Pythagorean theorem
Similarly, triangle CBH is similar to ABC. By introducing the notation 
1. Place four equal right triangles as shown in the figure. 
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 built squares on the sides of a right triangle and drew from the vertex 
Let's consider the drawing, as can be seen from the symmetry, the segment CI cuts the square ABHJ into two identical parts (since triangles ABC and JHI are equal in construction). Using a 90-degree counterclockwise rotation, we see the equality of the shaded figures CAJI and GDAB. Now it is clear that the area of the figure we have shaded is equal to the sum of half the areas of the 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 square built on the hypotenuse, plus the area of the original triangle. The last step in the proof is left to the reader.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 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 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 given equation and using the initial conditions, we get:
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:
Every schoolchild knows that the square of the hypotenuse is always equal to the sum of the legs, each of which is squared. This statement is called the Pythagorean theorem. It is one of the most famous theorems of trigonometry and mathematics in general. Let's take a closer look at it.
The concept of a right triangle
Before moving on to consider the Pythagorean theorem, in which the square of the hypotenuse is equal to the sum of the legs that are squared, we should consider the concept and properties of a right triangle for which the theorem is valid.
A triangle is a flat figure with three angles and three sides. A right triangle, as its name suggests, has one right angle, that is, this angle is equal to 90 o.
From general properties for all triangles, it is known that the sum of all three angles of this figure is 180 o, which means that for a right triangle, the sum of two angles that are not right angles is 180 o - 90 o = 90 o. Last fact means that any angle in a right triangle that is not right will always be less than 90 o.
The side that lies opposite the right angle is called the hypotenuse. The other two sides are the legs of the triangle, they can be equal to each other, or they can be different. From trigonometry it is known that the greater the angle against which a side in a triangle lies, the longer length this side. This means that in a right triangle the hypotenuse (lies opposite the 90 o angle) will always be greater than any of the legs (lie opposite the angles< 90 o).
Mathematical notation of Pythagorean theorem
This theorem states that the square of the hypotenuse is equal to the sum of the legs, each of which is previously squared. To write this formulation mathematically, consider a right triangle in which sides a, b and c are the two legs and the hypotenuse, respectively. In this case, the theorem, which is formulated as the square of the hypotenuse is equal to the sum of the squares of the legs, can be represented by the following formula: c 2 = a 2 + b 2. From here other formulas important for practice can be obtained: a = √(c 2 - b 2), b = √(c 2 - a 2) and c = √(a 2 + b 2).
Note that in the case of a rectangular equilateral triangle, that is, a = b, formulation: the square of the hypotenuse is equal to the sum of the legs, each of which is squared, mathematically written as follows: c 2 = a 2 + b 2 = 2a 2, which implies the equality: c = a√2.
Historical reference
The Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the legs, each of which is squared, was known long before the famous Greek philosopher paid attention to it. Many papyri Ancient Egypt, as well as clay tablets of the Babylonians confirm that these peoples used the noted property of the sides of a right triangle. For example, one of the first Egyptian pyramids, the Pyramid of Khafre, the construction of which dates back to the 26th century BC (2000 years before the life of Pythagoras), was built based on knowledge of the aspect ratio in a right triangle 3x4x5.
Why then does the theorem now bear the name of the Greek? The answer is simple: Pythagoras is the first to mathematically prove this theorem. The surviving Babylonian and Egyptian written sources only speak of its use, but do not provide any mathematical proof.
It is believed that Pythagoras proved the theorem in question by using the properties of similar triangles, which he obtained by drawing the height in a right triangle from an angle of 90 o to the hypotenuse.
An example of using the Pythagorean theorem
Let's consider simple task: it is necessary to determine the length of the inclined staircase L, if it is known that it has a height H = 3 meters, and the distance from the wall against which the staircase rests to its foot is P = 2.5 meters.
In this case, H and P are the legs, and L is the hypotenuse. Since the length of the hypotenuse is equal to the sum of the squares of the legs, we get: L 2 = H 2 + P 2, whence L = √(H 2 + P 2) = √(3 2 + 2.5 2) = 3.905 meters or 3 m and 90, 5 cm.
The potential for creativity is usually attributed to the humanities, leaving the natural science to analysis, a practical approach and the dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity you won’t go far in the “queen of all sciences” - people have known 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 cliches and elementary truths - only in such conditions are all great discoveries born.
Such discoveries include what we know today as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be exciting. And that this adventure is suitable not only for nerds with 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. Right triangle and its special properties were studied long before him. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.
Today you can no longer check who is right and who is wrong. What is known is that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid’s Elements may belong to Pythagoras, and Euclid only recorded it.
It is also known today that problems about a right triangle are found in Egyptian sources from the time of Pharaoh Amenemhat I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise “Sulva Sutra” and the ancient Chinese work “Zhou-bi suan jin”.
As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. This is confirmed by about 367 different pieces of evidence that exist today. In this, no other theorem can compete with it. Among the famous authors of proofs we can recall Leonardo da Vinci and the twentieth US President James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or are somehow connected with it.
Proofs of the Pythagorean theorem
School textbooks mostly give algebraic proofs. But the essence of the theorem is in geometry, so let’s first consider those proofs of the famous theorem that are based on this science.
Evidence 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 rectangular, but also isosceles. There is reason to believe that it was precisely this kind of triangle that ancient mathematicians initially considered.
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 construct a square consisting of four triangles equal to the original ABC. And on sides AB and BC a square is built, each of which contains two similar triangles.
By the way, this drawing formed the basis of numerous jokes and cartoons dedicated to the Pythagorean theorem. The most famous is probably "Pythagorean pants are equal in all directions":
Evidence 2
This method combines algebra and geometry and can be considered a variant of the ancient Indian proof of the mathematician Bhaskari.
Construct a right triangle with sides a, b and c(Fig. 1). Then construct 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 triangles similar to those in Figure 1. The result is two squares: 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 checked by calculating the area 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 triangles inscribed in the square from the area of a large square with a side (a+b).
Writing all this down, we have: a 2 +b 2 =(a+b) 2 – 2ab. Open the brackets, carry out all the necessary algebraic calculations and get that a 2 +b 2 = a 2 +b 2. In this case, the area inscribed in Fig. 3. square can also be calculated using the traditional formula S=c 2. Those. a 2 +b 2 =c 2– you have proven the Pythagorean theorem.
Evidence 3
The ancient Indian proof itself was 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 observation skills of students and followers: “Look!”
But we will analyze this proof in more detail:
Inside the square, build four right triangles as indicated in the drawing. Let us denote the side of the large square, also known as the hypotenuse, 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 formula for the area of a square S=c 2 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 areas of all four right triangles: (a-b) 2 2+4*1\2*a*b.
You can use both options for calculating the area of a square to make sure that they give the same result. And this 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 receive the formula of the Pythagorean theorem c 2 =a 2 +b 2. The theorem has been proven.
Proof 4
This curious ancient Chinese proof was called the “Bride’s Chair” - because of the chair-like figure that results from all the constructions:
It uses the drawing that we have already seen in Fig. 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 triangles from the drawing in Fig. 1, move them to opposite sides attach a square with side c and hypotenuses to the hypotenuses of 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 make sure 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 c 2 =a 2 +b 2.
Evidence 5
This is another way to find a solution to the Pythagorean theorem using geometry. It's called the Garfield Method.
Construct a right triangle ABC. We need to prove that BC 2 = AC 2 + AB 2.
To do this, continue the leg AC and construct a segment CD, which is equal to the leg AB. Lower the perpendicular AD line segment ED. Segments ED And AC are equal. Connect the dots E And IN, and E And WITH and get a drawing like the picture below:
To prove the tower, we again resort to the method we have already tried: 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 up 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=SE– this will allow us to simplify the recording and not overload it. So, S ABED =2*1/2(AB*AC)+1/2ВС 2.
At the same time, it is obvious that ABED- This is a trapezoid. Therefore, we calculate its area using the formula: S ABED =(DE+AB)*1/2AD. For our calculations, it is more convenient and clearer to represent the segment AD as the sum of segments AC And CD.
Let's write down both ways to calculate the area of a figure, 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 side of the notation: AB*AC+1/2BC 2 =1/2(AB+AC) 2. Now let’s open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having completed all the transformations, we get exactly what we need: BC 2 = AC 2 + AB 2. We have proven the theorem.
Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proven 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, you can 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 at all in the school curriculum. Meanwhile, he is very interesting and has great importance in geometry. Pythagorean triples are used to solve many mathematical problems. Understanding them may be useful to you in further education.
So what are Pythagorean triplets? This is the name for natural numbers collected in groups of three, 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);
- not primitive (if each number of a triple is multiplied by the same number, you get a new triple, which is not primitive).
Even before our era, the ancient Egyptians were fascinated by the mania for numbers of Pythagorean triplets: in problems they considered a right 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 rectangular by default.
Examples of Pythagorean triplets: (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 is used not only in mathematics, but also in architecture and construction, astronomy and even literature.
First about construction: the Pythagorean theorem finds in it wide application in tasks different levels difficulties. For example, look at a Romanesque window:
Let us denote the width of the window as b, then the radius of the major semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed through 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 is just useful to calculate R. To do this, we use a right 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 represents the 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 =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 terms by b, we present 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 tall the tower is mobile communications the signal needs to reach a certain settlement. And even install steadily christmas tree on the city square. As you can see, this theorem lives not only on the pages of textbooks, but is also often useful in real life.
In literature, the Pythagorean theorem has inspired writers since antiquity and continues to do so in our time. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired to write a sonnet:
The light of truth will not dissipate soon,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
It will not cause doubts or disputes.
The wisest when it touches your gaze
Light of truth, thank the gods;
And a hundred bulls, slaughtered, lie -
A return gift from the lucky Pythagoras.
Since then the bulls have been roaring desperately:
Forever alarmed the bull tribe
Event mentioned here.
It seems to them that the time is about to come,
And they will be sacrificed again
Some great theorem.
(translation by Viktor Toporov)
And in the twentieth century, the Soviet writer Evgeny Veltistov, in his book “The Adventures of Electronics,” devoted an entire chapter to proofs of the Pythagorean theorem. And another half chapter to the story about the two-dimensional world that could exist if the Pythagorean theorem became a fundamental law and even a religion for a single world. Living there would be much easier, 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 mathematics teacher Taratar, says: “The main thing in mathematics is the movement of thought, new ideas.” It is precisely this creative flight of thought that gives rise to the Pythagorean theorem - it is not for nothing that it has so many varied proofs. It helps you go beyond the boundaries of the familiar and look at familiar things in a new way.
Conclusion
This article is designed to help you look beyond 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 and other interesting 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 qualify for higher scores in mathematics lessons - information on the subject from additional sources is always highly appreciated.
Secondly, we wanted to help you get a feel for how mathematics interesting science. Make sure 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 independently explore and make 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 useful in your studies? Write to us 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 a simple 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 figures in geometry is a 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. Based on the type of angle, there are acute, obtuse and right triangles. A right triangle is like this 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 simple ways Calculate the hypotenuse of a right triangle.
The first way is to find the hypotenuse of a right triangle. Pythagorean theorem
Pythagorean theorem - the oldest way Calculate any side 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 should derive Square root from the sum of two legs squared. For clarity, formulas and a diagram are given.
Second way. Calculation of the hypotenuse using 2 known quantities: leg and adjacent angle
One of the properties of a right triangle states 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 known definition, you can easily formulate a formula for calculating the hypotenuse: Hypotenuse = leg/cos(α)
Third way. Calculation of the hypotenuse using 2 known quantities: leg and opposite angle
If the opposite angle is known, it is possible to again use the properties of a right triangle. The ratio of the length of the leg and the hypotenuse is equivalent to the sine of the opposite angle. Let us again call the known angle α. Now for the calculations we will use 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 you are given a right triangle, where there is the following data:
- Leg – 8 cm.
- The adjacent angle cosα1 is 0.8.
- The opposite angle sinα2 is 0.8.
According to the Pythagorean theorem: Hypotenuse = square root of (36+64) = 10 cm.
According to the size of the leg and adjacent angle: 8/0.8 = 10 cm.
According to the size of the leg and the opposite angle: 8/0.8 = 10 cm.
Once you understand the formula, you can easily calculate the hypotenuse with any data.
Video: Pythagorean Theorem
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