Different ways of proving the Pythagorean theorem: examples, description and feedback

In one you can be sure one hundred percent that any question on what the square of the hypotenuse is equal to, any adult person will answer boldly: "The sum of the squares of the legs". This theorem has firmly settled in the minds of every educated person, but it is enough just to ask someone to prove it, and there can be difficulties. Therefore, let us recall and consider different ways of proving the Pythagorean theorem.

Brief overview of biography

Pythagoras' theorem is familiar to almost everyone, but for some reason the biography of the person who produced it is not so popular. This is fixable. Therefore, before studying different ways of proving Pythagoras' theorem, one must briefly get acquainted with his personality.

Pythagoras is a philosopher, mathematician, thinker originally from Ancient Greece. Today it is very difficult to distinguish his biography from the legends that have developed in memory of this great man. But as follows from the writings of his followers, Pythagoras of Samos was born on the island of Samos. His father was a common stone-cutter, but his mother came from a noble family.

Judging by the legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor they called the boy. According to her prediction, a born boy had to bring many benefits and good to humanity. Which, in fact, he did.

The birth of the theorem

In his youth Pythagoras moved from the island of Samos to Egypt to meet there with the famous Egyptian sages. After a meeting with them, he was admitted to study, where he learned all the great achievements of Egyptian philosophy, mathematics and medicine.

Probably, it was in Egypt that Pythagoras was inspired by the majesty and beauty of the pyramids and created his great theory. This may shock readers, but modern historians believe that Pythagoras did not prove his theory. And only transferred his knowledge to followers who later completed all the necessary mathematical calculations.

Whatever it was, today is not one method of proof of this theorem known, but several. Today we can only guess how exactly the ancient Greeks made their calculations, so here we consider different ways of proving the Pythagorean theorem.

Pythagorean theorem

Before starting any calculations, you need to find out which theory to prove. Pythagoras' theorem reads: "In a triangle with one of the angles equal to 90 ^° , the sum of the squares of the legs is equal to the square of the hypotenuse."

In total there are 15 different ways of proving the Pythagorean theorem. This is a fairly large figure, so let's pay attention to the most popular of them.

Method one

First, let's denote what is given to us. These data will be extended to other methods of proof of the Pythagorean theorem, so it is worth remembering all the available notations.

Suppose a rectangular triangle is given, with legs a, b and hypotenuse equal to c. The first method of proof is based on the fact that a square is necessary to draw a rectangle from a right triangle.

To do this, it is necessary to draw a segment equal to the coutette to the length of a leg and vice versa. This should result in two equal sides of the square. It remains only to draw two parallel lines, and the square is ready.

Inside the resulting figure, you need to draw another square with the side equal to the hypotenuse of the original triangle. To do this, from the vertices a and c, we need to draw two parallel segments of equal c. Thus, it turns out three sides of the square, one of which is the hypotenuse of the original rectangular triangles. It remains only to subsidize the fourth segment.

On the basis of the resulting figure, we can conclude that the area of the outer square is (a + b) ² . If you look inside the figure, you can see that in addition to the inner square there are four rectangular triangles in it. The area of each is 0.5aV.

Therefore, the area is: 4 * 0.5aв + с ² = 2ав + с ²

Hence (a + b) ² = 2aB + c ²

And, consequently, with ² = a ² + in ²

The theorem is proved.

Method two: similar triangles

This formula for the proof of Pythagoras' theorem was derived on the basis of an assertion from a section of geometry on similar triangles. It says that the cathete of a right triangle is an average proportional for its hypotenuse and a segment of the hypotenuse originating from the apex of the angle of 90 ^° .

The initial data remains the same, so we will start right away with the proof. We draw the segment of the CD perpendicular to the side AB. Based on the above statement, the triangle legs are:

AC = √AB * AD, CB = √AB * DV.

To answer the question how to prove the Pythagorean theorem, the proof must be laid by squaring both inequalities.

AC ² = AB * AD and CB ² = AB * DV

Now we need to add the resulting inequalities.

AC ² + CB ² = AB * (AD * DV), where AD + DB = AB

It turns out that:

AC ² + CB ² = AB * AB

And, consequently:

AC ² + CB ² = AB ²

The proof of the Pythagorean theorem and the various methods for solving it require a versatile approach to this problem. However, this option is one of the simplest.

Another method of calculation

The description of the different methods of proving the Pythagorean theorem can not be said about anything, until unless one begins to practice on his own. Many methods provide not only mathematical calculations, but also the construction of new figures from the original triangle.

In this case, it is necessary to finish building another rectangular triangle of the VSD from the BC. Thus, now there are two triangles with a common leg BC.

Knowing that the areas of similar figures have a ratio as the squares of their similar linear dimensions, then:

S _avs * With ² - S _avd * in ² = S _avd * a ² - S _ave * a ²

S _avc * (c ² -c ² ) = a ² * (S _aug- S _vsd )

With ² -a2 = ^a2

With ² = a ² + in ²

Since from the different methods of proof of the Pythagorean theorem for grade 8 this variant is hardly suitable, one can use the following procedure.

The simplest way to prove the theorem of Pythagoras. Reviews

As historians believe, this method was first used to prove the theorem even in ancient Greece. It is the simplest, since it requires absolutely no calculations. If the drawing is drawn correctly, then the proof of the statement that a ² + in ² = c ² will be seen clearly.

The conditions for this method will be slightly different from the previous one. To prove the theorem, suppose that a right triangle ABC is an isosceles triangle.

We take the hypotenuse of the AS for the side of the square and we have three of its sides. In addition, it is necessary to draw two diagonal lines in the resulting square. Thus, to get four isosceles triangles inside it.

To the legs AB and CB, you also need to have a child in the square and draw one diagonal line in each of them. The first line is drawn from the vertex A, the second line is drawn from C.

Now you need to look closely at the resulting drawing. Since there are four triangles on the hypotenuse of the AS, equal to the original triangle, and on the legs by two, this indicates the truth of the theorem.

By the way, thanks to this method of proof of the theorem of Pythagoras, the famous phrase appeared: "Pythagorean pants are equal in all directions".

Proof of G. Garfield

James Garfield is the twentieth president of the United States of America. In addition, he left his mark in history as the ruler of the United States, he was also a gifted self-taught.

At the beginning of his career he was an ordinary teacher in the public school, but soon became the director of one of the higher educational institutions. The desire for self-development allowed him to propose a new theory of the proof of the Pythagorean theorem. The theorem and an example of its solution are as follows.

First you need to draw on a sheet of paper two rectangular triangles in such a way that the cathet of one of them was a continuation of the second one. The vertices of these triangles need to be joined, so that eventually a trapezoid turns out.

As is known, the area of the trapezoid is equal to the product of the half-sum of its bases to the height.

S = a + b / 2 * (a + b)

If we consider the resulting trapezoid as a figure consisting of three triangles, then its area can be found as follows:

S = av / 2 * 2 + s 2/2

Now it is necessary to equalize the two initial expressions

2ав / 2 + с / 2 = (а + в) 2/2

With ² = a ² + in ²

About the theorem of Pythagoras and the methods of its proof, you can write more than one volume of the training manual. But is there any sense in it when this knowledge can not be applied in practice?

Practical application of the theorem of Pythagoras

Unfortunately, in modern school programs, the use of this theorem is foreseen only in geometric problems. Graduates will soon leave the school walls, without knowing, and how they can apply their knowledge and skills in practice.

In fact, everyone can use the Pythagorean theorem in their daily lives. And not only in professional work, but also in ordinary domestic affairs. Let us consider several cases when the Pythagorean theorem and the methods of its proof may prove to be extremely necessary.

Relationship between theorem and astronomy

It would seem, how the stars and triangles can be connected on paper. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.

For example, consider the motion of a light beam in space. It is known that light moves in both directions at the same speed. The trajectory AB, which moves a ray of light is called l . And half the time that is necessary for the light to get from point A to point B is called T. And the speed of the beam is c . It turns out that: c * t = l

If we look at this same ray from another plane, for example, from a space liner that moves with a velocity v, then with such a monitoring of bodies their speed will change. In this case, even fixed elements will move with velocity v in the opposite direction.

Let's say a comic liner swims to the right. Then the points A and B, between which the ray rushes, will move to the left. And, when the beam moves from point A to point B, point A manages to move and, accordingly, light already arrives at a new point C. To find half of the distance to which point A has shifted, the speed of the liner must be multiplied by half of the travel time of the ray (t ').

D = t '* v

And in order to find out how far a ray of light could pass through this time, it is necessary to designate half the path of the new beech s and get the following expression:

S = c * t '

If we imagine that the points of light C and B, as well as the cosmic liner are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two rectangular triangles. Therefore, thanks to the Pythagorean theorem, one can find the distance that a ray of light could pass.

S ² = l ² + d ²

This example, of course, is not the most successful, since only units can be lucky enough to try it in practice. Therefore, consider more mundane versions of the application of this theorem.

Radius of mobile signal transmission

Modern life is impossible to imagine without the existence of smartphones. But how much would it be for them to proc, if they could not connect subscribers through mobile communication ?!

The quality of mobile communication directly depends on the height of the mobile operator's antenna. In order to calculate the distance from the mobile tower the phone can receive a signal, we can apply the Pythagorean theorem.

Suppose we need to find the approximate height of a stationary tower so that it can propagate the signal within a radius of 200 kilometers.

AB (tower height) = x;

BC (signal transmission radius) = 200 km;

OS (radius of the globe) = 6380 km;

From here

OB = OA + ABOV = r + x

Applying the theorem of Pythagoras, we will find out that the minimum height of the tower should be 2.3 kilometers.

The Pythagorean theorem in everyday life

Ironically, the Pythagorean theorem can prove useful even in everyday matters, such as determining the height of the closet, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements using roulette. But many wonder why in the assembly process there are certain problems, if all the measurements were taken more than accurately.

The fact is that the closet is assembled in a horizontal position and only then rises and mounts to the wall. Therefore, the sidewall of the cabinet during the lifting of the structure must pass freely both in height and diagonally of the room.

Suppose there is a closet with a depth of 800 mm. The distance from the floor to the ceiling is 2600 mm. An experienced furniture maker will say that the height of the cabinet should be 126 mm less than the height of the room. But why on 126 mm? Consider the example.

Let us check the effect of the Pythagorean theorem for ideal dimensions of the cabinet:

AC = √AB ² + √BC ²

AC = √2474 ² +800 ² = 2600 mm - everything converges.

Suppose the height of the cabinet is not 2474 mm, but 2505 mm. Then:

AC = √2505 ² + √800 ² = 2629 mm.

Therefore, this cabinet is not suitable for installation in this room. Since when lifting it to a vertical position, you can damage its hull.

Perhaps, having considered different ways of proving the theorem of Pythagoras by different scientists, we can conclude that it is more than truthful. Now you can use the information received in your daily life and be completely sure that all calculations will not only be useful, but also true.