Different ways to prove the Pythagorean theorem: examples, descriptions and reviews. Several ways to prove the Pythagorean theorem Practical application of the Pythagorean theorem
Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relation
between the sides of a right triangle.
It is believed that it was proven by the Greek mathematician Pythagoras, after whom it was named.
Geometric formulation of the Pythagorean theorem.
The theorem was originally formulated as follows:
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 legs.
Algebraic formulation of the Pythagorean theorem.
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
That is, denoting the length of the hypotenuse of the triangle by c, and the lengths of the legs through a And b:
Both formulations Pythagorean theorem are equivalent, but the second formulation is more elementary, it does not
requires the concept of area. That is, the second statement can be verified without knowing anything about the area and
by measuring only the lengths of the sides of a right triangle.
Converse Pythagorean theorem.
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then
right triangle.
Or, in other words:
For every triple of positive numbers a, b And c, such that
there is a right triangle with legs a And b and hypotenuse c.
Pythagorean theorem for an isosceles triangle.
Pythagorean theorem for an equilateral triangle.
Proofs of the Pythagorean theorem.
On this moment V scientific literature 367 proofs of this theorem have been recorded. Probably the theorem
Pythagoras is the only theorem with such an impressive number of proofs. Such diversity
can only be explained by the fundamental significance of the theorem for geometry.
Of course, conceptually all of them can be divided into a small number of classes. The most famous of them:
proof area method, axiomatic And exotic evidence(For example,
by using differential equations).
1. Proof of the Pythagorean theorem using similar triangles.
The following proof of the algebraic formulation is the simplest of the proofs constructed
directly from the axioms. In particular, it does not use the concept of area of a figure.
Let ABC there is a right triangle with a right angle C. Let's draw the height from C and denote
its foundation through H.
Triangle ACH similar to a triangle AB C at two corners. Likewise, triangle CBH similar ABC.
By introducing the notation:
we get:
,
which corresponds to -
Folded a 2 and b 2, we get:
or , which is what needed to be proven.
2. Proof of the Pythagorean theorem using the area method.
The proofs below, despite their apparent simplicity, are not so simple at all. All of them
use properties of area, the proofs of which are more complex than the proof of the Pythagorean theorem itself.
- Proof through equicomplementarity.
Let's arrange four equal rectangular
triangle as shown in the figure
on right.
Quadrangle with sides c- square,
since the sum of two sharp corners 90°, a
unfolded angle - 180°.
The area of the entire figure is, on the one hand,
area of a square with side ( a+b), and on the other hand, the sum of the areas of four triangles and
Q.E.D.
3. Proof of the Pythagorean theorem by the infinitesimal method.
Looking at the drawing shown in the figure and
watching the side changea, we can
write the following relation for infinitely
small side incrementsWith And a(using similarity
triangles):
Using the variable separation method, we find:
A more general expression for the change in the hypotenuse in the case of increments on both sides:
Integrating given equation and using the initial conditions, we get:
Thus we arrive at the desired answer:
As is easy to see, the quadratic dependence in the final formula appears due to the linear
proportionality between the sides of the triangle and the increments, while the sum is related to the independent
contributions from the increment of different legs.
A simpler proof can be obtained if we assume that one of the legs does not experience an increase
(in this case the leg b). Then for the integration constant we obtain:
One thing you can be one hundred percent sure of is that when asked what the square of the hypotenuse is, any adult will boldly answer: “The sum of the squares of the legs.” This theorem is firmly ingrained in the minds of every educated person, but you just need to ask someone to prove it, and difficulties can arise. So let's remember and consider different ways proof of the Pythagorean theorem.
Brief biography
The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who brought it into the world is not so popular. This can be fixed. Therefore, before exploring the different ways to prove Pythagoras’ theorem, you need to briefly get to know his personality.
Pythagoras - philosopher, mathematician, thinker originally from 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.
Judging by the legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, the born boy was supposed to bring a lot of benefit and good to humanity. Which is exactly what he did.
Birth of the theorem
In his youth, Pythagoras moved to Egypt to meet famous Egyptian sages there. After meeting with them, he was allowed to study, where he learned 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. But he only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.
Be that as it may, today not one method of proving this theorem is known, but several at once. Today we can only guess how exactly the ancient Greeks carried out their calculations, so here we will look at different ways to prove the Pythagorean theorem.
Pythagorean theorem
Before you begin any calculations, you need to figure out what theory you want to prove. The Pythagorean theorem goes like this: “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 a total of 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's define what we've been given. These data will also apply to other methods of proving the Pythagorean theorem, so it is worth immediately remembering all the available notations.
Suppose we are given a right triangle with legs a, b and a hypotenuse equal to c. The first method of proof is based on the fact that right triangle you need to complete the square.
To do this, you need to add a segment equal to leg b to the leg of length a, and vice versa. This should make two equal sides square. All that remains is 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 ас and св you need to draw two parallel segments equal to с. Thus, we get three sides of the square, one of which is the hypotenuse of the original right triangle. All that remains is to draw the fourth segment.
Based on 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 right triangles. The area of each is 0.5av.
Therefore, the area is equal to: 4 * 0.5ab + c 2 = 2av + c 2
Hence (a+c) 2 =2ab+c 2
And, therefore, c 2 =a 2 +b 2
The theorem is proven.
Method two: similar triangles
This formula for proving the Pythagorean theorem was derived based on a statement from the section of geometry about similar triangles. It states that the leg of a right triangle is the average proportional to its hypotenuse and the segment of the hypotenuse emanating from the vertex of the 90° angle.
The initial data remains the same, so let's start right away with the proof. Let us draw a segment CD perpendicular to side AB. Based on the above statement, the sides of the triangles are equal:
AC=√AB*AD, SV=√AB*DV.
To answer the question of how to prove the Pythagorean theorem, the proof must be completed by squaring both inequalities.
AC 2 = AB * AD and CB 2 = AB * DV
Now we need to add up the resulting inequalities.
AC 2 + CB 2 = AB * (AD * DV), where AD + DV = AB
It turns out that:
AC 2 + CB 2 =AB*AB
And therefore:
AC 2 + CB 2 = AB 2
Proof of the Pythagorean theorem and various ways its solutions require a multifaceted approach to this problem. However, this option is one of the simplest.
Another calculation method
Descriptions of different methods of proving the Pythagorean theorem may not mean anything until you start practicing on your own. Many techniques involve not only mathematical calculations, but also the construction of new figures from the original triangle.
In this case, it is necessary to complete another right triangle VSD from the side 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 * c 2 - S avd * in 2 = S avd * a 2 - S vsd * a 2
S avs *(from 2 - to 2) = a 2 *(S avd -S vsd)
from 2 - to 2 =a 2
c 2 =a 2 +b 2
Since out of the various methods of proving the Pythagorean theorem for grade 8, this option is hardly suitable, you can use the following method.
The easiest way to prove the Pythagorean Theorem. Reviews
According to historians, this method was first used to prove the theorem back in ancient Greece. It is the simplest, as it does not require absolutely any calculations. If you draw the picture correctly, then the proof of the statement that a 2 + b 2 = c 2 will be clearly visible.
The conditions for this method will be slightly different from the previous one. To prove the theorem, assume that right triangle ABC is isosceles.
We take the hypotenuse AC as the side of the square and draw its three sides. In addition, it is necessary to draw two diagonal lines in the resulting square. So that inside it you get four isosceles triangles.
You also need to draw a square to the legs AB and CB and draw one diagonal straight 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 on the hypotenuse AC there are four triangles equal to the original one, and on the sides there are two, this indicates the veracity of this theorem.
By the way, thanks to this method of proving the Pythagorean theorem, the famous phrase was born: “Pythagorean pants are equal in all directions.”
Proof by J. Garfield
James Garfield is the twentieth President of the United States of America. In addition to making his mark on history as the ruler of the United States, he was also a gifted autodidact.
At the beginning of his career he was an ordinary teacher in a public school, but soon became the director of one of the highest educational institutions. The desire for self-development allowed him to offer new theory proof of the Pythagorean theorem. The theorem and an example of its solution are as follows.
First you need to draw two right 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 to ultimately form a trapezoid.
As you know, the area of a trapezoid is equal to the product of half the sum of its bases and its 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 we need to equalize the two original expressions
2ab/2 + c/2=(a+b) 2 /2
c 2 =a 2 +b 2
More than one volume could be written about the Pythagorean theorem and methods of proving it. teaching aid. But is there any point in it when this knowledge cannot be applied in practice?
Practical application of the Pythagorean theorem
Unfortunately, in modern school programs This theorem is intended to be used only in geometric problems. Graduates will soon leave school without knowing how they can apply their knowledge and skills in practice.
In fact, use the Pythagorean theorem in your Everyday life everyone can. And not only in professional activity, but also in ordinary household chores. Let's consider several cases when the Pythagorean theorem and methods of proving it may be extremely necessary.
Relationship between the theorem 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. Let's call the trajectory AB along which the light ray moves l. And let's call half the time it takes light to get from point A to point B t. And the speed of the beam - 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 when observing bodies in this way, their speed will change. In this case, even stationary elements will begin to move at speed v in the opposite direction.
Let's say the comic liner is sailing to the right. Then points A and B, between which the beam rushes, will begin to move to the left. Moreover, when the beam moves from point A to point B, point A has time to move and, accordingly, the light will already arrive at a new point C. To find half the distance by which point A has moved, you need to multiply the speed of the liner by half the travel time of the beam (t ").
And to find how far a ray of light could travel during this time, you need to mark half the path with a new letter s and get the following expression:
If we imagine that points of light C and B, as well as the space liner, are the vertices isosceles triangle, then the segment from point A to the liner will divide it into two right triangles. Therefore, thanks to the Pythagorean theorem, you can find the distance that a ray of light could travel.
This example, of course, is not the most successful, since only a few will be lucky enough to try it out in practice. Therefore, let's consider more mundane applications of this theorem.
Mobile signal transmission range
Modern life can no longer be imagined without the existence of smartphones. But how much use would they be if they couldn’t connect subscribers via mobile communications?!
The quality of mobile communications directly depends on the height at which the mobile operator’s antenna is located. In order to calculate how far from a mobile tower a phone can receive a signal, you can apply the Pythagorean theorem.
Let's say you need to find the approximate height of a stationary tower so that it can distribute a signal within a radius of 200 kilometers.
AB (tower height) = x;
BC (signal transmission radius) = 200 km;
OS (radius globe) = 6380 km;
OB=OA+ABOB=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 everyday matters, 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 people wonder why certain problems arise during the assembly process if all measurements were taken more than accurately.
The fact is that the wardrobe is assembled in a horizontal position and only then raised and installed against the wall. Therefore, during the process of lifting the structure, the side of the cabinet must move freely both along the height and diagonally of the room.
Let's assume there is a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 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, let’s check the operation of the Pythagorean theorem:
AC =√AB 2 +√BC 2
AC=√2474 2 +800 2 =2600 mm - everything fits.
Let's say 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. Because lifting it into 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 confident that all calculations will be not only useful, but also correct.
Lesson on the topic: “Pythagorean Theorem”
Lesson type: lesson on learning new material. (according to the textbook “Geometry, 7–9”, textbook for educational institutions; L.S. Atanasyan et al. - 12th ed. - M.: Education, 2009).
Target:
introduce students to the Pythagorean theorem and historical information related to this theorem; develop interest in studying mathematics, logical thinking; attention.
During the classes:
1. Organizational moment.
SLIDE 2 Fairy tale “Home”.
The topic of our lesson is “Pythagorean Theorem”. Today in the lesson we will get acquainted with the biography of Pythagoras, study one of the most famous geometric theorems of antiquity, called the Pythagorean theorem, one of the main theorems of planimetry.
2. Updating knowledge.(Preparation for studying new material, repeating the material that will be needed to prove the theorem)
1) Questions:
Which quadrilateral is called a square?
How to find the area of a square?
Which triangle is called a right triangle?
What are the sides of a right triangle called?
How to find the area of a right triangle?
3. Studying new material.
1) Historical reference.
SLIDE 3 and 4.
The great scientist Pythagoras was born around 570 BC. on the island of Samos. Pythagoras's father was Mnesarchus, a gem cutter. The name of Pythagoras' mother is unknown. According to many ancient testimonies, the boy who was born was fabulously handsome, and soon showed his extraordinary abilities. Like any father, Mnesarchus dreamed that his son would continue his work - the craft of a goldsmith. Life decided otherwise. The future great mathematician and philosopher already in childhood showed great abilities for science.
Pythagoras is credited with studying the properties of integers and proportions, proving the Pythagorean theorem, etc. Pythagoras is not a name, but a nickname that the philosopher received because he always spoke correctly and convincingly, like a Greek oracle. (Pythagoras - “persuasive by speech.”)
With his speeches he acquired 2,000 students, who, together with their families, formed a school-state, where the laws and rules of Pythagoras were in effect. The school of Pythagoras, or, as it is also called, the Pythagorean Union, was at the same time a philosophical school, a political party, and a religious fraternity.
The favorite geometric figure of the Pythagoreans was the pentagram, also called the Pythagorean star. The Pythagoreans used this figure, drawing it in the sand, to greet and recognize each other. The pentagram served as their password and was a symbol of health and happiness.
Tradition says that when Pythagoras came to the theorem that bears his name, he brought 100 bulls to the gods. In the five hundred years BC, Pythagoras was killed in a street fight during a popular uprising. Currently, about 200 proofs of the Pythagorean theorem are known.
Statement of the theorem
2) Proof of the theorem.
Let's build the rectangle to a square with side a + b.
The children, with the help of the teacher, prove the theorem using a drawing, then write the proof in their notebooks.
Proof:
Square area
- the theorem is proven.
4. Primary consolidation of knowledge.
Work according to the textbook (Application of the Pythagorean theorem to solving problems).
Problems are solved on the board and in notebooks.
Conclusion: using the Pythagorean theorem you can solve two types of problems:
1. Find the hypotenuse of a right triangle if the legs are known.
2. Find a leg if the hypotenuse and the other leg are known.
.
5. Independent problem solving.
No. 483(b), 484(b)
6. Homework: P 54, No. 483 (g), 484 (g).
7. Lesson summary.
What new did you learn in class today?
For which triangles does the Pythagorean theorem apply?
End the lesson with a poem.
Many people know Chamisso’s sonnet:
The truth will remain eternal, as soon as
He will know her weak person!
And now the Pythagorean theorem
True, as in his distant age.
The sacrifice was abundant
To the gods from Pythagoras. Hundred bulls
He gave it up to be slaughtered and burned
Behind the light is a ray that came from the clouds.
Therefore, ever since then,
The truth is just being born,
The bulls roar, sensing her, following her.
They are unable to stop the light,
And they can only close their eyes and tremble
From the fear that Pythagoras instilled in them.
Question - answer An angle whose degree measure is 90° RIGHT The side lying opposite right angle triangle HYPOTENUSE Triangle, square, trapezoid, circle - these are geometric ... FIGURES Small side of a right triangle LATE A figure formed by two rays emanating from one point ANGLE A perpendicular segment drawn from the vertex of a triangle to a straight line containing the opposite side HEIGHT A triangle with two sides equal isosceles
Pythagoras of Samos (c. 580 - c. 500 BC) Ancient Greek mathematician and philosopher. Born on the island of Samos. He organized his own school - the school of Pythagoras (Pythagorean Union), which was at the same time a philosophical school, a political party, and a religious fraternity. He was the first to prove the relationship between the hypotenuse and the legs of a right triangle.
Problem of the 12th century Indian mathematician Bhaskara. A lonely poplar tree grew on the bank of the river. Suddenly a gust of wind broke its trunk. The poor poplar fell. And its trunk made a right angle with the flow of the river. Remember now that in this place the river was only four feet wide. The top leaned at the edge of the river. There are only three feet left from the trunk. I ask you, tell me soon: How tall is the poplar?”
