In one you can be sure of everythingpercent, that to the question of what the square of the hypotenuse is equal to, any adult person will respond 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
The theorem of Pythagoras is familiar to almost everyone, butfor 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 - philosopher, mathematician, thinker originally fromAncient Greece. Today it is very difficult to distinguish his biography from the legends that formed 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 appearance of Pythagoraspredicted 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 toEgypt, 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 inspiredmajesty 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. He only transferred his knowledge to followers, who later completed all the necessary mathematical calculations.
Whatever it was, today is known not onlyThe technique of the proof of this theorem, 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 sounds like this: "In a triangle with one of the angles equal to 90about, 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, given a rectangular triangle, with legs a, b and hypotenuse, equal to c. The first way of proof is based on the fact that a rectangle needs to draw a square.
To do this, you need a leg of length adraw a segment equal to the cathete in, and vice versa. This should result in two equal sides of the square. It only remains to draw two parallel straight lines, and the square is ready.
Inside the resulting figure, you need to draw moreone square with side equal to the hypotenuse of the original triangle. To do this, from the vertices ac and st must draw two parallel segments of equal c. Thus, we get three sides of the square, one of which is the hypotenuse of the original rectangular triangle. 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 + с2= 2ав + с2
Hence (a + b)2= 2ав + с2
And, consequently, with2= a2+ in2
The theorem is proved.
Method two: similar triangles
This formula for the proof of the theorem of Pythagoraswas derived on the basis of the statement from the geometry section about similar triangles. It says that the cathete of a right triangle is an average proportional for its hypotenuse and a segment of the hypotenuse issuing from the vertex of the corner 90about.
The initial data remains the same, so we will start right away with the proof. We draw perpendicular to the side AB a segment of the SD. Based on the above statement, the triangle legs are:
AC = √AB * AD, CB = √AB * DV.
To answer the question of how to prove the Pythagorean theorem, the proof must be constructed by squaring both inequalities.
AC2= AB * AD and CB2= AB * DV
Now we need to add up the resulting inequalities.
AC2+ CB2= AB * (АД * ДВ), where АД + ДВ = АВ
It turns out that:
AC2+ CB2= AB * AB
And, consequently:
AC2+ CB2= AB2
The proof of the Pythagorean theorem and the various ways of solving it require a versatile approach to this problem. However, this option is one of the simplest.
Another method of calculation
The description of different ways of proving the theoremPythagoras can not say anything, until as long as you yourself do not begin to practice. 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 complete 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:
FROMavs * from2- FROMavd*at2 = Cavd*a2- FROMup*a2
FROMavs*(from2-at2) = a2*(FROMavd-FROMup)
from2-at2= a2
from2= a2+ in2
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 the first timeused 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 assertion that a2+ in2= s2 , will be seen clearly.
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 AS for the side of the square anddaughters 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 proving Pythagoras' theorem, 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 ordinarya teacher in a 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 tworectangular triangle in such a way that the catet of one of them is a continuation of the second. The vertices of these triangles need to be connected so that the trapezium eventually 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 + s2/ 2
Now it is necessary to equalize the two initial expressions
2ав / 2 + с / 2 = (а + в)2/ 2
from2= a2+ in2
A theorem of Pythagoras and the methods of its proof can be written not just one volume of the textbook. But is there any sense in it when this knowledge can not be applied in practice?
Practical application of the Pythagorean theorem
Unfortunately, in modern school curriculaIt is intended to use this theorem 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, to use the Pythagorean theorem inEveryone can do their everyday. 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.
The connection 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 light needs to get from point A to point B, we will call it t. And the speed of the beam - from. It turns out that: c * t = l
If we look at this ray from anotherplane, for example, from a space liner that moves with a speed v, then with such observation of bodies their speed will change. In this case, even fixed elements will move at a speed 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 ray moves from point A to point B, point A manages to move and, accordingly, the light already arrives at the new point C. To find half the distance to which the point A has shifted, the speed of the liner must be multiplied by half the travel time of the beam (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, and alsothe space liner is the vertex 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.
from2 = l2 + d2
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 is directly dependent onThe height of the antenna of the mobile operator. In order to calculate the distance from the mobile tower, the phone can receive a signal, you 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
Strange as it may seem, the Pythagorean theorem may turn out to beuseful even in household 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 are wondering 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 inhorizontal 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 = √ AB2+ √BC2
AC = √24742+8002= 2600 mm - everything converges.
Suppose the height of the cabinet is not 2474 mm, but 2505 mm. Then:
AC = √25052+ √8002= 2629 mm.
Therefore, this cabinet is not suitable for installation in this room. As when lifting it to a vertical position, you can damage its body.
Perhaps, having considered different methods of proofPythagorean theorem 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 confident that all calculations will not only be useful, but also true.
