One can be sure one hundred percent that the question of what is the square of the hypotenuse, any adult will safely answer: "The sum of the squares of the legs." This theorem is firmly entrenched in the minds of every educated person, but just ask someone to prove it, and this can cause difficulties. Therefore, let us recall and consider different ways of proving the Pythagorean theorem.
Biography Overview
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 exploring different ways of proving the Pythagorean theorem, you need to briefly get acquainted with his personality.
Pythagoras is a philosopher, mathematician, and 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 works of his followers, Pythagoras of Samos was born on the island of Samos. His father was an ordinary stone-cutter, but his mother came from a noble family.
According to legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, a born boy was supposed to bring much benefit and good to humanity. What actually he did.
The birth of a theorem
In his youth, Pythagoras moved from Samos to Egypt to meet famous Egyptian sages there. After meeting with them, he was allowed to study, where he knew all the great achievements of Egyptian philosophy, mathematics and medicine.
It was probably 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. He only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.
Whatever the case, today not one technique for proving this theorem is known, but several at once. Today, one can only guess how exactly the ancient Greeks made their calculations, so here we will consider different ways of proving the Pythagorean theorem.
Pythagorean theorem
Before you begin any calculations, you need to figure out what theory to prove. The Pythagorean theorem reads as follows: "In a triangle, in which one of the angles is 90 ° , the sum of the squares of the legs is equal to the square of the hypotenuse."
There are 15 different ways to prove the Pythagorean theorem. This is a fairly large number, so we will pay attention to the most popular of them.
Method one
First, let us know what is given to us. These data will also apply to other methods of proof of the Pythagorean theorem, so it is worth immediately remembering all the notation available.
Suppose a right 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 should be drawn from a right triangle.
To do this, you need to draw to a leg with length a and draw a segment equal to leg in, and vice versa. So it should turn out 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 a side equal to the hypotenuse of the original triangle. To do this, from the vertices ac and cv, draw two parallel segments equal to c. Thus, we get three sides of the square, one of which is the hypotenuse of the original right triangle. It remains only to draw the fourth segment.
Based on the resulting figure, we can conclude that the area of the outer square is (a + c) 2 . If you look inside the figure, you can see that in addition to the inner square, it has four rectangular triangles. The area of each is 0.5av.
Therefore, the area is: 4 * 0.5av + s 2 = 2av + s 2
Hence (a + c) 2 = 2av + c 2
And therefore, with 2 = a 2 + in 2
The theorem is proved.
Method two: similar triangles
This formula for the proof of the Pythagorean theorem was derived on the basis of the statement from the geometry section on similar triangles. It says that the leg of a right triangle is proportional to its hypotenuse and the segment of the hypotenuse starting from the top of the angle of 90 ° .
The initial data remain the same, so we start right away with the proof. Draw a segment perpendicular to the side AB of the LED. Based on the above statement, the legs of the triangles are equal:
AC = √ AB * HELL, CB = √ AB * DW.
To answer the question of how to prove the Pythagorean theorem, the proof must be laid by squaring both inequalities.
AC 2 = AB * HELL and CB 2 = AB * LW
Now we need to add up the resulting inequalities.
AC 2 + CB 2 = AB * (HELL * LW), where HELL + LW = AB
It turns out that:
AC 2 + CB 2 = AB * AB
And therefore:
AC 2 + CB 2 = AB 2
The proof of the Pythagorean theorem and various methods for its solution require a versatile approach to this problem. However, this option is one of the simplest.
Another calculation method
The description of different ways of proving the Pythagorean theorem can not say anything, until then until you independently begin to practice. Many methods include not only mathematical calculations, but also the construction of new shapes from the original triangle.
In this case, it is necessary to build another rectangular triangle of the IRR from the side of the aircraft. Thus, now there are two triangles with a common side leg.
Knowing that the areas of such figures are related as the squares of their similar linear sizes, then:
S ABC * s 2 - S avd * in 2 = S avd * a 2 - S vsd * a 2
S avs * (s 2 -v 2 ) = a 2 * (S avd -S vsd )
with 2 -in 2 = a 2
c 2 = a 2 + to 2
Since this option is hardly suitable for the 8th grade from different methods of proving the Pythagorean theorem, we can use the following methodology.
The easiest way to prove the Pythagorean theorem. Reviews
According to historians, this method was first used to prove the theorem in ancient Greece. It is the simplest, since it requires absolutely no calculations. If you draw the drawing correctly, then the proof of the statement that a 2 + in 2 = c 2 will be clearly visible.
The conditions for this method will be slightly different from the previous one. To prove the theorem, suppose that the right triangle ABC is isosceles.
We take the hypotenuse AC as the side of the square and daughter three of its sides. In addition, it is necessary to draw two diagonal lines in the resulting square. Thus, in order to get four isosceles triangles inside it.
To the legs AB and CB, it is also necessary to draw a square and draw one diagonal line in each of them. We draw the first line from vertex A, the second - from C.
Now you need to carefully look at the resulting drawing. Since there are four triangles on the hypotenuse of the AS equal to the original, and two on the legs, this indicates the veracity of this theorem.
By the way, thanks to this technique of proving the Pythagorean theorem, the famous phrase was born: "Pythagorean trousers are equal in all directions."
Proof of J. Garfield
James Garfield is the twentieth president of the United States of America. In addition to leaving his mark on history as the ruler of the United States, he was also a gifted self-taught man.
At the beginning of his career, he was an ordinary teacher in a public school, but soon became the director of one of the higher educational institutions. The desire for self-development and allowed him to propose a new theory of proof of the Pythagorean theorem. The theorem and an example of its solution are as follows.
First you need to draw two rectangular triangles on a piece of paper so that the leg of one of them is a continuation of the second. The vertices of these triangles need to be connected in order to end up with a trapezoid.
As you know, the area of a trapezoid is equal to the product of half the sum of its bases by 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 = AB / 2 * 2 + s 2/2
Now you need to equalize the two source expressions
2av / 2 + s / 2 = (a + c) 2/2
c 2 = a 2 + to 2
More than one volume of the textbook can be written on the Pythagorean theorem and methods of its proof. But does it make sense when this knowledge cannot be put into practice?
The practical application of the Pythagorean theorem
Unfortunately, modern school curricula provide for the use of this theorem only in geometric problems. Graduates will soon leave the school walls, never knowing how they can put their knowledge and skills into practice.
In fact, everyone can use the Pythagorean theorem in their daily lives. And not only in professional activities, but also in ordinary household chores. We consider several cases where the Pythagorean theorem and methods of its proof can be extremely necessary.
The connection between theorems and astronomy
It would seem how stars and triangles on paper can be connected. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.
For example, consider the movement of a light beam in space. It is known that light moves in both directions at the same speed. The trajectory AB, by which the light beam moves, is called l . And half the time that the light needs 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 you look at this same ray from another plane, for example, from a space liner that moves with speed v, then with such observation of bodies their speed will change. In this case, even the stationary elements will begin to move at a speed v in the opposite direction.
Let's say a comic airliner floats to the right. Then points A and B, between which the beam is tagged, will begin to move to the left. Moreover, when the beam moves from point A to point B, point A manages to move and, accordingly, the light arrives at a new point C. To find half the distance by which point A has shifted, you need to multiply the liner speed by half the beam travel time (t ').
d = t '* v
And in order to find how far a ray of light could go during this time, you need 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 space liner, are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two right-angled triangles. Therefore, thanks to the Pythagorean theorem, one can find the distance that a ray of light could travel.
s 2 = l 2 + d 2
This example, of course, is not the most successful, since only a few can be lucky enough to try it in practice. Therefore, we consider more mundane applications of this theorem.
Mobile Signal Transmission Radius
Modern life is impossible to imagine without the existence of smartphones. But how much use would there be if they could not connect subscribers through mobile communications ?!
The quality of mobile communications directly depends on how high the antenna of the mobile operator is. In order to calculate how far a telephone can receive a signal from a mobile tower, the Pythagorean theorem can be applied.
Suppose you need to find the approximate height of a stationary tower so that it can propagate a 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 + ABOW = r + x
Applying the Pythagorean theorem, we find out that the minimum height of the tower should be 2.3 kilometers.
Pythagorean theorem in everyday life
Oddly enough, the Pythagorean theorem can be useful even in household affairs, such as determining the height of a wardrobe, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements using a tape measure. But many are surprised why during the assembly process there are certain problems if all the measurements were taken more than accurately.
The fact is that the wardrobe is assembled in a horizontal position and only then rises and installs against the wall. Therefore, the sidewall of the cabinet in the process of lifting the structure must freely pass both in height and in the diagonal of the room.
Suppose you have a sliding wardrobe 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 exactly 126 mm? Let's look at an example.
With ideal cabinet dimensions, we verify the action of the Pythagorean theorem:
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 lifting it to a vertical position can cause damage to its body.
Perhaps, having considered different ways of proving the Pythagorean theorem by different scientists, we can conclude that it is more than true. Now you can use the information received in your daily life and be completely sure that all the calculations will be not only useful, but also correct.