Pythagorean area theorem. Pythagorean theorem: history, proof, examples of practical application
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. The right triangle and its special properties were 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. 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
IN school textbooks They mainly 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 right triangle need to be asked 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 rectangular triangles from the drawing in Fig. 1, move 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 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? That's what they call it integers, collected in threes, the sum of the squares of two of which is equal to the third number in the square.
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 is widely used in problems of various levels of complexity. 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 gable roof. Determine how high a mobile communications tower is needed for the signal to reach a certain populated area. 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 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 doubt or controversy.
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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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 various sources, from 11 to 22 years, 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 succeeded in proving 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.
Construction right angle with the help of the Egyptian triangle it was also stated 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 the large square is calculated from the sum of the areas of the 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 + 2av. 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 the converse of the Pythagorean theorem):
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 whole number sides. 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 calculation of 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 
Usage 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.
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. Label the legs as “a” and “b” (legs are sides intersecting at right angles), and the hypotenuse as “c” (hypotenuse is the largest side of a right triangle, lying 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) you need to find.
- For example, given a hypotenuse equal to 5, and given a leg equal to 3. In this case, it is necessary to find the second leg. We'll come back to this example later.
- If the other two sides are unknown, you need to find the length of one of the unknown sides 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 oblique angles).
Substitute the values given to you (or the values you found) into the formula a 2 + b 2 = c 2. 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 powers - 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, move 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 you don't need to do anything).
- In our example, move 9 to right side equations to isolate the unknown b². You will get b² = 16.
Take the square root of both sides of the equation after you have the unknown (squared) on one side of the equation and the intercept (a number) on the other side.
- In our example, b² = 16. Take the square root of both sides of the equation and get b = 4. Thus, the second leg is 4.
Use the Pythagorean theorem in Everyday life, since it can be used in large number 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 staircase leaning against a building. The bottom of the stairs is 5 meters from the base of the wall. Top part The stairs are located 20 meters from the ground (up the wall). What is the length of the stairs?
- “5 meters from the base of the wall” means that a = 5; “located 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 staircase 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.
- “5 meters from the base of the wall” means that a = 5; “located 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 staircase is the length of the hypotenuse, which is unknown.
MEASUREMENT OF AREA OF GEOMETRIC FIGURES.
§ 58. PYTHAGOREAN THEOREM 1.
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1 Pythagoras is a Greek scientist who lived about 2500 years ago (564-473 BC).
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Let us be given a right triangle whose sides A, b And With(drawing 267).
Let's build squares on its sides. The areas of these squares are respectively equal A 2 , b 2 and With 2. Let's prove that With 2 = a 2 +b 2 .
Let's construct two squares MKOR and M"K"O"R" (drawings 268, 269), taking as the side of each of them a segment equal to the sum of the legs of the right triangle ABC.
Having completed the constructions shown in drawings 268 and 269 in these squares, we will see that the MCOR square is divided into two squares with areas A 2 and b 2 and four equal right triangles, each of which is equal to right triangle ABC. The square M"K"O"R" was divided into a quadrangle (it is shaded in drawing 269) and four right triangles, each of which is also equal to triangle ABC. A shaded quadrilateral is a square, since its sides are equal (each is equal to the hypotenuse of triangle ABC, i.e. With), and the angles are right / 1 + / 2 = 90°, from where / 3 = 90°).
Thus, the sum of the areas of the squares built on the legs (in drawing 268 these squares are shaded) is equal to the area of the square MCOR without the sum of the areas of four equal triangles, and the area of the square built on the hypotenuse (in drawing 269 this square is also shaded) is equal to the area of the square M"K"O"R", equal to the square of MCOR, without the sum of the areas of four similar triangles. Therefore, 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 the legs.
We get the formula With 2 = a 2 +b 2 where With- hypotenuse, A And b- legs of a right triangle.
The Pythagorean theorem is usually formulated briefly as follows:
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
From the formula With 2 = a 2 +b 2 you can get the following formulas:
A 2 = With 2 - b 2 ;
b 2 = With 2 - A 2 .
These formulas can be used to find the unknown side of a right triangle from its two given sides.
For example:
a) if the legs are given A= 4 cm, b=3 cm, then you can find the hypotenuse ( With):
With 2 = a 2 +b 2, i.e. With 2
= 4 2 + 3 2 ; with 2 = 25, whence With= √25 =5 (cm);
b) if the hypotenuse is given With= 17 cm and leg A= 8 cm, then you can find another leg ( b):
b 2 = With 2 - A 2, i.e. b 2 = 17 2 - 8 2 ; b 2 = 225, from where b= √225 = 15 (cm).
Consequence:
If two right triangles ABC and A have 1 B 1 C 1 hypotenuse With And With 1 are equal, and leg b triangle ABC is longer than the leg b 1 triangle A 1 B 1 C 1,
then the leg A triangle ABC is smaller than the leg A 1 triangle A 1 B 1 C 1. (Make a drawing illustrating this consequence.)
In fact, based on the Pythagorean theorem we obtain:
A 2 = With 2 - b 2 ,
A 1 2 = With 1 2 - b 1 2
In the written formulas, the minuends are equal, and the subtrahend in the first formula is greater than the subtrahend in the second formula, therefore, the first difference is less than the second,
i.e. A 2 < A 12 . Where A< A 1 .
Exercises.
1. Using drawing 270, prove the Pythagorean theorem for an isosceles right triangle.
2. One leg of a right triangle is 12 cm, the other is 5 cm. Calculate the length of the hypotenuse of this triangle.
3. The hypotenuse of a right triangle is 10 cm, one of the legs is 8 cm. Calculate the length of the other leg of this triangle.
4. The hypotenuse of a right triangle is 37 cm, one of its legs is 35 cm. Calculate the length of the other leg of this triangle.
5. Construct a square with an area twice the size of the given one.
6. Construct a square with an area half the size of the given one. Note. Carry out in given square diagonals. The squares constructed on the halves of these diagonals will be the ones we are looking for.
7. The legs of a right triangle are respectively 12 cm and 15 cm. Calculate the length of the hypotenuse of this triangle with an accuracy of 0.1 cm.
8. The hypotenuse of a right triangle is 20 cm, one of its legs is 15 cm. Calculate the length of the other leg to the nearest 0.1 cm.
9. How long must the ladder be so that it can be attached to a window located at a height of 6 m, if the lower end of the ladder must be 2.5 m from the building? (Chart 271.)