proof of the Pythagorean theorem itself.

Proof through equicomplementarity.

Let's arrange four equal rectangular

triangle as shown in the figure

on right.

Quadrangle with sides c- square,

since the sum of two acute angles is 90°, and

unfolded angle - 180°.

The area of the entire figure is equal, on the one hand,

area of a square with side ( a+b), and on the other hand, the sum of the areas of four triangles and

Q.E.D.

3. Proof of the Pythagorean theorem by the infinitesimal method.

Looking at the drawing shown in the figure and

watching the side changea, we can

write the following relation for infinitely

small side incrementsWith And a(using similarity

triangles):

Using the variable separation method, we find:

A more general expression for the change in the hypotenuse in the case of increments on both sides:

Integrating this equation and using the initial conditions, we obtain:

Thus we arrive at the desired answer:

As is easy to see, the quadratic dependence in the final formula appears due to the linear

proportionality between the sides of the triangle and the increments, while the sum is related to the independent

contributions from the increment of different legs.

A simpler proof can be obtained if we assume that one of the legs does not experience an increase

(in this case the leg b). Then for the integration constant we obtain:

Story

Chu-pei 500-200 BC. On the left is the inscription: the sum of the squares of the lengths of the height and base is the square of the length of the hypotenuse.

In the ancient Chinese book Chu-pei ( English) (Chinese 周髀算經) 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.

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:

The theorem was originally formulated as follows:

Algebraic formulation:

That is, denoting the length of the hypotenuse of the triangle by , and the lengths of the legs by and :

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.

Converse Pythagorean theorem:

For every triple of positive numbers , and , such that , there exists a right triangle with legs and and hypotenuse .

Proof

At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. 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 there is a right triangle with a right angle C. Let's draw the height from C and denote its base by H. Triangle ACH similar to a triangle ABC at two corners. Likewise, triangle CBH similar ABC. By introducing the notation

we get

What is equivalent

Adding it up, we get

, which is what needed to be proven

Proofs using the area method

The proofs below, despite their apparent simplicity, are not so simple at all. They all use properties of area, the proof of which is more complex than the proof of the Pythagorean theorem itself.

Proof via equicomplementarity

Let's arrange four equal right triangles as shown in Figure 1.
Quadrangle with sides c is a square, since the sum of two acute angles is 90°, and the 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 have proven 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.

Proof of Leonardo da Vinci

The main elements of the proof are symmetry and motion.

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.

Proof by the infinitesimal method

The following proof using differential equations is often attributed to the famous English mathematician Hardy, who lived in the first half of the 20th century.

Looking at the drawing shown in the figure and observing the change in side a, we can write the following relation for infinitesimal side increments With And a(using triangle similarity):

Using the method of separation of variables, we find

A more general expression for the change in the hypotenuse in the case of increments on both sides

Integrating this equation and using the initial conditions, we obtain

Thus we arrive at the desired answer

As is easy to see, the quadratic dependence in the final formula appears due to the linear proportionality between the sides of the triangle and the increments, while the sum is associated with independent contributions from the increment of different legs.

A simpler proof can be obtained if we assume that one of the legs does not experience an increment (in this case leg). Then for the integration constant we obtain

Variations and generalizations

Similar geometric shapes on three sides

Generalization for similar triangles, area of green shapes A + B = area of blue C

Pythagorean theorem using similar right triangles

Euclid generalized the Pythagorean theorem in his work Beginnings, expanding the areas of the squares on the sides to the areas of similar geometric figures:

If we construct similar geometric figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the two smaller figures will be equal to the area of the larger figure.

The main idea of this generalization is that the area of such a geometric figure is proportional to the square of any of its linear dimensions and, in particular, to the square of the length of any side. Therefore, for similar figures with areas A, B And C built on sides with length a, b And c, we have:

But, according to the Pythagorean theorem, a 2 + b 2 = c 2 then A + B = C.

Conversely, if we can prove that A + B = C for three similar geometric figures without using the Pythagorean theorem, then we can prove the theorem itself, moving in the opposite direction. For example, the starting center triangle can be reused as a triangle C on the hypotenuse, and two similar right triangles ( A And B), built on the other two sides, which are formed by dividing the central triangle by its height. The sum of the two smaller triangles' areas is then obviously equal to the area of the third, thus A + B = C and, performing the previous proof in reverse order, we obtain the Pythagorean theorem a 2 + b 2 = c 2 .

Cosine theorem

The Pythagorean theorem is a special case of the more general cosine theorem, which relates the lengths of the sides in an arbitrary triangle:

where θ is the angle between the sides a And b.

If θ is 90 degrees then cos θ = 0 and the formula simplifies to the usual Pythagorean theorem.

Free Triangle

To any selected corner of an arbitrary triangle with sides a, b, c Let us inscribe an isosceles triangle in such a way that the equal angles at its base θ are equal to the chosen angle. Let us assume that the selected angle θ is located opposite the side designated c. As a result, we got triangle ABD with angle θ, which is located opposite the side a and parties r. The second triangle is formed by the angle θ, which is located opposite the side b and parties With length s, as it shown on the picture. Thabit Ibn Qurra argued that the sides in these three triangles are related as follows:

As the angle θ approaches π/2, the base of the isosceles triangle becomes smaller and the two sides r and s overlap each other less and less. When θ = π/2, ADB becomes a right triangle, r + s = c and we obtain the initial Pythagorean theorem.

Let's consider one of the arguments. Triangle ABC has the same angles as triangle ABD, but in reverse order. (The two triangles have a common angle at vertex B, both have an angle θ and also have the same third angle, based on the sum of the angles of the triangle) Accordingly, ABC is similar to the reflection ABD of triangle DBA, as shown in the lower figure. Let us write down the relationship between opposite sides and those adjacent to the angle θ,

Also a reflection of another triangle,

Let's multiply the fractions and add these two ratios:

Q.E.D.

Generalization for arbitrary triangles via parallelograms

Generalization for arbitrary triangles,
green area plot = area blue

Proof of the thesis that in the figure above

Let's make a further generalization for non-right triangles by using parallelograms on three sides instead of squares. (squares are a special case.) The top figure shows that for an acute triangle, the area of the parallelogram on the long side is equal to the sum of the parallelograms on the other two sides, provided that the parallelogram on the long side is constructed as shown in the figure (the dimensions indicated by the arrows are the same and determine sides of the lower parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the initial theorem of Pythagoras, thought to have been formulated by Pappus of Alexandria in 4 AD. e.

The bottom figure shows the progress of the proof. Let's look at the left side of the triangle. The left green parallelogram has the same area as the left side of the blue parallelogram because they have the same base b and height h. Additionally, the left green parallelogram has the same area as the left green parallelogram in the top picture because they share a common base (the top left side of the triangle) and a common height perpendicular to that side of the triangle. Using similar reasoning for the right side of the triangle, we will prove that the lower parallelogram has the same area as the two green parallelograms.

Complex numbers

The Pythagorean theorem is used to find the distance between two points in a Cartesian coordinate system, and this theorem is valid for all true coordinates: distance s between two points ( a, b) And ( c, d) equals

There are no problems with the formula if complex numbers are treated as vectors with real components x + i y = (x, y). . For example, distance s between 0 + 1 i and 1 + 0 i calculated as the modulus of the vector (0, 1) − (1, 0) = (−1, 1), or

However, for operations with vectors with complex coordinates, it is necessary to make certain improvements to the Pythagorean formula. Distance between points with complex numbers ( a, b) And ( c, d); a, b, c, And d all complex, we formulate using absolute values. Distance s based on vector difference (a − c, b − d) in the following form: let the difference a − c = p+i q, Where p- real part of the difference, q is the imaginary part, and i = √(−1). Likewise, let b − d = r+i s. Then:

where is the complex conjugate number for . For example, the distance between points (a, b) = (0, 1) And (c, d) = (i, 0) , let's calculate the difference (a − c, b − d) = (−i, 1) and the result would be 0 if complex conjugates were not used. Therefore, using the improved formula, we get

The module is defined as follows:

Stereometry

A significant generalization of the Pythagorean theorem for three-dimensional space is de Goy's theorem, named after J.-P. de Gois: if a tetrahedron has a right angle (as in a cube), then the square of the area of the face opposite the right angle is equal to the sum of the squares of the areas of the other three faces. This conclusion can be summarized as " n-dimensional Pythagorean theorem":

The Pythagorean theorem in three-dimensional space relates the diagonal AD to three sides.

Another generalization: The Pythagorean Theorem can be applied to stereometry in the following form. Consider a rectangular parallelepiped as shown in the figure. Let's find the length of the diagonal BD using the Pythagorean theorem:

where the three sides form a right triangle. We use the horizontal diagonal BD and the vertical edge AB to find the length of the diagonal AD, for this we again use the Pythagorean theorem:

or, if we write everything in one equation:

This result is a three-dimensional expression for determining the magnitude of the vector v(diagonal AD), expressed in terms of its perpendicular components ( v k ) (three mutually perpendicular sides):

This equation can be considered as a generalization of the Pythagorean theorem for multidimensional space. However, the result is actually nothing more than repeated application of the Pythagorean theorem to a sequence of right triangles in successively perpendicular planes.

Vector space

In the case of an orthogonal system of vectors, there is an equality, which is also called the Pythagorean theorem:

If - these are projections of the vector onto the coordinate axes, then this formula coincides with the Euclidean distance - and means that the length of the vector is equal to the square root of the sum of the squares of its components.

The analogue of this equality in the case of an infinite system of vectors is called Parseval's equality.

Non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry and, in fact, is not valid for non-Euclidean geometry, in the form in which it is written above. (That is, the Pythagorean theorem turns out to be a kind of equivalent to Euclid’s postulate of parallelism) In other words, in non-Euclidean geometry the relationship between the sides of a triangle will necessarily be in a form different from the Pythagorean theorem. For example, in spherical geometry, all three sides of a right triangle (say a, b And c), which limit the octant (eighth part) of the unit sphere, have a length of π/2, which contradicts the Pythagorean theorem, because a 2 + b 2 ≠ c 2 .

Let us consider here two cases of non-Euclidean geometry - spherical and hyperbolic geometry; in both cases, as for Euclidean space for right triangles, the result, which replaces the Pythagorean theorem, follows from the cosine theorem.

However, the Pythagorean theorem remains valid for hyperbolic and elliptic geometry if the requirement that the triangle is rectangular is replaced by the condition that the sum of two angles of the triangle must be equal to the third, say A+B = C. Then the relationship between the sides looks like this: the sum of the areas of circles with diameters a And b equal to the area of a circle with diameter c.

Spherical geometry

For any right triangle on a sphere with radius R(for example, if the angle γ in a triangle is right) with sides a, b, c The relationship between the parties will look like this:

This equality can be derived as a special case of the spherical cosine theorem, which is valid for all spherical triangles:

where cosh is the hyperbolic cosine. This formula is a special case of the hyperbolic cosine theorem, which is valid for all triangles:

where γ is the angle whose vertex is opposite to the side c.

Where g ij called a metric tensor. It may be a function of position. Such curved spaces include Riemannian geometry as a general example. This formulation is also suitable for Euclidean space when using curvilinear coordinates. For example, for polar coordinates:

Vector artwork

The Pythagorean theorem connects two expressions for the magnitude of a vector product. One approach to defining a cross product requires that it satisfy the equation:

this formula uses the dot product. The right side of the equation is called the Gram determinant for a And b, which is equal to the area of the parallelogram formed by these two vectors. Based on this requirement, as well as the requirement that the vector product is perpendicular to its components a And b it follows that, except for trivial cases from 0- and 1-dimensional space, the cross product is defined only in three and seven dimensions. We use the definition of the angle in n-dimensional space:

This property of a cross product gives its magnitude as follows:

Through the fundamental trigonometric identity of Pythagoras we obtain another form of writing its value:

An alternative approach to defining a cross product is to use an expression for its magnitude. Then, reasoning in reverse order, we obtain a connection with the scalar product:

Notes

History topic: Pythagoras’s theorem in Babylonian mathematics
( , p. 351) p. 351
( , Vol I, p. 144)
A discussion of historical facts is given in (, P. 351) P. 351
Kurt Von Fritz (Apr., 1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics, Second Series(Annals of Mathematics) 46 (2): 242–264.
Lewis Carroll, “The Story with Knots”, M., Mir, 1985, p. 7
Asger Aaboe Episodes from the early history of mathematics. - Mathematical Association of America, 1997. - P. 51. - ISBN 0883856131
Python Proposition by Elisha Scott Loomis
Euclid's Elements: Book VI, Proposition VI 31: “In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.”
Lawrence S. Leff cited work. - Barron's Educational Series. - P. 326. - ISBN 0764128922
Howard Whitley Eves§4.8:...generalization of Pythagorean theorem // Great moments in mathematics (before 1650). - Mathematical Association of America, 1983. - P. 41. - ISBN 0883853108
Tâbit ibn Qorra (full name Thābit ibn Qurra ibn Marwan Al-Ṣābiʾ al-Ḥarrānī) (826-901 AD) was a physician living in Baghdad who wrote extensively on Euclid’s Elements and other mathematical subjects.
Aydin Sayili (Mar. 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem." Isis 51 (1): 35–37. DOI:10.1086/348837.
Judith D. Sally, Paul Sally Exercise 2.10 (ii) // Cited work. - P. 62. - ISBN 0821844032
For the details of such a construction, see George Jennings Figure 1.32: The generalized Pythagorean theorem // Modern geometry with applications: with 150 figures. - 3rd. - Springer, 1997. - P. 23. - ISBN 038794222X
Arlen Brown, Carl M. Pearcy Item C: Norm for an arbitrary n-tuple ... // An introduction to analysis . - Springer, 1995. - P. 124. - ISBN 0387943692 See also pages 47-50.
Alfred Gray, Elsa Abbena, Simon Salamon Modern differential geometry of curves and surfaces with Mathematica. - 3rd. - CRC Press, 2006. - P. 194. - ISBN 1584884487
Rajendra Bhatia Matrix analysis. - Springer, 1997. - P. 21. - ISBN 0387948465
Stephen W. Hawking cited work. - 2005. - P. 4. - ISBN 0762419229

Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (Fig. 1):

$c^(2)=a^(2)+b^(2)$

Proof of the Pythagorean Theorem

Let triangle $A B C$ be a right triangle with right angle $C$ (Fig. 2).

Let us draw the height from the vertex $C$ to the hypotenuse $A B$, and denote the base of the height as $H$.

Right triangle $A C H$ is similar to triangle $A B C$ at two angles ($\angle A C B=\angle C H A=90^(\circ)$, $\angle A$ is common). Likewise, triangle $C B H$ is similar to $A B C$ .

By introducing the notation

$$B C=a, A C=b, A B=c$$

from the similarity of triangles we get that

$$\frac(a)(c)=\frac(H B)(a), \frac(b)(c)=\frac(A H)(b)$$

From here we have that

$$a^(2)=c \cdot H B, b^(2)=c \cdot A H$$

Adding the resulting equalities, we get

$$a^(2)+b^(2)=c \cdot H B+c \cdot A H$$

$$a^(2)+b^(2)=c \cdot(H B+A H)$$

$$a^(2)+b^(2)=c \cdot A B$$

$$a^(2)+b^(2)=c \cdot c$$

$$a^(2)+b^(2)=c^(2)$$

Q.E.D.

Geometric formulation of the Pythagorean theorem

Theorem

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 (Fig. 2):

Examples of problem solving

Example

Exercise. Given a right triangle $A B C$, the sides of which are 6 cm and 8 cm. Find the hypotenuse of this triangle.

Solution. According to the leg condition $a=6$ cm, $b=8$ cm. Then, according to the Pythagorean theorem, the square of the hypotenuse

$c^(2)=a^(2)+b^(2)=6^(2)+8^(2)=36+64=100$

From this we obtain that the desired hypotenuse

$c=\sqrt(100)=10$ (cm)

Answer. 10 cm

Example

Exercise. Find the area of a right triangle if it is known that one of its legs is 5 cm larger than the other and the hypotenuse is 25 cm.

Solution. Let $x$ cm be the length of the smaller leg, then $(x+5)$ cm be the length of the larger one. Then, according to the Pythagorean theorem, we have:

$$x^(2)+(x+5)^(2)=25^(2)$$

We open the brackets, reduce similar ones and solve the resulting quadratic equation:

$x^(2)+5 x-300=0$

According to Vieta's theorem, we obtain that

$x_(1)=15$ (cm) , $x_(2)=-20$ (cm)

The value $x_(2)$ does not satisfy the conditions of the problem, which means that the smaller leg is 15 cm, and the larger leg is 20 cm.

The area of a right triangle is equal to half the product of the lengths of its legs, that is

$$S=\frac(15 \cdot 20)(2)=15 \cdot 10=150\left(\mathrm(cm)^(2)\right)$$

Answer.$S=150\left(\mathrm(cm)^(2)\right)$

Historical reference

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle.

The ancient Chinese book "Zhou Bi Xuan Jing" talks about a Pythagorean triangle with sides 3, 4 and 5. The leading German historian of mathematics, Moritz Cantor (1829 - 1920), believes that the equality $3^(2)+4^(2)=5^ (2) $ was already known to the Egyptians around 2300 BC. According to the scientist, builders then built right angles using right triangles with sides 3, 4 and 5. Somewhat more is known about the Pythagorean theorem among the Babylonians. One text gives an approximate calculation of the hypotenuse of an isosceles right triangle.

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.

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Introduction

In a school geometry course, only mathematical problems are solved using the Pythagorean theorem. Unfortunately, the question of the practical application of the Pythagorean theorem is not considered.

In this regard, the purpose of my work was to find out the areas of application of the Pythagorean theorem.

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

I will consider examples of the practical application of the Pythagorean theorem. I will not try to give all examples of the use of the theorem - this would hardly be possible. The scope of the theorem is quite extensive and generally cannot be indicated with sufficient completeness.

Hypothesis:

Using the Pythagorean theorem, you can solve not only mathematical problems.

For this research work, the following goal is defined:

Find out the areas of application of the Pythagorean theorem.

Based on the above goal, the following tasks were identified:

Collect information about the practical application of the Pythagorean theorem in various sources and determine the areas of application of the theorem.

Study some historical information about Pythagoras and his theorem.

Show the application of the theorem in solving historical problems.

Process the collected data on the topic.

I was engaged in searching and collecting information - studying printed material, working with material on the Internet, processing the collected data.

Research methodology:

Studying theoretical material.

Study of research methods.

Practical implementation of the study.

Communicative (measurement method, questionnaire).

Project type: information and research. The work was done in free time.

About Pythagoras.

Pythagoras - ancient Greek philosopher, mathematician, astronomer. He substantiated many properties of geometric figures, developed a mathematical theory of numbers and their proportions. He made significant contributions to the development of astronomy and acoustics. Author of the Golden Verses, founder of the Pythagorean school in Croton.

According to legend, Pythagoras was born around 580 BC. e. on the island of Samos in a wealthy merchant family. His mother, Pyphasis, received her name in honor of Pythia, a priestess of Apollo. Pythia predicted to Mnesarchus and his wife the birth of a son, the son was also named after Pythia. According to many ancient testimonies, the boy was fabulously beautiful and soon showed his extraordinary abilities. He received his first knowledge from his father Mnesarchus, a jeweler and precious stone carver, who dreamed that his son would continue his business. But life decided otherwise. The future philosopher showed great abilities for science. Among Pythagoras' teachers were Pherecydes of Syros and Elder Hermodamant. The first instilled in the boy a love of science, and the second - of music, painting and poetry. Subsequently, Pythagoras met the famous philosopher and mathematician Thales of Miletus and, on his advice, went to Egypt, the center of scientific and research activity at that time. After living 22 years in Egypt and 12 years in Babylon, he returned to the island of Samos, then left it for unknown reasons and moved to the city of Croton, in southern Italy. Here he created the Pythagorean school (union), in which various issues of philosophy and mathematics were studied. At the age of approximately 60, Pythagoras married Theano, one of his students. They have three children, all of whom become followers of their father. The historical conditions of that time are characterized by a broad movement of the demos against the power of the aristocrats. Fleeing from the waves of popular anger, Pythagoras and his students moved to the city of Tarentum. According to one version: Kilon, a rich and evil man, came to him, wanting to join the brotherhood while drunk. Having been refused, Cylon began to fight Pythagoras. During the fire, the students saved the teacher’s life at their own cost. Pythagoras became sad and soon committed suicide.

It should be noted that this is one of the options for his biography. The exact dates of his birth and death have not been established; many facts about his life are contradictory. But one thing is clear: this man lived and left his descendants with a great philosophical and mathematical heritage.

Pythagorean theorem.

The Pythagorean theorem is the most important statement of geometry. The theorem is formulated as follows: 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.

The discovery of this statement is attributed to Pythagoras of Samos (12th century BC)

A study of Babylonian cuneiform tablets and ancient Chinese manuscripts (copies of even more ancient manuscripts) showed that the famous theorem was known long before Pythagoras, perhaps several thousand years before him.

(But there is an assumption that Pythagoras gave a complete proof of it)

But there is another opinion: in the Pythagorean school there was a wonderful custom of attributing all the merits to Pythagoras and not attributing to themselves the glory of the discoverers, except perhaps in a few cases.

(Iamblichus-Syrian Greek-speaking writer, author of the treatise “The Life of Pythagoras.” (2nd century AD)

Thus, the German mathematical historian Cantor believes that the equality 3 2 + 4 2 = 5 2 was

known to the Egyptians around 2300 BC. e. during the time of King Amenehmet (according to papyrus 6619 of the Berlin Museum). Some believe that Pythagoras gave the theorem a complete proof, while others deny him this merit.

Some attribute to Pythagoras the proof that Euclid gave in his Elements. On the other hand, Proclus (mathematician, 5th century) claims that the proof in the Elements belonged to Euclid himself, that is, the history of mathematics has preserved almost no reliable data about the mathematical activity of Pythagoras. In mathematics, perhaps, there is no other theorem that deserves all kinds of comparisons.

In some lists of Euclid's Elements, this theorem was called the “nymph theorem” for the similarity of the drawing with a bee, a butterfly (“butterfly theorem”), which in Greek was called a nymph. The Greeks used this word to name some other goddesses, as well as young women and brides. The Arabic translator did not pay attention to the drawing and translated the word “nymph” as “bride.” This is how the affectionate name “bride’s theorem” appeared. There is a legend that when Pythagoras of Samos proved his theorem, he thanked the gods by sacrificing 100 bulls. Hence another name - “the theorem of a hundred bulls”.

In English-speaking countries it was called: “windmill”, “peacock tail”, “bride’s chair”, “donkey bridge” (if the student could not “cross” it, then he was a real “donkey”)

In pre-revolutionary Russia, the drawing of the Pythagorean theorem for the case of an isosceles triangle was called “Pythagorean pants.”

These “pants” appear when you build squares on each side of a right triangle to the outside.

How many different proofs of Pythagoras' theorem are there?

Since the time of Pythagoras, more than 350 of them have appeared. The theorem was included in the Guinness Book of Records. If we analyze the proofs of the theorem, they use few fundamentally different ideas.

Areas of application of the theorem.

It is widely used in solving geometric tasks.

It is with its help that you can geometrically find the values of square roots of integers:

To do this, we build a right triangle AOB (angle A is 90°) with unit legs. Then its hypotenuse is √2. Then we construct a unit segment BC, BC is perpendicular to OB, the length of the hypotenuse OC = √3, etc.

(we meet this method in Euclid and F. Kirensky).

Tasks in the know physicists High schools require knowledge of the Pythagorean theorem.

These are problems related to the addition of velocities.

Pay attention to the slide: a problem from a 9th grade physics textbook. In a practical sense, it can be formulated as follows: at what angle to the river flow should a boat transporting passengers between piers move in order to meet the schedule? (the piers are on opposite banks of the river)

When a biathlete shoots at a target, he makes an “adjustment for the wind.” If the wind is blowing from the right, and the athlete shoots straight, the bullet will go to the left. To hit the target, you need to move the sight to the right by the distance the bullet is displaced. Special tables have been compiled for them (based on corollaries from Pythagoras). The biathlete knows at what angle to move the sight when the wind speed is known.

Astronomy - also a wide area for application of the theorem Path of the light beam. The figure shows the path of a light ray from A to B and back. The beam path is shown with a curved arrow for clarity; in fact, the light beam is straight.

What path does the ray take?? Light travels the same path back and forth. What is half the distance the ray travels? If we designate the segment AB symbol l, half the time like t, and also denoting the speed of light with the letter c, then our equation will take the form

c * t = l

This is the product of the time spent and the speed!

Now let's try to look at the same phenomenon from a different frame of reference, for example, from a spaceship flying past a running beam at a speed v. With such an observation, the velocities of all bodies will change, and stationary bodies will begin to move at a speed v in the opposite direction. Let's assume that the ship is moving to the left. Then the two points between which the bunny runs will begin to move to the right at the same speed. Moreover, while the bunny runs its way, the starting point A shifts and the beam returns to a new point C.

Question: how much does the point have time to move (to turn into point C) while the light beam travels? More precisely: what is half of this displacement? If we denote half the travel time of the beam by the letter t", and half the distance A.C. letter d, then we get our equation in the form:

v * t" = d

Letter v indicates the speed of the spacecraft.

Another question: how far will the light beam travel?(More precisely, what is half of this path? What is the distance to the unknown object?)

If we denote half the length of the light path by the letter s, we get the equation:

c * t" = s

Here c is the speed of light, and t"- this is the same time as discussed above.

Now consider the triangle ABC. This is an isosceles triangle whose height is l, which we introduced when considering the process from a fixed point of view. Since the movement is perpendicular l, then it could not affect her.

Triangle ABC composed of two halves - identical right-angled triangles, the hypotenuses of which AB And B.C. must be connected to the legs according to the Pythagorean theorem. One of the legs is d, which we just calculated, and the second leg is s, which the light passes through, and which we also calculated. We get the equation:

s 2 = l 2 + d 2

This is Pythagorean theorem!

Phenomenon stellar aberration, discovered in 1729, is that all the stars on the celestial sphere describe ellipses. The semimajor axis of these ellipses is observed from Earth at an angle of 20.5 degrees. This angle is associated with the movement of the Earth around the Sun at a speed of 29.8 km per hour. In order to observe a star from a moving Earth, it is necessary to tilt the telescope tube forward according to the movement of the star, since while the light travels the length of the telescope, the eyepiece moves forward along with the earth. The addition of the speeds of light and the Earth is done vectorially, using the so-called.

Pythagoras. U 2 =C 2 +V 2

C-speed of light

V-ground speed

Telescope tube

At the end of the nineteenth century, various assumptions were made about the existence of human-like inhabitants of Mars, this was a consequence of the discoveries of the Italian astronomer Schiaparelli (he discovered canals on Mars that had long been considered artificial). Naturally, the question of whether it is possible to communicate with these hypothetical creatures using light signals has caused a lively discussion. The Paris Academy of Sciences even established a prize of 100,000 francs for the first person to establish contact with any inhabitant of another celestial body; this prize is still waiting for the lucky winner. As a joke, although not entirely without reason, it was decided to transmit a signal to the inhabitants of Mars in the form of the Pythagorean theorem.

It is not known how to do this; but it is obvious to everyone that the mathematical fact expressed by the Pythagorean theorem holds everywhere, and therefore inhabitants of another world similar to us must understand such a signal.

mobile connection

Who in the modern world does not use a cell phone? Every mobile phone subscriber is interested in its quality. And the quality, in turn, depends on the height of the mobile operator’s antenna. To calculate the radius within which transmission can be received, we use Pythagorean theorem.

What is the maximum height of a mobile operator's antenna so that transmission can be received within a radius of R=200 km? (The radius of the Earth is 6380 km.)

Solution:

Let AB=x , BC=R=200 km , OC= r =6380 km.

OB=OA+ABOB=r + x.

Using the Pythagorean theorem, we get Answer: 2.3 km.

When building houses and cottages, the question often arises about the length of the rafters for the roof if the beams have already been made. For example: it is planned to build a gable roof on a house (sectional shape). What length should the rafters be if the beams are made AC=8 m, and AB=BF.

Solution:

Triangle ADC is isosceles AB=BC=4 m, BF=4 m. If we assume that FD=1.5 m, then:

A) From triangle DBC: DB=2.5 m.

B) From triangle ABF:

Window

In buildings Gothic and Romanesque style the upper parts of the windows are divided by stone ribs, which not only play the role of ornament, but also contribute to the strength of the windows. The figure shows a simple example of such a window in the Gothic style. The method of constructing it is very simple: From the figure it is easy to find the centers of six arcs of circles whose radii are equal

window width (b) for external arches

half the width, (b/2) for internal arcs

There remains a complete circle touching four arcs. Since it is enclosed between two concentric circles, its diameter is equal to the distance between these circles, i.e. b/2 and, therefore, the radius is b/4. And then it becomes clear and

the position of its center.

IN romanesque architecture The motif shown in the figure is often found. If b still denotes the width of the window, then the radii of the semicircles will be R = b / 2 and r = b / 4. The radius p of the inner circle can be calculated from the right triangle shown in Fig. dotted line The hypotenuse of this triangle, passing through the point of tangency of the circles, is equal to b/4+p, one side is equal to b/4, and the other is b/2-p. According to the Pythagorean theorem we have:

(b/4+p) 2 =(b/4) 2 +(b/4-p) 2

b 2 /16+ bp/2+p 2 =b 2 /16+b 2 /4 - bp/2 +p 2 ,

Dividing by b and bringing similar terms, we get:

(3/2)p=b/4, p=b/6.

In the forest industry: for construction needs, logs are cut into beams, and the main task is to obtain as little waste as possible. The least amount of waste will occur when the timber has the largest volume. What should be in the section? As can be seen from the solution, the cross section must be square, and Pythagorean theorem and other considerations allow us to draw such a conclusion.

Largest volume timber

Task

From a cylindrical log you need to cut a rectangular beam of the largest volume. What shape should its cross-section be (Fig. 23)?

Solution

If the sides of a rectangular section are x and y, then by the Pythagorean theorem

x 2 + y 2 = d 2,

where d is the diameter of the log. The volume of the beam is greatest when its cross-sectional area is greatest, that is, when xy reaches its greatest value. But if xy is greatest, then the product x 2 y 2 will also be greatest. Since the sum x 2 + y 2 is unchanged, then, according to what was proven earlier, the product x 2 y 2 is greatest when

x 2 = y 2 or x = y.

So, the cross-section of the beam should be square.

Transport tasks(so-called optimization problems; problems, the solution of which allows us to answer the question: how to allocate funds to achieve great benefits)

At first glance, nothing special: take measurements of the height from floor to ceiling at several points, subtract a few centimeters so that the cabinet does not rest against the ceiling. By doing this, difficulties may arise in the process of assembling furniture. After all, furniture makers assemble the frame by placing the cabinet in a horizontal position, and when the frame is assembled, they lift it to a vertical position. Let's look at the side wall of the cabinet. The height of the cabinet should be 10 cm less than the distance from floor to ceiling, provided that this distance does not exceed 2500 mm. And the depth of the cabinet is 700 mm. Why 10 cm, and not 5 cm or 7, and what does the Pythagorean theorem have to do with it?

So: side wall 2500-100=2400 (mm) - maximum height of the structure.

During the process of lifting the frame, the side wall must pass freely both vertically and diagonally. By Pythagorean theorem

AC = √ AB 2 + BC 2

AC = √ 2400 2 + 700 2 = 2500 (mm)

What happens if the height of the cabinet is reduced by 50 mm?

AC = √ 2450 2 + 700 2 = 2548 (mm)

Diagonal 2548 mm. This means you can’t install a cabinet (you could ruin the ceiling).

Lightning rod.

It is known that a lightning rod protects from lightning all objects whose distance from its base does not exceed twice its height. It is necessary to determine the optimal position of the lightning rod on a gable roof, ensuring its lowest accessible height.

According to the Pythagorean theorem h 2 ≥ a 2 +b 2 means h≥(a 2 +b 2) 1/2

We urgently need to make a greenhouse for seedlings at our summer cottage.

A 1m1m square is made from boards. There are film remnants measuring 1.5m1.5m. At what height in the center of the square should the strip be attached so that the film completely covers it?

1) Greenhouse diagonal d==1.4;0.7

2) Film diagonal d 1= 2,12 1,06

3) Rail height x= 0,7

Conclusion

As a result of the research, I found out some areas of application of the Pythagorean theorem. I have collected and processed a lot of material from literary sources and the Internet on this topic. I studied some historical information about Pythagoras and his theorem. Yes, indeed, using the Pythagorean theorem you can solve not only mathematical problems. The Pythagorean theorem has found its application in construction and architecture, mobile communications, and literature.

Study and analysis of sources of information about the Pythagorean theorem

showed that:

A) the exclusive attention of mathematicians and mathematics lovers to the theorem is based on its simplicity, beauty and significance;

b) for many centuries, the Pythagorean theorem has served as an impetus for interesting and important mathematical discoveries (Fermat’s theorem, Einstein’s theory of relativity);

V) Pythagorean theorem - is the embodiment of the universal language of mathematics, valid throughout the world;

G) the scope of the theorem is quite extensive and generally cannot be indicated with sufficient completeness;

d) the secrets of the Pythagorean theorem continue to excite humanity and therefore each of us is given a chance to be involved in their discovery.

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