Geometry lesson grade 8.
"Pythagorean theorem"
Teacher: Naumenko N.M.
- Educational Purpose:get acquainted with the biography of Pythagoras,study of the Pythagorean theorem, its role in geometry; use of the theorem in solving problems.
- Development goal:
- educational goal:culture of mathematical speech.
Lesson plan:
- Organizing time.
- Knowledge update.
- Learning new material
- Historical background on Pythagoras (presentation)
- Primary consolidation of knowledge.
- Lesson results.
- Homework.
- Merry minute
Equipment: portrait of Pythagoras, blackboard, multimedia equipment (PC, projector, screen), presentation material, handouts (according to the number of students).
During the classes:
(Annex 1 )
I. Organizational moment.
Hello guys, sit down
And do not be lazy to work.
Take notebooks and pens
Number in notebooks 11/19/15. wrote instantly.
Today we have guests at the lesson. And I would like them to have a good time with us. And it depends on us. I hope that we will do our best to make our guests leave us with good impressions.
Let's start the lesson with a review of the studied material.
II. Updating of basic knowledge.
slide 2 - right triangle.
slide 3 - equality of triangles on two legs
slide 4 – area property
slide 5 -finding the angle
Slide 6 is a task.
Slide 7
And in order for us to decide
What should be learned in the classroom
Verbally consider the drawing on the board,
Find the area of each figure.
1. Given ∆ABC - rectangular, hypotenuse AB = 12 cm, leg CB-3 cm.
Find S∆.
2. What figure is shown?
What is the S of the trapezoid - ?
What do we not know? (height)
How to find height?
(problem is posed)
We are given ∆ABC-rectangular, hypotenuse AB=5m., leg CB-3m.
Find S∆.
What is S ∆ -?
What do we know? (rolls, hypotenuse, angle 90 0 )
In this problem, can we find the leg AC?
Can we or can't we?
For today's lesson, we don't know how to find it.
So what is our task today? To find out that? (Find the unknown side right triangle).
That. we have formulated the goal of our lesson: Learn to find the unknown side of a right triangle.
III. Learning new material.
Student:
The stories of the veil are opened and
We immediately find ourselves in the ancient world
4th century BC goes,
And in ancient greece the learned Pythagoras neither eats nor sleeps nor drinks.
Teacher:
Oh, gods, my mind beg you to bestow.
So that the truth, which is dearer to me to open,
I am ready to sacrifice 100 bulls,
to prove this theorem.
I'm not alone? Did people come here?
Then, friends, help me,
So that I found the truth, which is dearer than all.
And if I'm wrong, please correct me.
Slide 8
All triangles are equal, I will distribute rectangular ones,
I ask myself and you a question -
Is it possible to arrange them in such a way as to end up with a square?
Please take white sheets, 4 triangles, and try to make a square out of them on a white sheet. From 4 triangles should make a square.
There are options?
Everything, we got a square,
And I am very happy about this!
On the board, the teacher lays out a square with the help of 4 triangles and magnets.
Now look at the board carefully
And find the area of the resulting square.
All ways you find are good!
I wish you all success from the bottom of my heart!
Lay and glue the resulting square on a white sheet. Sign where the legs are, and where is the hypotenuse (legs - a, b, hypotenuse - c), vertices A, B, C, D.
We work quickly and accurately.
Why is this figure a square? (definition)
- Angles 90 0 ;
- The sides are equal (a + b);
- So how do you find the S of the square ABCD?
S sq. = square side. What is the length of the side of our square?
S ABSD \u003d (a + c) 2 - write.
What is the square of the sum?Call the student to the board.
S ABSD \u003d (a + b) 2 \u003d a 2 + 2av + b 2 (1)
How else can you find S sq. ? We think. This figure is made up of what figures?
From 4 triangles and MNLK figure (sign vertices), i.e.
S ABSD = 4 S tr + S MNLK
What is S ∆ -? S = ∆av
That. S ABSD \u003d 4 av + S MNLK \u003d 2av + S MNLK
Why is MNLK a square?
The sides are equal, but it can also be a rhombus. How is a rhombus different from a square? (corners)
Why is the angle 90 0 ? Because the amount sharp corners right triangle is 90 0 and the triangles are equal in 2 legs.
What is S MNLK equal to? S MNLK = s 2
Received, S ABSD \u003d 2av + s 2 (2)
What can we do with you now? Can we equate equalities (1) and (2)? 2av + s 2 \u003d a 2 + 2av + in 2 How can we simplify this equality? ( student to the blackboard)
c 2 \u003d a 2 + b 2
WITH - ? A - ? V - ? (hypotenuse, leg, leg)
Without naming letters, name what we got for a right triangle.
The square of the hypotenuse is equal to the sum of the squares of the legs.
Slide 9
Proved everything! Praise the gods!
What you promised, you have to give
And 100 bulls all as a sacrifice to you,
Let the theorem be called by my name!
We write down the topic of the lesson: "The Pythagorean theorem."
Many people believe that Pythagoras is a myth, that he was invented, and that he is a man - a legend. But we proceed from the position that the real is a real person, a great person in the history of all mankind.
slide 10. Let's listen to a story about this mathematician, after whom the theorem is named ( student). The message was prepared for us by Daria Orlova.
PYTHAGORUS OF SAMOS (c. 580 - c. 500 BC)
Little is known about the life of Pythagoras. He was born in 580 BC. e. in ancient Greece on the island of Samos, which is located in the Aegean Sea off the coast of Asia Minor, so he is called Pythagoras of Samos.
Pythagoras was born in the family of a stone carver who found fame rather than wealth. As a child, he showed extraordinary abilities, and when he grew up, the restless imagination of the young man became crowded on a small island.
He went to Egypt. An unknown country opened up before Pythagoras. He comprehended the science of the Egyptian priests, and was going home to create his own school there. But the priests did not want their knowledge to spread beyond the territory of their temples and did not want to let him go. With great difficulty he managed to overcome this obstacle.
However, on the way home, Pythagoras was captured and ended up in Babylon. The Babylonians appreciated smart people so he found his place among the Babylonian sages. The science of Babylon was more developed than in Egypt. The Babylonians invented and applied the positional number system in counting, they were able to solve linear, quadratic and some cubic equations.
Pythagoras lived in Babylon for 10 years and returned to his homeland. But on the island of Samos, he did not stay long, and settled in one of the Greek colonies of southern Italy. There Pythagoras organized a secret youth union.
Slide 11. New members were admitted to this union with great ceremonies after long trials. The Pythagoreans, as they were later called, were engaged in mathematics, philosophy, and the natural sciences. Pythagoreans did a lot important discoveries in arithmetic and geometry, including:
Geometric solutions of quadratic equations;
Division of numbers into even and odd, prime and composite;
Theorem on the sum of the angles of a triangle and many others. others
Pythagoras participated in the Olympic Games and won two fisticuffs.
The scientist devoted about forty years to the school he created, and at the age of eighty, according to one version, Pythagoras was killed in a street fight during a popular uprising.
slide 12. The proof of the Pythagorean theorem was considered very difficult in the circles of students of the Middle Ages and was sometimes called Pons Asinorum."donkey bridge" or elefuga - "flight of the poor"because some "wretched" students, who did not have a serious mathematical background, fled from geometry.
Weak students who memorized theorems by heart, without understanding, and therefore called "donkeys", were not able to overcome the Pythagorean theorem, which served for them like an insurmountable bridge.
Pythagoras made many important discoveries, but the most famous scientist was the theorem he proved, which now bears his name.
Slide 13. (teacher) So, the Pythagorean theorem.
Slide 14. (student). Prepared by Bulgakov
Teacher:
slide 15. It is believed that in the time of Pythagoras the theorem sounded differently:
"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 its legs."
Look, and here is "Pythagorean pants are equal in all directions."
Such rhymes were invented by students of the Middle Ages when studying the theorem; drew cartoons. Here, for example, are. slide 16.
The Pythagorean theorem is one of the main theorems of geometry, because it can be used to prove many other theorems and solve many problems.
We will solve several problems.
slide 17. Problem number 483. Let's take a handout and consider the solution of this problem together.
∆ABC - rectangular with hypotenuse AB.
According to the Pythagorean theorem, AB² = AC² + BC²
C²=a²+b²
С²=6²+8²
С²=36+64
С²=100
C=10
Answer: 10
Slide 18. Task No. 483. (on my own)
Slide 19. Problem number 484.
slide 20. Problem number 486.
Slide 21. Problem number 487.
slide 22.
Reflection .(2 min)
- What new did you learn at the lesson today?(Today, for the lesson, we got acquainted with the Pythagorean theorem, with some information from the life of a scientist. We solved several simple problems)
- For which triangles does the Pythagorean theorem apply?
- What is the Pythagorean theorem?
Well done boys. You did a great job today
slide 23. Homework.
So, today in the lesson we got acquainted with one of the main theorems of geometry, the Pythagorean theorem and its proof, with some information from the life of the scientist, whose name it bears, we solved several simple problems.
The significance of the Pythagorean theorem lies in the fact that many theorems of geometry can be deduced from it or with its help and many problems can be solved.
By the next lesson, you should have learned the Pythagorean theorem with proof, as we will learn how to apply it to more complex problems.
- P.54, problems 483 (c), 484 (b, d), 486 (b).
- Prepare the message "Egyptian Triangle".
slide 22. Merry minute (with a question for the attentive and observant - where is the mistake?)– application 2 .
Emotional release:
- frown like an autumn cloud, an angry person, an evil sorceress
- smile like a cat in the sun, Pinocchio, a cunning fox, a child who saw a miracle
- tired like dad after work, a man who lifted a load, an ant who dragged a big fly
- relax like a tourist who took off a heavy backpack, a child who worked hard, a tired warrior.
Preview:
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Slides captions:
Pythagorean theorem Geometry Grade 8 Naumenko N.M.
What is shown? Questions What is the sum of the acute angles in a right triangle? A + B = 90° What is the area of this triangle? What are the names of sides AC and BC? C A B a b c
B C A C 1 A 1 B 1 Prove that the triangles are congruent.
A B C D E S ABCDE = S ABC + S ADC + S ADE
1 3 2 Find 3 if 1+ 2 = 90°.
Solve orally C A B Given: ∆ ABC, C=90°, AB=18 cm, BC=9 cm Find: B, A 1. 18 9 60 12 10
Orally consider the drawing on the board, find the area of \u200b\u200bthe figure of each.
Pythagoras of Samos Samos
The Pythagoreans made many important discoveries in arithmetic and geometry. Pythagoras of Samos
"Donkey Bridge" The proof of the Pythagorean theorem was considered very difficult in the circles of students of the Middle Ages and was sometimes called Pons Asinorum "donkey bridge" or elefuga - "flight of the wretched", since some "wretched" students who did not have serious mathematical training fled geometry. Weak students who memorized theorems by heart, without understanding, and therefore called "donkeys", were not able to overcome the Pythagorean theorem, which served for them like an insurmountable bridge.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Pythagorean theorem with b and c ² = a² + b² So, If we are given a triangle, And, moreover, with a right angle, Then the square of the hypotenuse We will always easily find: We square the legs, We find the sum of degrees - And in such a simple way We will come to the result. c²=a²+b²
History of the Pythagorean Theorem Pythagoras of Samos c. 580 - approx. 500 BC
It is believed that in the time of Pythagoras the theorem sounded differently: "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 its legs."
Because of the drawings accompanying the Pythagorean theorem, the students called it the same “ windmill”, composed poems like “Pythagorean pants are equal on all sides”, drew caricatures. Cartoons from a 16th century textbook Student cartoon of the 19th century
No. 483 6 8 ? C A B Given: ∆ABC, C=90 º, a=6, b=8 Find: p. Solution: ∆ABC - rectangular with hypotenuse AB. According to the Pythagorean theorem, AB ² \u003d AC ² + BC ² s ² \u003d a ² + b² s ² = 6² + 8² s ² = 36 + 64 s ² = 100 c \u003d 10 Answer: 10
c ² \u003d a 2 + b 2 8 6 5 10 8 6 c b a a b c C A B No. 483 √61 c \u003d √ a 2 + b 2
c ² \u003d a 2 + b 2 a b c C A B No. 484 2 3b 2b 12 13 5 12 c b a 13 ² \u003d 12 2 + b 2 169 \u003d 144 + b 2 b 2 \u003d 169-144 \u003d 25 b \u003d 5 4 b ² = 12 2 + b 2 3b ² = 144 b ² = 48 b = √ 48 √ 48 a 2 + b 2 = c ² a 2 = c ²-b² b 2 = c ²-a² a = √ c ² -b² b = √ c ²-a² Let's write down the formulas for finding the legs of a right triangle:
c ² \u003d a 2 + b 2 No. 48 6 A C B D 5 13 AD ² \u003d AC²-CD² AD \u003d 12
№ 487 Given: ∆ABC, AB=BC=17 cm, AC=16 cm, BD AC Find: BD. Solution. 1. AD=DC=AC: 2=8 s m 2. Consider ∆ADB . BD²=AB²-AD² BD=√289-64 BD=15 (cm) Answer: 15 cm A C B D
Conduct a self-assessment learning activities according to the table Activity high average low topic Learned well Learned partially Learned poorly Explain to a friend I can do it myself I can, but I find it difficult to give hints
Homework P.54, tasks 483 (c), 484 (b, d), 486 (b). Prepare the message "Egyptian Triangle".
THANK YOU FOR THE LESSON!!!
Preview:
Slide 13 (student). The history of the Pythagorean theorem is interesting.
Although this theorem is associated with the name of Pythagoras, it was known long before him. In Babylonian texts, she occurs 1200 years before Pythagoras. Apparently, he was the first to find its proof. An ancient legend has been preserved that, in honor of his discovery, Pythagoras sacrificed a bull to the gods, according to other testimonies, even a hundred bulls. But this contradicts the information about the moral and religious views of Pythagoras. It is said that he "forbidden even to kill animals, and even more so to feed on them, for animals have a soul, like us." In this regard, the following entry can be considered more plausible: “... when he discovered that in a right triangle the hypotenuse corresponds to the legs, he sacrificed a bull made of wheat dough.”
Preview:
Handout
c²=a²+b²
№ 483
Solution :
Conclusion:
№ 484
Solution:
C²=a²+b² | C²=a²+b² | C²=a²+b² | a²+b²=c² |
13²=12²+b² | a² = c² - b² |
||
b2 = | b²=c² -a² |
||
Conclusion:
C²=a²+b² No. 486
Given: ABCD is a rectangle,
AB=5 cm, AC=13 cm
Find: AD.
Solution:
№ 487
Given: ∆ABC, AB=BC=17 cm,
AC=16 cm, BD⊥ AC
Find: BD.
Solution:
Preview:
Conduct a self-assessment of your own learning activities according to the table.
Conduct a self-assessment of your own learning activities according to the table.
Activity | high | average | low |
topic | learned well | Learned partially | Learned poorly |
Explain to a friend | I can do it myself | I can, but with hints | find it difficult |
Activity | high | average | low |
topic Lesson fit thematic planning work program in geometry of the 8th grade, developed according to the author's program of L.S. Atanasyan. The lesson is closely related to the previously studied material, is held immediately after studying the topic “Area of a parallelogram, a triangle and a trapezoid” and is the first on this topic, in each next class students will apply the knowledge gained in grade 8. The Pythagorean theorem is one of the important theorems of geometry. The Pythagorean theorem allows you to significantly expand the range of problems solved in the course of geometry. The further exposition of the theoretical course is largely based on it. The type of lesson is the study and primary consolidation of new knowledge. The goal of the teacher: Organize the activities of students together with the teacher for the derivation, proof and primary consolidation of the Pythagorean theorem The structure of the lesson is aimed at creating favorable conditions for the study of this topic. Update stageknowledge is organized in the form of a presentation, which allows students to vividly and figuratively repeat the studied material, which prepares them for learning new topic, allows you to quickly get to work. At the next stage I create a problematic situation to determine the purpose of the lesson. At the stage learning new material, I organize the activities of students to prove the Pythagorean theorem (drawing up a model and discussing the proof). At the stage of primaryapplication of the Pythagorean theorem, the simplest problems were analyzed, returned to solving the problem that caused difficulties at the beginning of the lesson. The goal of the lesson set by me was fully achieved, the students were motivated and involved in educational and cognitive activities in the lesson. The interaction in the lesson was productive, the students showed independence, interest and ability to solve geometric problems. All assignments have been reviewed and completed. Techniques and teaching methods were applied in a logical sequence, clearly fitting into the structure of the lesson. In this lesson, I did not aim to solve more complex problems, because. this is the first lesson of three in the program and the whole variety of lessons where the Pythagorean theorem is used. The reflexive stage of the lesson was conducted in the form of frontal questions:Explain to a friend | I can do it myself | I can, but with hints | find it difficult |
1
Shapovalova L.A. (station Egorlykskaya, MBOU ESOSH No. 11)
1. Glazer G.I. History of mathematics at school VII - VIII grades, a guide for teachers, - M: Education, 1982.
2. Dempan I.Ya., Vilenkin N.Ya. "Behind the pages of a mathematics textbook" Handbook for students in grades 5-6. – M.: Enlightenment, 1989.
3. Zenkevich I.G. "Aesthetics of the Mathematics Lesson". – M.: Enlightenment, 1981.
4. Litzman V. The Pythagorean theorem. - M., 1960.
5. Voloshinov A.V. "Pythagoras". - M., 1993.
6. Pichurin L.F. "Beyond the Pages of an Algebra Textbook". - M., 1990.
7. Zemlyakov A.N. "Geometry in the 10th grade." - M., 1986.
8. Newspaper "Mathematics" 17/1996.
9. Newspaper "Mathematics" 3/1997.
10. Antonov N.P., Vygodskii M.Ya., Nikitin V.V., Sankin A.I. "Collection of Problems in Elementary Mathematics". - M., 1963.
11. Dorofeev G.V., Potapov M.K., Rozov N.Kh. "Mathematics Handbook". - M., 1973.
12. Shchetnikov A.I. "The Pythagorean doctrine of number and magnitude". - Novosibirsk, 1997.
13. " Real numbers. Irrational expressions» Grade 8. publishing house Tomsk University. – Tomsk, 1997.
14. Atanasyan M.S. "Geometry" grade 7-9. – M.: Enlightenment, 1991.
15. URL: www.moypifagor.narod.ru/
16. URL: http://www.zaitseva-irina.ru/html/f1103454849.html.
This academic year, I got acquainted with an interesting theorem, known, as it turned out, from ancient times:
"The square built on the hypotenuse of a right triangle is equal to the sum of the squares built on the legs."
Usually the discovery of this statement is attributed to the ancient Greek philosopher and mathematician Pythagoras (VI century BC). But the study of ancient manuscripts showed that this statement was known long before the birth of Pythagoras.
I wondered why, in this case, it is associated with the name of Pythagoras.
Relevance of the topic: The Pythagorean theorem is of great importance: it is used in geometry literally at every step. I believe that the works of Pythagoras are still relevant, because wherever we look, everywhere we can see the fruits of his great ideas, embodied in various branches of modern life.
The purpose of my research was: to find out who Pythagoras was, and what relation he has to this theorem.
Studying the history of the theorem, I decided to find out:
Are there other proofs of this theorem?
What is the significance of this theorem in people's lives?
What role did Pythagoras play in the development of mathematics?
From the biography of Pythagoras
Pythagoras of Samos is a great Greek scientist. Its fame is associated with the name of the Pythagorean theorem. Although now we already know that this theorem was known in ancient Babylon 1200 years before Pythagoras, and in Egypt 2000 years before him, a right-angled triangle with sides 3, 4, 5 was known, we still call it by the name of this ancient scientist.
Almost nothing is reliably known about the life of Pythagoras, but it is associated with his name a large number of legends.
Pythagoras was born in 570 BC on the island of Samos.
Pythagoras had a handsome appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received for always speaking correctly and convincingly, like a Greek oracle. (Pythagoras - "persuasive speech").
In 550 BC, Pythagoras makes a decision and goes to Egypt. So, an unknown country and an unknown culture opens up before Pythagoras. Much amazed and surprised Pythagoras in this country, and after some observations of the life of the Egyptians, Pythagoras realized that the path to knowledge, protected by the caste of priests, lies through religion.
After eleven years of study in Egypt, Pythagoras goes to his homeland, where along the way he falls into Babylonian captivity. There he gets acquainted with the Babylonian science, which was more developed than the Egyptian. The Babylonians knew how to solve linear, quadratic and some types of cubic equations. Having escaped from captivity, he could not stay long in his homeland because of the atmosphere of violence and tyranny that reigned there. He decided to move to Croton (a Greek colony in northern Italy).
It is in Croton that the most glorious period in the life of Pythagoras begins. There he established something like a religious-ethical brotherhood or a secret monastic order, whose members were obliged to lead the so-called Pythagorean way of life.
Pythagoras and the Pythagoreans
Pythagoras organized in the Greek colony in the south of the Apennine Peninsula a religious and ethical brotherhood, such as a monastic order, which would later be called the Pythagorean Union. The members of the union had to adhere to certain principles: firstly, to strive for the beautiful and glorious, secondly, to be useful, and thirdly, to strive for high pleasure.
The system of moral and ethical rules, bequeathed by Pythagoras to his students, was compiled into a kind of moral code of the Pythagoreans "Golden Verses", which were very popular in the era of Antiquity, the Middle Ages and the Renaissance.
The Pythagorean system of studies consisted of three sections:
Teachings about numbers - arithmetic,
Teachings about figures - geometry,
Teachings about the structure of the universe - astronomy.
The education system laid down by Pythagoras lasted for many centuries.
The school of Pythagoras did much to give geometry the character of a science. The main feature of the Pythagorean method was the combination of geometry with arithmetic.
Pythagoras dealt a lot with proportions and progressions and, probably, with the similarity of figures, since he is credited with solving the problem: “Based on the given two figures, construct a third, equal in size to one of the data and similar to the second.”
Pythagoras and his students introduced the concept of polygonal, friendly, perfect numbers and studied their properties. Arithmetic, as a practice of calculation, did not interest Pythagoras, and he proudly declared that he "put arithmetic above the interests of the merchant."
Members of the Pythagorean Union were residents of many cities in Greece.
The Pythagoreans also accepted women into their society. The Union flourished for more than twenty years, and then the persecution of its members began, many of the students were killed.
There were many different legends about the death of Pythagoras himself. But the teachings of Pythagoras and his disciples continued to live.
From the history of the creation of the Pythagorean theorem
It is currently known that this theorem was not discovered by Pythagoras. However, some believe that it was Pythagoras who first gave its full proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Elements. On the other hand, Proclus claims that the proof in the Elements is due to Euclid himself. As we can see, the history of mathematics has almost no reliable concrete data on the life of Pythagoras and his mathematical activity.
Let's start our historical review of the Pythagorean theorem with ancient China. Here Special attention attracted by the mathematical book of Chu-pei. This essay says this about the Pythagorean triangle with sides 3, 4 and 5:
"If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4."
It is very easy to reproduce their method of construction. Take a rope 12 m long and tie it to it along a colored strip at a distance of 3 m. from one end and 4 meters from the other. A right angle will be enclosed between sides 3 and 4 meters long.
Geometry among the Hindus was closely connected with the cult. It is highly probable that the hypotenuse squared theorem was already known in India around the 8th century BC. Along with purely ritual prescriptions, there are works of a geometrically theological nature. In these writings, dating back to the 4th or 5th century BC, we meet with the construction right angle using a triangle with sides 15, 36, 39.
In the Middle Ages, the Pythagorean theorem defined the limit, if not of the greatest possible, then at least of good mathematical knowledge. The characteristic drawing of the Pythagorean theorem, which is now sometimes turned by schoolchildren, for example, into a top hat dressed in a mantle of a professor or a man, was often used in those days as a symbol of mathematics.
In conclusion, we present various formulations of the Pythagorean theorem translated from Greek, Latin and German.
Euclid's theorem reads (literal translation):
"In a right triangle, the square of the side spanning the right angle is equal to the squares on the sides that enclose the right angle."
As we see, in different countries And different languages There are various versions of the formulation of the familiar theorem. Created at different times and in different languages, they reflect the essence of one mathematical pattern, the proof of which also has several options.
Five Ways to Prove the Pythagorean Theorem
ancient chinese proof
In an ancient Chinese drawing, four equal right-angled triangles with legs a, b and hypotenuse c are stacked so that their outer contour forms a square with side a + b, and the inner one forms a square with side c, built on the hypotenuse
a2 + 2ab + b2 = c2 + 2ab
Proof by J. Gardfield (1882)
Let us arrange two equal right-angled triangles so that the leg of one of them is a continuation of the other.
The area of the trapezoid under consideration is found as the product of half the sum of the bases and the height
On the other hand, the area of the trapezoid is equal to the sum of the areas of the resulting triangles:
Equating these expressions, we get:
The proof is simple
This proof is obtained in the simplest case of an isosceles right triangle.
Probably, the theorem began with him.
Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true.
For example, for the triangle ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two. The theorem has been proven.
Proof of the ancient Hindus
A square with a side (a + b), can be divided into parts either as in fig. 12. a, or as in fig. 12b. It is clear that parts 1, 2, 3, 4 are the same in both figures. And if equals are subtracted from equals (areas), then equals will remain, i.e. c2 = a2 + b2.
Euclid's proof
For two millennia, the most common was the proof of the Pythagorean theorem, invented by Euclid. It is placed in his famous book "Beginnings".
Euclid lowered the height BH from the vertex of the right angle to the hypotenuse and proved that its extension divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs.
The drawing used in the proof of this theorem is jokingly called "Pythagorean pants". For a long time he was considered one of the symbols of mathematical science.
Application of the Pythagorean Theorem
The significance of the Pythagorean theorem lies in the fact that most of the theorems of geometry can be derived from it or with its help and many problems can be solved. Besides, practical value the Pythagorean theorem and its inverse theorem is that with their help you can find the lengths of the segments without measuring the segments themselves. This, as it were, opens the way from a straight line to a plane, from a plane to volumetric space and beyond. It is for this reason that the Pythagorean theorem is so important for humanity, which seeks to discover more dimensions and create technologies in these dimensions.
Conclusion
The Pythagorean theorem is so famous that it is difficult to imagine a person who has not heard about it. I learned that there are several ways to prove the Pythagorean theorem. I studied a number of historical and mathematical sources, including information on the Internet, and realized that the Pythagorean theorem is interesting not only for its history, but also because it occupies an important place in life and science. This is evidenced by the results presented by me in this work. various interpretations the text of this theorem and the way of its proofs.
So, the Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable in that in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly on the drawing. But no matter how much you look at a right triangle, you will never see that there is a simple relation between its sides: c2 = a2 + b2. Therefore, visualization is often used to prove it. The merit of Pythagoras was that he gave a complete scientific proof this theorem. The personality of the scientist himself, whose memory is not accidentally preserved by this theorem, is interesting. Pythagoras is a wonderful speaker, teacher and educator, the organizer of his school, focused on the harmony of music and numbers, goodness and justice, knowledge and a healthy lifestyle. He may well serve as an example for us, distant descendants.
Bibliographic link
Tumanova S.V. SEVERAL WAYS TO PROVE THE PYTHAGOREAN THEOREM // Start in science. - 2016. - No. 2. - P. 91-95;URL: http://science-start.ru/ru/article/view?id=44 (date of access: 01/10/2020).
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 proved by the Greek mathematician Pythagoras, after whom it is 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 catheters.
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 through c, and the lengths of the legs through a And b:
Both formulations pythagorean theorems 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.
The inverse 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
triangle is rectangular.
Or, in other words:
For any triple of positive numbers a, b And c, such that
there is a right triangle with legs a And b and hypotenuse c.
The Pythagorean theorem for an isosceles triangle.
Pythagorean theorem for an equilateral triangle.
Proofs of the Pythagorean theorem.
At the moment, 367 proofs of this theorem have been recorded in the scientific literature. 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 in terms of 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 the area of a figure.
Let ABC there is a right angled triangle C. Let's draw a height from C and denote
its foundation through H.
Triangle ACH similar to a triangle AB C on two corners. Likewise, the triangle CBH similar ABC.
By introducing the notation:
we get:
,
which matches -
Having folded a 2 and b 2 , we get:
or , which was to be proved.
2. Proof of the Pythagorean theorem by the area method.
The following proofs, despite their apparent simplicity, are not so simple at all. All of them
use the properties of the area, the proof of which is more complicated than the proof of the Pythagorean theorem itself.
- Proof through equicomplementation.
Arrange four equal rectangular
triangle as shown in the picture
on right.
Quadrilateral with sides c- square,
since the sum of two acute angles is 90°, and
the developed angle is 180°.
The area of the whole 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.
Considering the drawing shown in the figure, and
watching the side changea, we can
write the following relation for infinite
small side incrementsWith And a(using similarity
triangles):
Using the method of separation of variables, we find:
A more general expression for changing the hypotenuse in the case of increments of both legs:
Integrating this equation and using the initial conditions, we obtain:
Thus, we arrive at the desired answer:
As it 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 increment
(in this case, the leg b). Then for the integration constant we get:
In one thing, you can be one hundred percent sure 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 planted in the minds of every educated person, but it is enough just to ask someone to prove it, and then difficulties can arise. Therefore, let's remember and consider different ways of proving the Pythagorean theorem.
Brief overview of the biography
The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who produced it is not so popular. We'll fix it. Therefore, before studying the different ways of proving the Pythagorean theorem, you need to briefly get acquainted with his personality.
Pythagoras - a 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 writings 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 to bring many benefits and good to mankind. Which is what he actually did.
The birth of a theorem
In his youth, Pythagoras moved to Egypt to meet the famous Egyptian sages there. After 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. 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 technique for proving this theorem is known, but several at once. Today we 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 start any calculations, you need to figure out which theory to prove. The Pythagorean theorem sounds like this: "In a triangle in which one of the angles is 90 o, 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 in total. This is a fairly large number, so let's pay attention to the most popular of them.
Method one
Let's first define what we have. This data will also apply to other ways of proving the Pythagorean theorem, so you should immediately remember all the available notation.
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 must be drawn from a right-angled triangle.
To do this, you need to draw a segment equal to the leg in to the leg length a, 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 sv, you need to 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-angled triangle. It remains only to draw the fourth segment.
Based on the resulting figure, we can conclude that the area of \u200b\u200bthe outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, it has four right-angled triangles. The area of each is 0.5 av.
Therefore, the area is: 4 * 0.5av + s 2 \u003d 2av + s 2
Hence (a + c) 2 \u003d 2av + c 2
And, therefore, with 2 \u003d a 2 + in 2
The theorem has been proven.
Method two: similar triangles
This formula for the proof of the Pythagorean theorem was derived on the basis of a statement from the section of geometry about similar triangles. It says that the leg of a right triangle is the mean proportional to its hypotenuse and the hypotenuse segment emanating from the vertex of an angle of 90 o.
The initial data remain the same, so let's start right away with the proof. Let us draw a segment CD perpendicular to the side AB. Based on the above statement, the legs of the triangles are equal:
AC=√AB*AD, SW=√AB*DV.
To answer the question of how to prove the Pythagorean theorem, the proof must be laid by squaring both inequalities.
AC 2 \u003d AB * HELL and SV 2 \u003d AB * DV
Now we need to add the resulting inequalities.
AC 2 + SV 2 \u003d AB * (AD * DV), where AD + DV \u003d AB
It turns out that:
AC 2 + CB 2 \u003d AB * AB
And therefore:
AC 2 + CB 2 \u003d AB 2
The proof of the Pythagorean theorem and various ways of solving it require a versatile approach to this problem. However, this option is one of the simplest.
Another calculation method
Description of different ways of proving the Pythagorean theorem may not say anything, until you start practicing on your own. Many methods 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-angled triangle VSD from the leg of the aircraft. 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 * s 2 - S avd * in 2 \u003d S avd * a 2 - S vd * a 2
S avs * (from 2 to 2) \u003d a 2 * (S avd -S vvd)
from 2 to 2 \u003d a 2
c 2 \u003d a 2 + in 2
Since this option is hardly suitable from different methods of proving the Pythagorean theorem for grade 8, you can use the following technique.
The easiest way to prove the Pythagorean theorem. Reviews
Historians believe that this method was first used to prove a theorem in ancient Greece. It is the simplest, since it does not require absolutely any calculations. If you draw a picture correctly, then the proof of the statement that a 2 + b 2 \u003d 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 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.
To the legs AB and CB, you also need 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 picture. Since there are four triangles on the hypotenuse AC, equal to the original one, and two on the legs, 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 20th 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.
At the beginning of his career, he was an ordinary teacher at a folk school, but soon became the director of one of the higher educational institutions. The desire for self-development and allowed him to offer 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 right-angled 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 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 and 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 \u003d av / 2 * 2 + s 2 / 2
Now we need to equalize the two original expressions
2av / 2 + s / 2 \u003d (a + c) 2 / 2
c 2 \u003d a 2 + in 2
More than one volume can be written about the Pythagorean theorem and how to prove it study guide. But does it make sense when this knowledge cannot be put into practice?
Practical application of the Pythagorean theorem
Unfortunately, in modern school programs the use of this theorem is provided only in geometric problems. Graduates will soon leave the school walls without knowing how they can apply their knowledge and skills in practice.
In fact, everyone can use the Pythagorean theorem in their daily life. And not only in professional activity but also in normal household chores. Let's consider several cases when the Pythagorean theorem and methods of its proof can be extremely necessary.
Connection of the theorem and astronomy
It would seem how stars and triangles can be connected on paper. In fact, astronomy is scientific field, which makes extensive use of the Pythagorean theorem.
For example, consider the motion of a light beam in space. We know that light travels in both directions at the same speed. We call the trajectory AB along which the light ray moves l. And half the time it takes for light to get from point A to point B, let's call t. And the speed of the beam - c. It turns out that: c*t=l
If you look at this same beam from another plane, for example, from a space liner that moves at a speed v, then with such an observation of the bodies, their speed will change. In this case, even stationary elements will move with a 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 ray rushes, will 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 that point A has shifted, you need to multiply the speed of the liner by half the travel time of the beam (t ").
And in order to find how far a ray of light could travel during this time, you need to designate half the path of the new beech s and get the following expression:
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 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 can be lucky enough to try it out in practice. Therefore, we 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 would they be of use if they could not connect subscribers via mobile communications?!
The quality of mobile communications directly depends on the height at which the antenna of the mobile operator 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 propagate a signal within a radius of 200 kilometers.
AB (tower height) = x;
BC (radius of signal transmission) = 200 km;
OS (radius the 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 closet, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements with a tape measure. But many are surprised why certain problems arise during the assembly process 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 is installed against the wall. Therefore, the sidewall of the cabinet in the process of lifting the structure must freely pass both along the height and diagonally of the room.
Suppose 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 dimensions of the cabinet, let's check the operation of the Pythagorean theorem:
AC \u003d √AB 2 + √BC 2
AC \u003d √ 2474 2 +800 2 \u003d 2600 mm - everything converges.
Let's say the height of the cabinet is not 2474 mm, but 2505 mm. Then:
AC \u003d √2505 2 + √800 2 \u003d 2629 mm.
Therefore, this cabinet is not suitable for installation in this room. Since when lifting it to a vertical position, damage to its body can be caused.
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 calculations will be not only useful, but also correct.
Class: 8
Lesson Objectives:
- Educational: to achieve the assimilation of the Pythagorean theorem, to instill the skills of calculating the unknown side of a right-angled triangle using two known ones, to teach how to apply the Pythagorean theorem to solving simple problems
- Developing: contribute to the development of the ability to compare, observation, attention, the development of the ability to analytical and synthetic thinking, broadening one's horizons
- Educational: formation of the need for knowledge, interest in mathematics
Lesson type: new material presentation lesson
Equipment: computer, multimedia projector, presentation for the lesson ( Annex 1)
Lesson plan:
- Organizing time
- oral exercises
- Research, putting forward a hypothesis and testing it on particular cases
- Explanation of new material
a) About Pythagoras
b) Statement and proof of the theorem - Consolidation of the above through problem solving
- Homework, summarizing the lesson.
During the classes
Slide 2: Do the exercises
- Expand brackets: (3 + x) 2
- Calculate 3 2 + x 2 for x = 1, 2, 3, 4
- Does it exist natural number, whose square is 10, 13, 18, 25? - Find the area of a square with sides 11 cm, 50 cm, 7 dm.
What is the formula for the area of a square?
How to find the area of a right triangle?
Slide 3: Question answer
- Corner, degree measure which is equal to 90°. (Straight)
The side opposite the right angle of the triangle. (Hypotenuse)
- Triangle, square, trapezium, circle - these are geometric ... (Shapes)
- The smaller side of a right triangle. (Katet)
- A figure formed by two rays emanating from one point. (Corner)
- A segment of a perpendicular drawn from the vertex of a triangle to the line containing the opposite side. (Height)
- A triangle with two equal sides . (Isosceles)
Slide 4: Task
Construct a right triangle with sides 3 cm, 4 cm and 6 cm.
The task is divided into rows.
| 1 row | 2 row | 3 row | |
| leg a | 3 | 3 | |
| leg b | 4 | 4 | |
| Hypotenuse With | 6 | 6 |
Questions:
- Did anyone get a triangle with given parties?
- What can be the conclusion? (A right triangle cannot be arbitrarily defined. There is a dependency between its sides).
- Measure the resulting sides. ( The approximate average result from each row is entered in the table)
| 1 row | 2 row | 3 row | |
| leg a | 3 | 3 | ~4,5 |
| leg b | 4 | ~5,2 | 4 |
| Hypotenuse With | ~5 | 6 | 6 |
- Try to establish a relationship between the legs and the hypotenuse in each of the cases.
(It is proposed to recall oral exercises and check the same relationship between other numbers).
- Attention is drawn to the fact that the exact result will not work, because. measurements cannot be considered accurate.
The teacher asks for guesses (hypotheses): students formulate.
- Yes, indeed, there is a relationship between the hypotenuse and the legs, and the first to prove it was the scientist, whose name you will name yourself. This theorem is named after him.
Slide 5: Decipher
Slide 6: Pythagoras of Samos
Who will name the topic of today's lesson?
Students in notebooks write down the topic of the lesson: “The Pythagorean Theorem”
The Pythagorean theorem is one of the main theorems of geometry. With its help, many other theorems are proved and problems from various fields are solved: physics, astronomy, construction, etc. It was known long before Pythagoras proved it. The ancient Egyptians used it when building a right triangle with sides of 3, 4 and 5 units using a rope to build right angles when laying buildings, pyramids. Therefore, such a triangle is called Egyptian triangle.
There are over three hundred ways to prove this theorem. We will look at one of them today.
Slide 7: Pythagorean theorem
Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
Given:
Right triangle,
a, b - legs, With- hypotenuse
Prove:
Proof.
1. We continue the legs of a right triangle: leg A- for length b, leg b- for length A.
What shape can a triangle be built to? Why up to a square? What will be the side of the square?
2. We complete the triangle to a square with a side a + b.
How can you find the area of this square?
3. The area of the square is
- Let's break the square into parts: 4 triangles and a square with side c.
How else can you find the area of the original square?
Why are the resulting right triangles congruent?
4. On the other hand,
5. Equate the resulting equalities:
The theorem has been proven.
There is a comic formulation of this theorem: “Pythagorean pants are equal in all directions.” Probably, such a formulation is due to the fact that this theorem was originally established for an isosceles right triangle. Moreover, it sounded a little different: “The area of a square built on the hypotenuse of a right triangle is equal to the sum of the areas of squares built on its legs.”
Slide 8: Another formulation of the Pythagorean theorem
And I will give you another formulation of this theorem in verse:
If we are given a triangle
And, moreover, with a right angle,
That is the square of the hypotenuse
We can always easily find:
We build the legs in a square,
We find the sum of degrees
And in such a simple way
We will come to the result.
- So, today you got acquainted with the most famous theorem of planimetry - the Pythagorean theorem. How is the Pythagorean theorem formulated? How else can it be formulated?
Primary fixation of the material
Slide 9: Solution of problems according to ready-made drawings.
Slide 10: Solving problems in a notebook
Three students are called to the board at the same time to solve problems.
Slide 11: Problem of the 12th century Indian mathematician Bhaskara
Summing up the lesson:
What new did you learn at the lesson today?
- Formulate the Pythagorean theorem.
- What did you learn to do in the lesson?
Homework:
– Learn the Pythagorean theorem with proof
- Tasks from textbook No. 483 c, d; No. 484 in, city of
– For more advanced students: find other proofs of the Pythagorean theorem, learn one of them.
The work of the class as a whole is evaluated, highlighting individual students.