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 proved by the Greek mathematician Pythagoras, after whom it is 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 catheters.

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 through c, and the lengths of the legs through a and b:

Both formulations pythagorean theorems 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.

The inverse 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

triangle is rectangular.

Or, in other words:

For any triple of positive numbers a, b and c, such that

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

The Pythagorean theorem for an isosceles triangle.

Pythagorean theorem for an equilateral triangle.

Proofs of the Pythagorean theorem.

At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably the 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 of area method, axiomatic and exotic evidence(for example,

by using differential equations).

1. Proof of the Pythagorean theorem in terms of 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 the area of ​​a figure.

Let ABC there is a right angled triangle C. Let's draw a height from C and denote

its foundation through H.

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

By introducing the notation:

we get:

,

which matches -

Having folded a 2 and b 2 , we get:

or , which was to be proved.

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

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

use the properties of the area, the proof of which is more complicated than the proof of the Pythagorean theorem itself.

• Proof through equicomplementation.

Arrange four equal rectangular

triangle as shown in the picture

on right.

Quadrilateral with sides c- square,

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

the developed angle is 180°.

The area of ​​the whole figure is, 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.

​

Considering the drawing shown in the figure, and

watching the side changea, we can

write the following relation for infinite

small side incrementsWith and a(using similarity

triangles):

Using the method of separation of variables, we find:

A more general expression for changing the hypotenuse in the case of increments of both legs:

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

Thus, we arrive at the desired answer:

As it 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 increment

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

Never forget the Pythagorean theorem. The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. In other words, 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 its legs.

Denoting the length of the hypotenuse of the triangle through c, and the lengths of the legs through a and b:

Hypotenuse is one of the sides of a right triangle. Also in this triangle there are two leg.

In this case, the hypotenuse is the side that is opposite the right angle. And the legs are the sides that form a given angle.

According to the Pythagorean theorem, the square of the hypotenuse will be equal to the sum of the squares of the legs.

That is, AB = AC + BC.

The converse is also true - if this equality holds in a triangle, then this triangle is right-angled.

This property helps to solve many geometric problems.

There is a slightly different formulation of this theorem: the area of ​​the square, which is built on the hypotenuse, is equal to the sum of the areas of the squares, built on the legs.

The square of the hypotenuse is equal to the sum of the squares of the legs ... from school by heart. This is one of those rules that will be remembered forever.)))

The square of the hypotenuse is equal to the sum of the squares of the legs

That's right, the square of the hypotenuse is equal to the sum of the squares of the legs. Of course, this was taught to us, and that this Pythagorean theorem leaves no doubt, it is so nice to remember what was taught a long time ago among the usual routine.

It depends on the length of this hypotenuse. If it is equal to one meter, then e square is one square meter. And if, for example, it is equal to 39.37 inches, then e square is equal to 1550 square inches, nothing can be done about it.

The square of the hypotenuse is equal to the sum of the squares of the legs - the Pythagorean theorem (by the way, the easiest paragraph in the geometry textbook)

Yes, the square of the hypotenuse is equal to the sum of the squares of the legs. It's like we were taught in school. How many years have passed, and we still remember this theorem, beloved by us. Probably, strain and prove I can, as in the school curriculum.

They also said a counting Pythagorean pants, equal in all directions;

The teacher told us that if you are sleeping and suddenly there is a fire - you must know the Pythagorean theorem))) It is equal to the sum of the squares of the legs

The square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle (legs).

You can remember this, or you can understand once and for all why this is so.

to begin with, consider a right-angled triangle with identical legs and place it inside a square with a side equal to the hypotenuse.

The area of ​​the large square will be equal to the area of ​​four identical triangles inside it.

We quickly calculate everything and get the result we need.



In case the legs are not the same, everything is also quite simple:

the area of ​​the large square is equal to the sum of the areas of four identical triangles plus the area of ​​the square in the middle.



Whatever one may say, we always get equality

the sum of the squares of the legs is equal to the square of the hypotenuse.

One of the most famous in geometry, the Pythagorean theorem states:

This theorem concerns a right triangle, that is, one whose angle is 90 degrees. The sides of a right angle are called the legs, and the oblique sides are called the hypotenuse. So, if you draw three squares with a base at each side of the triangle, then the area of ​​​​the two squares near the leg is equal to the area of ​​​​the square near the hypotenuse.

The potential for creativity is usually attributed to the humanities, leaving the natural scientific analysis, practical approach and dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity in the "queen of all sciences" you will not go far - people have known about this for a long time. Since the time of Pythagoras, for example.

School textbooks, unfortunately, usually do not explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and feel its fundamental principles. And at the same time, try to free your mind from clichés and elementary truths - only in such conditions are all great discoveries born.

Such discoveries include the one that today we know as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be fun. And that this adventure is suitable not only for nerds in thick glasses, but for everyone who is strong in mind and strong in spirit.

From the history of the issue

Strictly speaking, although the theorem is called the "Pythagorean theorem", Pythagoras himself did not discover it. The right triangle and its special properties have been studied long before it. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.

Today you can no longer check who is right and who is wrong. It is only known that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid's Elements may belong to Pythagoras, and Euclid only recorded it.

It is also known today that problems about a right-angled triangle are found in Egyptian sources from the time of Pharaoh Amenemhet I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise Sulva Sutra and the ancient Chinese work Zhou-bi suan jin.

As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. Approximately 367 various pieces of evidence that exist today serve as confirmation. No other theorem can compete with it in this respect. Notable evidence authors include Leonardo da Vinci and the 20th President of the United States, James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or, in one way or another, connected with it.

Proofs of the Pythagorean Theorem

School textbooks mostly give algebraic proofs. But the essence of the theorem is in geometry, so let's first of all consider those proofs of the famous theorem that are based on this science.

Proof 1

For the simplest proof of the Pythagorean theorem for a right triangle, you need to set ideal conditions: let the triangle be not only right-angled, but also isosceles. There is reason to believe that it was such a triangle that was originally considered by ancient mathematicians.

Statement "a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs" can be illustrated with the following drawing:

Look at the isosceles right triangle ABC: On the hypotenuse AC, you can build a square consisting of four triangles equal to the original ABC. And on the legs AB and BC built on a square, each of which contains two similar triangles.

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

Proof 2

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

Construct a right triangle with sides a, b and c(Fig. 1). Then build two squares with sides equal to the sum of the lengths of the two legs - (a+b). In each of the squares, make constructions, as in figures 2 and 3.

In the first square, build four of the same triangles as in Figure 1. As a result, two squares are obtained: one with side a, the second with side b.

In the second square, four similar triangles constructed form a square with a side equal to the hypotenuse c.

The sum of the areas of the constructed squares in Fig. 2 is equal to the area of ​​the square we constructed with side c in Fig. 3. This can be easily verified by calculating the areas of the squares in Fig. 2 according to the formula. And the area of ​​​​the inscribed square in Figure 3. by subtracting the areas of four equal right-angled triangles inscribed in the square from the area of ​​\u200b\u200ba large square with a side (a+b).

Putting all this down, we have: a 2 + b 2 \u003d (a + b) 2 - 2ab. Expand the brackets, do all the necessary algebraic calculations and get that a 2 + b 2 = a 2 + b 2. At the same time, the area of ​​the inscribed in Fig.3. square can also be calculated using the traditional formula S=c2. Those. a2+b2=c2 You have proved the Pythagorean theorem.

Proof 3

The very same ancient Indian proof is described in the 12th century in the treatise “The Crown of Knowledge” (“Siddhanta Shiromani”), and as the main argument the author uses an appeal addressed to the mathematical talents and powers of observation of students and followers: “Look!”.

But we will analyze this proof in more detail:

Inside the square, build four right-angled triangles as indicated in the drawing. The side of the large square, which is also the hypotenuse, is denoted With. Let's call the legs of the triangle a and b. According to the drawing, the side of the inner square is (a-b).

Use the square area formula S=c2 to calculate the area of ​​the outer square. And at the same time calculate the same value by adding the area of ​​​​the inner square and the area of ​​\u200b\u200ball four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options to calculate the area of ​​a square to make sure they give the same result. And that gives you the right to write down that c 2 =(a-b) 2 +4*1\2*a*b. As a result of the solution, you will get the formula of the Pythagorean theorem c2=a2+b2. The theorem has been proven.

Proof 4

This curious ancient Chinese evidence was called the "Bride's Chair" - because of the chair-like figure that results from all the constructions:

It uses the drawing we have already seen in Figure 3 in the second proof. And the inner square with side c is constructed in the same way as in the ancient Indian proof given above.

If you mentally cut off two green right-angled triangles from the drawing in Fig. 1, transfer them to opposite sides of the square with side c and attach the hypotenuses to the hypotenuses of the lilac triangles, you will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will see that the "bride's chair" is formed by two squares: small ones with a side b and big with a side a.

These constructions allowed the ancient Chinese mathematicians and us following them to come to the conclusion that c2=a2+b2.

Proof 5

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

Construct a right triangle ABC. We need to prove that BC 2 \u003d AC 2 + AB 2.

To do this, continue the leg AC and build a segment CD, which is equal to the leg AB. Lower Perpendicular AD line segment ED. Segments ED and AC are equal. connect the dots E and AT, as well as E and FROM and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tested: we find the area of ​​the resulting figure in two ways and equate the expressions to each other.

Find the area of ​​a polygon ABED can be done by adding the areas of the three triangles that form it. And one of them ERU, is not only rectangular, but also isosceles. Let's also not forget that AB=CD, AC=ED and BC=CE- this will allow us to simplify the recording and not overload it. So, S ABED \u003d 2 * 1/2 (AB * AC) + 1 / 2BC 2.

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

Let's write both ways to calculate the area of ​​​​a figure by putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify the right-hand side of the notation: AB*AC+1/2BC 2 =1/2(AB+AC) 2. And now we open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having finished all the transformations, we get exactly what we need: BC 2 \u003d AC 2 + AB 2. We have proved the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proved using vectors, complex numbers, differential equations, stereometry, etc. And even physicists: if, for example, liquid is poured into square and triangular volumes similar to those shown in the drawings. By pouring liquid, it is possible to prove the equality of areas and the theorem itself as a result.

A few words about Pythagorean triplets

This issue is little or not studied in the school curriculum. Meanwhile, it is very interesting and is of great importance in geometry. Pythagorean triples are used to solve many mathematical problems. The idea of ​​them can be useful to you in further education.

So what are Pythagorean triplets? So called natural numbers, collected in threes, the sum of the squares of two of which is equal to the third number squared.

Pythagorean triples can be:

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

Even before our era, the ancient Egyptians were fascinated by the mania for the numbers of Pythagorean triplets: in tasks they considered a right-angled triangle with sides of 3.4 and 5 units. By the way, any triangle whose sides are equal to the numbers from the Pythagorean triple is by default rectangular.

Examples of Pythagorean triples: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20) ), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (10, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (27, 36, 45), (14 , 48, 50), (30, 40, 50) etc.

Practical application of the theorem

The Pythagorean theorem finds application not only in mathematics, but also in architecture and construction, astronomy, and even literature.

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

Let's denote the width of the window as b, then the radius of the great semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed in terms of b: r=b/4. In this problem, we are interested in the radius of the inner circle of the window (let's call it p).

The Pythagorean theorem just comes in handy to calculate R. To do this, we use a right-angled triangle, which is indicated by a dotted line in the figure. The hypotenuse of a triangle consists of two radii: b/4+p. One leg is a radius b/4, another b/2-p. Using the Pythagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Next, we open the brackets and get b 2 /16+ bp / 2 + p 2 \u003d b 2 / 16 + b 2 / 4-bp + p 2. Let's transform this expression into bp/2=b 2 /4-bp. And then we divide all the terms into b, we give similar ones to get 3/2*p=b/4. And in the end we find that p=b/6- which is what we needed.

Using the theorem, you can calculate the length of the rafters for a gable roof. Determine how high a mobile tower is needed for the signal to reach a certain settlement. And even steadily install a Christmas tree in the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

As far as literature is concerned, the Pythagorean theorem has inspired writers since antiquity and continues to do so today. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired by her to write a sonnet:

The light of truth will not soon dissipate,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
Will not cause doubts and disputes.

The wisest when it touches the eye
Light of truth, thank the gods;
And a hundred bulls, stabbed, lie -
The return gift of the lucky Pythagoras.

Since then, the bulls have been roaring desperately:
Forever aroused the bull tribe
event mentioned here.

They think it's about time
And again they will be sacrificed
Some great theorem.

(translated by Viktor Toporov)

And in the twentieth century, the Soviet writer Yevgeny Veltistov in his book "The Adventures of Electronics" devoted a whole chapter to the proofs of the Pythagorean theorem. And half a chapter of the story about the two-dimensional world that could exist if the Pythagorean theorem became the fundamental law and even religion for a single world. It would be much easier to live in it, but also much more boring: for example, no one there understands the meaning of the words “round” and “fluffy”.

And in the book “The Adventures of Electronics”, the author, through the mouth of the mathematics teacher Taratara, says: “The main thing in mathematics is the movement of thought, new ideas.” It is this creative flight of thought that generates the Pythagorean theorem - it is not for nothing that it has so many diverse proofs. It helps to go beyond the usual, and look at familiar things in a new way.

Conclusion

This article was created so that you can look beyond the school curriculum in mathematics and learn not only those proofs of the Pythagorean theorem that are given in the textbooks "Geometry 7-9" (L.S. Atanasyan, V.N. Rudenko) and "Geometry 7 -11” (A.V. Pogorelov), but also other curious ways to prove the famous theorem. And also see examples of how the Pythagorean theorem can be applied in everyday life.

Firstly, this information will allow you to claim higher scores in math classes - information on the subject from additional sources is always highly appreciated.

Secondly, we wanted to help you get a feel for how interesting mathematics is. To be convinced by specific examples that there is always a place for creativity in it. We hope that the Pythagorean theorem and this article will inspire you to do your own research and exciting discoveries in mathematics and other sciences.

Tell us in the comments if you found the evidence presented in the article interesting. Did you find this information helpful in your studies? Let us know what you think about the Pythagorean theorem and this article - we will be happy to discuss all this with you.

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Geometry is not an easy science. It can be useful both for the school curriculum and in real life. Knowledge of many formulas and theorems will simplify geometric calculations. One of the simplest shapes in geometry is the triangle. One of the varieties of triangles, equilateral, has its own characteristics.

Features of an equilateral triangle

By definition, a triangle is a polyhedron that has three angles and three sides. This is a flat two-dimensional figure, its properties are studied in high school. According to the type of angle, acute-angled, obtuse-angled and right-angled triangles are distinguished. A right triangle is a geometric figure where one of the angles is 90º. Such a triangle has two legs (they create a right angle), and one hypotenuse (it is opposite the right angle). Depending on what quantities are known, there are three easy ways to calculate the hypotenuse of a right triangle.

The first way is to find the hypotenuse of a right triangle. Pythagorean theorem

The Pythagorean theorem is the oldest way to calculate any of the sides of a right triangle. It sounds like this: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.” Thus, to calculate the hypotenuse, one must derive the square root of the sum of the two legs squared. For clarity, formulas and a diagram are given.

The second way. Calculation of the hypotenuse using 2 known values: the leg and the adjacent angle

One of the properties of a right triangle says that the ratio of the length of the leg to the length of the hypotenuse is equivalent to the cosine of the angle between this leg and the hypotenuse. Let's call the angle known to us α. Now, thanks to the well-known definition, we can easily formulate a formula for calculating the hypotenuse: Hypotenuse = leg/cos(α)

The third way. Calculating the hypotenuse using 2 known values: the leg and the opposite angle

If the opposite angle is known, it is possible to use the properties of a right triangle again. The ratio of the length of the leg and the hypotenuse is equivalent to the sine of the opposite angle. Let's call the known angle α again. Now for the calculations we apply a slightly different formula:
Hypotenuse = leg/sin (α)

Examples to help you understand formulas

For a deeper understanding of each of the formulas, you should consider illustrative examples. So, suppose a right triangle is given, where there is such data:

• Leg - 8 cm.
• The adjoining angle cosα1 is 0.8.
• The opposite angle sinα2 is 0.8.

According to the Pythagorean theorem: Hypotenuse \u003d square root of (36 + 64) \u003d 10 cm.
By the size of the leg and the included angle: 8 / 0.8 \u003d 10 cm.
By the size of the leg and the opposite angle: 8 / 0.8 \u003d 10 cm.

Having understood the formula, you can easily calculate the hypotenuse with any data.

Video: Pythagorean theorem

Make sure the triangle you are given is a right triangle, as the Pythagorean theorem only applies to right triangles. In right triangles, one of the three angles is always 90 degrees.

• A right angle in a right triangle is indicated by a square instead of a curve, which represents non-right angles.

Label the sides of the triangle. Designate the legs as "a" and "b" (the legs are sides intersecting at right angles), and the hypotenuse as "c" (the hypotenuse is the largest side of a right triangle that lies opposite the right angle).

Determine which side of the triangle you want to find. The Pythagorean theorem allows you to find any side of a right triangle (if the other two sides are known). Determine which side (a, b, c) needs to be found.

• For example, given a hypotenuse equal to 5, and given a leg equal to 3. In this case, you need to find the second leg. We will return to this example later.
• If the other two sides are unknown, it is necessary to find the length of one of the unknown sides in order to be able to apply the Pythagorean theorem. To do this, use the basic trigonometric functions (if you are given the value of one of the non-right angles).

Substitute in the formula a 2 + b 2 \u003d c 2 the values ​​\u200b\u200bgiven to you (or the values ​​\u200b\u200bfound by you). Remember that a and b are legs, and c is the hypotenuse.

• In our example, write: 3² + b² = 5².

Square each known side. Or leave the degrees - you can square the numbers later.

• In our example, write: 9 + b² = 25.

Isolate the unknown side on one side of the equation. To do this, transfer the known values ​​to the other side of the equation. If you find the hypotenuse, then in the Pythagorean theorem it is already isolated on one side of the equation (so nothing needs to be done).

• In our example, move 9 to the right side of the equation to isolate the unknown b². You will get b² = 16.

Take the square root of both sides of the equation after there is an unknown (squared) on one side of the equation and an intercept (number) on the other side.

• In our example, b² = 16. Take the square root of both sides of the equation and get b = 4. So the second leg is 4.

Use the Pythagorean theorem in everyday life, as it can be applied in a large number of practical situations. To do this, learn to recognize right triangles in everyday life - in any situation in which two objects (or lines) intersect at right angles, and a third object (or line) connects (diagonally) the tops of the first two objects (or lines), you can use the Pythagorean theorem to find the unknown side (if the other two sides are known).

• Example: Given a ladder leaning against a building. The bottom of the stairs is 5 meters from the base of the wall. The top of the stairs is 20 meters from the ground (up the wall). What is the length of the ladder?
  • "5 meters from the base of the wall" means that a = 5; "is 20 meters from the ground" means that b = 20 (that is, you are given two legs of a right triangle, since the wall of the building and the surface of the Earth intersect at right angles). The length of the ladder is the length of the hypotenuse, which is unknown.
    • a² + b² = c²
    • (5)² + (20)² = c²
    • 25 + 400 = c²
    • 425 = c²
    • c = √425
    • c = 20.6. Thus, the approximate length of the stairs is 20.6 meters.