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 true.

Triangle - flat figure having 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 the general properties of all triangles, it is known that the sum of all three angles of this figure is equal to 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. This 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 against right angle, is usually 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 we know that the greater the angle against which a side of a triangle lies, the greater the length of that 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 right-angled equilateral triangle, that is, a = b, the formulation: the square of the hypotenuse is equal to the sum of the legs, each of which is squared, will be 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Historical reference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 of 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

Consider a simple problem: it is necessary to determine the length of an 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 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, from where L = √(H 2 + P 2) = √(3 2 + 2.5 2) = 3.905 meters or 3 m and 90, 5 cm.

Make sure that 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 icon rather than the curve that represents oblique angles. Label the sides of the triangle.

Instructions

If you need to calculate using the Pythagorean theorem, use the following algorithm: - Determine in a triangle which sides are the legs and which are the hypotenuse. The two sides forming an angle of ninety degrees are the legs, the remaining third is the hypotenuse. (cm) - Raise each leg to the second power given triangle, that is, multiply by yourself. Example 1. Suppose we need to calculate the hypotenuse if one leg in a triangle is 12 cm and the other is 5 cm. First, the squares of the legs are equal: 12 * 12 = 144 cm and 5 * 5 = 25 cm. Next, determine the sum of the squares legs. A certain number is hypotenuse, you need to get rid of the second power of the number to find length this side of the triangle. To do this, remove from under square root the value of the sum of squares of the legs. Example 1. 144+25=169. The square root of 169 is 13. Therefore, the length of this hypotenuse equal to 13 cm.

Another way to calculate length hypotenuse lies in the terminology of sine and angles in a triangle. By definition: the sine of the angle alpha - the opposite leg to the hypotenuse. That is, looking at the figure, sin a = CB / AB. Hence, hypotenuse AB = CB / sin a. Example 2. Let the angle be 30 degrees, and the opposite side be 4 cm. We need to find the hypotenuse. Solution: AB = 4 cm / sin 30 = 4 cm / 0.5 = 8 cm. Answer: length hypotenuse equal to 8 cm.

A similar way to find hypotenuse from the definition of cosine of an angle. The cosine of an angle is the ratio of the side adjacent to it and hypotenuse. That is, cos a = AC/AB, hence AB = AC/cos a. Example 3. In triangle ABC, AB is the hypotenuse, angle BAC is 60 degrees, leg AC is 2 cm. Find AB.
Solution: AB = AC/cos 60 = 2/0.5 = 4 cm Answer: The hypotenuse is 4 cm in length.

Helpful advice

When finding the value of the sine or cosine of an angle, use either the table of sines and cosines or the Bradis table.

Tip 2: How to find the length of the hypotenuse in a right triangle

The hypotenuse is the longest side in a right triangle, so it is not surprising that Greek language this word is translated as “tight”. This side always lies opposite the 90° angle, and the sides forming this angle are called legs. Knowing the lengths of these sides and the magnitudes sharp corners in different combinations of these values, the length of the hypotenuse can be calculated.

Instructions

If the lengths of both triangles (A and B) are known, then use the lengths of the hypotenuse (C), perhaps the most famous mathematical postulate - the Pythagorean theorem. It states that the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs, from which it follows that you should calculate the root of the sum of the squared lengths of the two sides: C = √ (A² + B²). For example, if the length of one leg is 15 and - 10 centimeters, then the length of the hypotenuse will be approximately 18.0277564 centimeters, since √(15²+10²)=√(225+100)= √325≈18.0277564.

If the length of only one of the legs (A) in a right triangle is known, as well as the value of the angle opposite it (α), then the length of the hypotenuse (C) can be used using one of the trigonometric functions - the sine. To do this, divide the length of the known side by the sine of the known angle: C=A/sin(α). For example, if the length of one of the legs is 15 centimeters, and the angle at the opposite vertex of the triangle is 30°, then the length of the hypotenuse will be equal to 30 centimeters, since 15/sin(30°)=15/0.5=30.

If in a right triangle the size of one of the acute angles (α) and the length of the adjacent leg (B) are known, then to calculate the length of the hypotenuse (C) you can use another trigonometric function- cosine. You should divide the length of the known leg by the cosine of the known angle: C=B/ cos(α). For example, if the length of this leg is 15 centimeters, and the acute angle adjacent to it is 30°, then the length of the hypotenuse will be approximately 17.3205081 centimeters, since 15/cos(30°)=15/(0.5* √3)=30/√3≈17.3205081.

Length is usually used to denote the distance between two points on a line segment. It can be straight, broken or closed line. You can calculate the length quite simply if you know some other indicators of the segment.

Instructions

If you need to find the length of the side of a square, then it will not be , if you know its area S. Due to the fact that all sides of the square have

The history of the Pythagorean theorem goes back several thousand years. A statement that states that it was known long before the birth of the Greek mathematician. However, the Pythagorean theorem, the history of its creation and its proof are associated for the majority with this scientist. According to some sources, the reason for this was the first proof of the theorem, which was given by Pythagoras. However, some researchers deny this fact.

Music and logic

Before telling how the history of the Pythagorean theorem developed, let us briefly look at the biography of the mathematician. He lived in the 6th century BC. The date of birth of Pythagoras is considered to be 570 BC. e., the place is the island of Samos. Little is known reliably about the life of the scientist. Biographical data in ancient Greek sources is intertwined with obvious fiction. On the pages of the treatises, he appears as a great sage with excellent command of words and the ability to persuade. By the way, this is why the Greek mathematician was nicknamed Pythagoras, that is, “persuasive speech.” According to another version, the birth of the future sage was predicted by Pythia. The father named the boy Pythagoras in her honor.

The sage learned from the great minds of the time. Among the teachers of the young Pythagoras are Hermodamantus and Pherecydes of Syros. The first instilled in him a love of music, the second taught him philosophy. Both of these sciences will remain the focus of the scientist throughout his life.

30 years of training

According to one version, being an inquisitive young man, Pythagoras left his homeland. He went to seek knowledge in Egypt, where he stayed, according to different sources, from 11 to 22 years old, and then was captured and sent to Babylon. Pythagoras was able to benefit from his position. For 12 years he studied mathematics, geometry and magic in ancient state. Pythagoras returned to Samos only at the age of 56. The tyrant Polycrates ruled here at that time. Pythagoras could not accept such political system and soon went to the south of Italy, where the Greek colony of Croton was located.

Today it is impossible to say for sure whether Pythagoras was in Egypt and Babylon. He may have left Samos later and went straight to Croton.

Pythagoreans

The history of the Pythagorean theorem is connected with the development of the school created by the Greek philosopher. This religious and ethical brotherhood preached observance of a special way of life, studied arithmetic, geometry and astronomy, and was engaged in the study of the philosophical and mystical side of numbers.

All the discoveries of the students of the Greek mathematician were attributed to him. However, the history of the emergence of the Pythagorean theorem is associated by ancient biographers only with the philosopher himself. It is assumed that he passed on to the Greeks the knowledge gained in Babylon and Egypt. There is also a version that he actually discovered the theorem on the relationship between the legs and the hypotenuse, without knowing about the achievements of other peoples.

Pythagorean theorem: history of discovery

Some ancient Greek sources describe Pythagoras' joy when he managed to prove the theorem. In honor of this event, he ordered a sacrifice to the gods in the form of hundreds of bulls and held a feast. Some scientists, however, point out the impossibility of such an act due to the peculiarities of the views of the Pythagoreans.

It is believed that in the treatise “Elements”, created by Euclid, the author provides a proof of the theorem, the author of which was the great Greek mathematician. However, not everyone supported this point of view. Thus, even the ancient Neoplatonist philosopher Proclus pointed out that the author of the proof given in the Elements was Euclid himself.

Be that as it may, the first person to formulate the theorem was not Pythagoras.

Ancient Egypt and Babylon

The Pythagorean theorem, the history of which is discussed in the article, according to the German mathematician Cantor, was known back in 2300 BC. e. in Egypt. The ancient inhabitants of the Nile Valley during the reign of Pharaoh Amenemhat I knew the equality 3 2 + 4 ² = 5 ². It is assumed that with the help of triangles with sides 3, 4 and 5, the Egyptian “rope pullers” built right angles.

They also knew the Pythagorean theorem in Babylon. On clay tablets dating back to 2000 BC. and dating back to the reign, an approximate calculation of the hypotenuse of a right triangle was discovered.

India and China

The history of the Pythagorean theorem is also connected with the ancient civilizations of India and China. The treatise “Zhou-bi suan jin” contains indications that (its sides are related as 3:4:5) was known in China back in the 12th century. BC e., and by the 6th century. BC e. The mathematicians of this state knew the general form of the theorem.

The construction of a right angle using the Egyptian triangle was also outlined in the Indian treatise “Sulva Sutra”, dating back to the 7th-5th centuries. BC e.

Thus, the history of the Pythagorean theorem by the time of the birth of the Greek mathematician and philosopher was already several hundred years old.

Proof

During its existence, the theorem became one of the fundamental ones in geometry. The history of the proof of the Pythagorean theorem probably began with the consideration of an equilateral square. Squares are constructed on its hypotenuse and legs. The one that “grew” on the hypotenuse will consist of four triangles equal to the first. The squares on the sides consist of two such triangles. Simple graphic image clearly demonstrates the validity of the statement formulated in the form of the famous theorem.

Another simple proof combines geometry with algebra. Four identical right triangles with sides a, b, c are drawn so that they form two squares: the outer one with side (a + b) and the inner one with side c. In this case, the area of ​​the smaller square will be equal to c 2. The area of ​​a large one is calculated from the sum of the areas small square and all triangles (the area of ​​a right triangle, recall, is calculated by the formula (a * b) / 2), that is, c 2 + 4 * ((a * b) / 2), which is equal to c 2 + 2ab. The area of ​​a large square can be calculated in another way - as the product of two sides, that is, (a + b) 2, which is equal to a 2 + 2ab + b 2. It turns out:

a 2 + 2ab + b 2 = c 2 + 2ab,

a 2 + b 2 = c 2.

There are many versions of the proof of this theorem. Euclid, Indian scientists, and Leonardo da Vinci worked on them. Often the ancient sages cited drawings, examples of which are located above, and did not accompany them with any explanations other than the note “Look!” The simplicity of the geometric proof, provided that some knowledge was available, did not require comments.

The history of the Pythagorean theorem, briefly outlined in the article, debunks the myth about its origin. However, it is difficult to even imagine that the name of the great Greek mathematician and philosopher will ever cease to be associated with it.

Animated proof of the Pythagorean theorem - one of fundamental theorems of Euclidean geometry establishing the relationship between the sides of a right triangle. It is believed that it was proven by the Greek mathematician Pythagoras, after whom it is named (there are other versions, in particular the alternative opinion that this theorem in general view was formulated by the Pythagorean mathematician Hippasus).
The theorem states:

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.

Determining the length of the hypotenuse of the triangle c, and the lengths of the legs are like a And b, we get the following formula:

Thus, the Pythagorean theorem establishes a relationship that allows you to determine the side of a right triangle, knowing the lengths of the other two. The Pythagorean theorem is a special case of the cosine theorem, which determines the relationship between the sides of an arbitrary triangle.
The converse statement has also been proven (also called inverse theorem Pythagoras):

For any three positive numbers a, b and c such that a ? + b ? = c ?, there is a right triangle with legs a and b and hypotenuse c.

Visual evidence for the triangle (3, 4, 5) from the book "Chu Pei" 500-200 BC. The history of the theorem can be divided into four parts: knowledge about Pythagorean numbers, knowledge about the ratio of sides in a right triangle, knowledge about the ratio adjacent corners and proof of the theorem.
Megalithic structures around 2500 BC. in Egypt and Northern Europe, contain right triangles with sides made of integers. Bartel Leendert van der Waerden hypothesized that at that time Pythagorean numbers were found algebraically.
Written between 2000 and 1876 BC. papyrus from the Middle Egyptian Kingdom Berlin 6619 contains a problem whose solution is Pythagorean numbers.
During the reign of Hammurabi the Great, Babylonian tablet Plimpton 322, written between 1790 and 1750 BC contains many entries closely related to Pythagorean numbers.
In the Budhayana sutras, which date from different versions eighth or second centuries BC in India, contains Pythagorean numbers derived algebraically, a statement of the Pythagorean theorem and a geometric proof for a equilateral right triangle.
The Apastamba Sutras (circa 600 BC) contain a numerical proof of the Pythagorean theorem using area calculations. Van der Waerden believes that it was based on the traditions of its predecessors. According to Albert Burco, this is the original proof of the theorem and he suggests that Pythagoras visited Arakon and copied it.
Pythagoras, whose years of life are usually indicated as 569 - 475 BC. uses algebraic methods for calculating Pythagorean numbers, according to Proklov's commentaries on Euclid. Proclus, however, lived between 410 and 485 AD. According to Thomas Guise, there is no indication of the authorship of the theorem until five centuries after Pythagoras. However, when authors such as Plutarch or Cicero attribute the theorem to Pythagoras, they do so as if the authorship was widely known and certain.
Around 400 BC According to Proclus, Plato gave a method for calculating Pythagorean numbers that combined algebra and geometry. Around 300 BC, in Beginnings Euclid we have the oldest axiomatic proof that has survived to this day.
Written sometime between 500 BC. and 200 BC, the Chinese mathematical book Chu Pei (? ? ? ?), gives a visual proof of the Pythagorean theorem, called the Gugu theorem (????) in China, for a triangle with sides (3, 4, 5). During the Han Dynasty, from 202 BC. to 220 AD Pythagorean numbers appear in the book "Nine Branches of the Mathematical Art" along with a mention of right triangles.
The first recorded use of the theorem was in China, where it is known as the Gugu (????) theorem, and in India, where it is known as Bhaskar's theorem.
It has been widely debated whether Pythagoras' theorem was discovered once or repeatedly. Boyer (1991) believes that the knowledge found in the Shulba Sutra may be of Mesopotamian origin.
Algebraic proof
Squares are formed from four right triangles. More than a hundred proofs of the Pythagorean theorem are known. Here is a proof based on the existence theorem of the area of ​​a figure:

Let's place four identical right triangles as shown in the figure.
Quadrangle with sides c is a square, since the sum of two acute angles is , and a straight angle is .
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, to the sum of the areas of four triangles and the inner square.

Which is what needs to be proven.
By similarity of triangles
Using similar triangles. Let ABC- a right triangle in which the angle C straight as shown in the picture. Let's draw the height from the point C, and let's call H point of intersection with the side AB. A triangle is formed ACH similar to a triangle ABC, since they are both rectangular (by definition of height) and they have a common angle A, Obviously the third angle in these triangles will also be the same. Similar to peace, triangle CBH also similar to a triangle ABC. With similarity of triangles: If

This can be written as

If we add these two equalities, we get

HB + c times AH = c times (HB + AH) = c ^ 2, ! Src = "http://upload.wikimedia.org/math/7/0/9/70922f59b11b561621c245e11be0b61b.png" />

In other words, the Pythagorean theorem:

Euclid's proof
Euclid's proof in Euclidean "Elements", the Pythagorean theorem is proven by the method of parallelograms. Let A, B, C vertices of a right triangle, with right angle A. Let's drop a perpendicular from the point A to the side opposite the hypotenuse in a square built on the hypotenuse. The line divides the square into two rectangles, each of which has the same area as the squares built on the sides. main idea in the proof is that the upper squares turn into parallelograms of the same area, and then return and turn into rectangles in the lower square and again with the same area.

Let's draw segments CF And A.D. we get triangles BCF And B.D.A.
Angles CAB And BAG– straight; respectively points C, A And G– collinear. Also B, A And H.
Angles CBD And FBA– both are straight lines, then the angle ABD equal to angle FBC, since both are the sum of a right angle and an angle ABC.
Triangle ABD And FBC level on two sides and the angle between them.
Since the points A, K And L– collinear, the area of ​​the rectangle BDLK is equal to two areas of the triangle ABD (BDLK = BAGF = AB 2)
Similarly, we obtain CKLE = ACIH = AC 2
On one side the area CBDE equal to the sum of the areas of the rectangles BDLK And CKLE, and on the other side the area of ​​the square BC 2, or AB 2 + AC 2 = BC 2.

Using differentials
Use of differentials. The Pythagorean theorem can be arrived at by studying how the increase in side affects the size of the hypotenuse as shown in the figure on the right and applying a little calculation.
As a result of the increase in side a, of similar triangles for infinitesimal increments

Integrating we get

If a= 0 then c = b, so "constant" is b 2. Then

As can be seen, the squares are due to the proportion between the increments and the sides, while the sum is the result of the independent contribution of the increments of the sides, not obvious from the geometric evidence. In these equations da And dc– correspondingly infinitesimal increments of sides a And c. But what do we use instead? a And? c, then the limit of the ratio if they tend to zero is da / dc, derivative, and is also equal to c / a, ratio of the lengths of the sides of the triangles, as a result we get differential equation.
In the case of an orthogonal system of vectors, the equality holds, 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.