Reverse Pythagore Theorem Proof. Theorem, reverse theorem Pythagora

Subject: Theorem, reverse theorem. Pythagora.

Objectives lesson: 1) consider the theorem inverse Pythagora theorem; its use in the process of solving problems; Fix the Pythagora theorem and improve the skills to solve problems for its use;

2) develop logical thinking, creative search, cognitive interest;

3) bring up students with a responsible attitude towards the teachings, culture of mathematical speech.

Type of lesson. Lesson assimilation of new knowledge.

During the classes

І. Organizing time

ІІ. Actualization Knowledge

Lesson mewould bei wantedstart with quatrain.

Yes, the path of knowledge is not glad

But we know with school years,

Riddles more than imagners

And there is no search for the limit!

So, in the past, the lesson you learned the Pythagore's theorem. Questions:

Pythagora theorem is valid for which figure?

What triangle is called rectangular?

Formulate Pythagore's theorem.

How will the Pythagora theorem for each triangle be written?

What triangles are called equal?

Word the signs of the equality of triangles?

And now we will spend a small independent work:

Solving tasks according to drawings.

№1

(1 b.) Find: av.

№2

(1 b.) Find: Sun.

№3

( 2 b.)Find: AC.

№4

(1 b.)Find: AC.

№5 Dano: ABCD. rhombus

(2 b.) Av \u003d 13 cm

Ac \u003d 10 cm

Find inD.

Self-test number 1. five

№2. 5

№3. 16

№4. 13

№5. 24

ІІІ. Study New material.

The ancient Egyptians built straight corners on the ground in this way: shared the rope for 12 equal partsThey associated her ends, after which the rope was stretched so on Earth, so that a triangle was formed with parties 3, 4 and 5 divisions. The angle of the triangle, which lay against the side with 5 divisions was straight.

Can you explain the correctness of this judgment?

As a result of the search for a response to the question, students should understand that from a mathematical point of view the question is set: whether the triangle is rectangular.

We put the problem: how, without making measurements, determine whether the triangle with the specified sides are rectangular. The solution to this problem is the purpose of the lesson.

Write down theme lesson.

Theorem. If the sum of the squares of the two sides of the triangle is equal to the third party square, then such a triangle is rectangular.

Independently prove the theorem (compile a plan for proof on the textbook).

From this theorem it follows that the triangle with the parties 3, 4, 5 is rectangular (Egyptian).

In general, the numbers for which equality is performed , Call Pythagora Troika. And the triangles, the lengths of the sides of which are expressed by Pythagora Troops (6, 8, 10), - Pythagora triangles.

Fastening.

Because , then the triangle with the parties 12, 13, 5 is not rectangular.

Because , then the triangle with the parties 1, 5, 6 is rectangular.

№ 430 (A, B, B)

( - is not)

Pythagorean theorem - one of the fundamental theorems of Euclidean geometry establishing the ratio

between the sides of the rectangular triangle.

It is believed to be proved by the Greek mathematician Pythagore, in honor of which and named.

Geometric formulation of the Pythagorean theorem.

Initially, the theorem was formulated as follows:

In a rectangular triangle, the square of the square built on the hypotenuse is equal to the sum of the squares of the squares,

built on catetes.

Algebraic formulation of the Pythagorean theorem.

In a rectangular triangle, the square of the length of the hypotenuse equal to sum Squares of cathet lengths.

That is, denoting the length of the triangle hypotenuse through c., and the length of the cathets through a. and b.:

Both wording pythagora theoremsequivalent, but the second wording is more elementary, it is not

requires the concept of area. That is, the second statement can be checked, nothing know about the area and

measuring only the length of the sides of the rectangular triangle.

Pythagorean reverse theorem.

If the square of one side of the triangle is equal to the sum of the squares of the two other sides, then

the triangle is rectangular.

Or, in other words:

For all Troika positive numbers a., b. and c., such that

exists right triangle With category a. and b.and hypotenuse c..

Pythagora theorem for an equifiable triangle.

Pythagora theorem for an equilateral triangle.

Proof of the Pythagorean theorem.

At the moment in scientific literature Fixed 367 evidence of this theorem. Probably theorem

Pythagora is the only theorem with such an impressive number of evidence. Such a variety

can be explained only by the fundamental value of the geometry theorem.

Of course, it is conceptually all of them can be divided into a small number of classes. The most famous of them:

proof of method of space, axiomatic and exotic evidence (eg,

via differential equations).

1. Proof of Pythagore's theorem through such triangles.

The following evidence of algebraic wording is the simplest of the proofs under construction.

directly from the axiom. In particular, it does not use the concept of the figure of the figure.

Let be ABC there is a rectangular triangle with a straight angle C.. Let's spend the height of C. And denote

its foundation through H..

Triangle Ach. Like a triangle ABC for two corners. Similarly, triangle CBH. Like ABC.

Entering notation:

we get:

,

what corresponds to -

Matching a. 2 I. b. 2, we get:

or, which was required to prove.

2. Proof of the Pythagore Theorem by the area of \u200b\u200bthe area.

Below, the evidence, despite their seeming simplicity, not so simple. All of them

use the properties of the area, the evidence of which is more complicated by the proof of the theorem of Pythagora itself.

• Proof through the equodockility.

Place four equal rectangular

triangle as shown in the picture

on right.

Quadril with sides c. - Square,

since the sum of two sharp corners of 90 °, and

deployed angle - 180 °.

The area of \u200b\u200bthe whole figure is equal to one hand,

square area with side ( a + B.), and on the other hand, the sum of the area of \u200b\u200bfour triangles and

Q.E.D.

3. Proof of the Pythagore Theorem by the method of infinitely small.

​

Considering the drawing shown in the figure and

observing a change of sidea., we can

record the following ratio for infinite

small increments of sidefrom and a. (Using the semblance

triangles):

Using the variable separation method, we find:

More general expression To change the hypotenuse in the event of increments of both cathets:

Integrating this equation and using the initial conditions, we get:

Thus, we come to the desired answer:

As it is not difficult to see, the quadratic dependence in the final formula appears due to the linear

proportionality between the sides of the triangle and increments, while the amount is associated with independent

deposits from the increment of different cathets.

More simple proof can be obtained, if we assume that one of the cathets does not experience increment

(in this case catat b.). Then, for the integration constant, we get:

Objectives lesson:

general:

• check theoretical knowledge of students (properties of a rectangular triangle, Pythagorean theorem), the ability to use them when solving tasks;
• creation problem situation, Test students to the "opening" of the Pythagorean reverse theorem.

developing:

• development of skills to apply theoretical knowledge in practice;
• development of ability to formulate conclusions during observations;
• memory development, attention, observation:
• development of the motivation of teachings through emotional satisfaction from discoveries, through the introduction of elements of the history of the development of mathematical concepts.

educational:

• raise sustainable interest in the subject through the study of the vital activity of Pythagore;
• education of mutual assistance and objective assessment of knowledge of classmates through the mutual test.

Form of the lesson: cool-class.

Lesson plan:

• Organizing time.
• Check your homework. Actualization of knowledge.
• Solving practical tasks using the Pythagorean theorem.
• New topic.
• Primary consolidation of knowledge.
• Homework.
• The results of the lesson.
• Independent work (according to individual cards with the guessing of Pythagora aphorisms).

During the classes.

Organizing time.

Check your homework. Actualization of knowledge.

Teacher: What task did you perform at home?

Pupils: According to two data to the sides of the rectangular triangle, find the third direction, the answers to appease in the form of a table. Repeat the properties of the rhombus and rectangle. Repeat what is called condition, and that the conclusion of the theorem. Prepare reports about the life and activities of Pythagora. Bring the rope with the 12 knots tied on it.

Teacher: Answers to your home job Check on the table

(Black color highlighted data, red - answers).

Teacher: On the board recorded approval. If you agree with them on the leaves opposite the corresponding question number, put "+", if you do not agree, then put "-".

On the board are written in advance approval.

1. Hypotenuse more category.
2. The sum of sharp corners of the rectangular triangle is 180 0.
3. Square of a rectangular triangle with customs butand in Calculated by formula S \u003d AB / 2.
4. Pythagore's theorem is true for all equal triangles.
5. In a rectangular triangle, the catat lying opposite the angle of 30 0 is equal to half the hypotenuse.
6. The sum of the squares of the cathets is equal to the square of the hypotenuse.
7. The square of the category is equal to the difference in the squares of the hypotenuse and the second category.
8. The side of the triangle is equal to the sum of the two other sides.

Checking work with the help of mutual test. Approvals that caused disputes are discussed.

The key to theoretical issues.

Students put each other assessments on the following system:

8 correct answers "5";
6-7 correct answers "4";
4-5 correct answers "3";
Less than 4 correct answers "2".

Teacher: What were we talking about in the past lesson?

Pupil: About Pythagore and its theorem.

Teacher: Formulate Pythagore's theorem. (Several students read the wording, at this time 2-3 student prove it at the board, 6 students - behind the first parties on leaves).

On the magnetic board on cards written mathematical formulas. Choose those of them that reflect the meaning of the Pythagora theorem, where but and in - Kartets, from - hypotenuse.

1) C 2 \u003d a 2 + in 2	2) C \u003d a + in	3) A 2 \u003d C 2 - in 2
4) C 2 \u003d a 2 - in 2	5) at 2 \u003d C 2 - A 2	6) A 2 \u003d C 2 + B 2

While students proving the theorem at the board and on the ground are not ready, the word is provided to those who have prepared reports on the life and activities of Pythagora.

Schoolchildren working in the field give leaves and listen to the evidence of those who worked at the board.

Solving practical tasks using the Pythagorean theorem.

Teacher: I offer you practical tasks using the theorem studied. We first in the forest, after the storm, then on the country site.

Task 1.. After the storm broke her fir. The height of the remaining part is 4.2 m. The distance from the base to the fallen crown of 5.6 m. Find the height of the storm.

Task 2.. The height of the house is 4.4 m The width of the lawn around the house is 1.4 m. What length should we make a staircase so that it does not stand on the lawn and delivered to the roof of the house?

New topic.

Teacher: (music sounds) Close your eyes, for a few minutes we will plunge into history. We are with you in ancient Egypt. Here on the shipyards of the Egyptians build their famous ships. But the landemers, they measure the plots of land, whose boundaries were washed after the spill of the Nile. Builders are building grand pyramids, which still amazing us with their magnificence. In all these activities, Egyptians needed direct corners. They knew how to build them with the help of a rope with 12 yo tied at the same distance from each other with nodules. Try both you, arguing as ancient Egyptians, build rectangular triangles using your ropes. (Solving this problem, the guys work in groups of 4 people. After some time, on the tablet at the board, someone shows the construction of a triangle).

The sides of the resulting triangle 3, 4 and 5. If it is tied between these nodes another one by one node, then its parties will be 6, 8 and 10. If two - 9, 12 and 15. All these triangles are rectangular t.

5 2 \u003d 3 2 + 4 2, 10 2 \u003d 6 2 + 8 2, 15 2 \u003d 9 2 + 12 2, etc.

What property should triangle have to be rectangular? (Students are trying to formulate the inverse theorem of Pythagora, finally, at someone it turns out).

How does this theorem differ from the Pythagorean theorem?

Pupil: Condition and conclusion changed places.

Teacher: At home you repeated how such theorems are called. So what are we met now?

Pupil: From the reverse Pythagores Theorem.

Teacher: We write the topic of the lesson in the notebook. Open tutorials on page 127 Read this approval again, write it down in a notebook and disassemble the proof.

(After several minutes of independent work with a textbook, at will, one person at the board leads proof of the theorem).

1. What is the name of the triangle with the parties 3, 4 and 5? Why?
2. What triangles are called Pythagorov?
3. What triangles did you work with your homework? And in tasks with a pine and stairs?

Primary consolidation of knowledge

.

This theorem helps to solve the tasks in which it is necessary to find out whether the triangles will be rectangular.

Tasks:

1) Find out whether the triangle is rectangular if its parties are equal:

a) 12.37 and 35; b) 21, 29 and 24.

2) Calculate the height of the triangle with the parties 6, 8 and 10 cm.

Homework

.

Pp.127: Pythagorean reverse theorem. № 498 (A, B, B) No. 497.

The results of the lesson.

What new learned in the lesson?

How was the inverse theorem of Pythagora used in Egypt?

When solving what tasks does it apply?

What triangles got acquainted?

What was remembered most and liked?

Independent work (conducted by individual cards).

Teacher:At home you repeated the properties of the rhombus and rectangle. List them (there is a conversation with class). At the last lesson, we talked about the fact that Pythagoras was a versatile person. He was engaged in medicine and music, and astronomy, as well as was an athlete and participated in the Olympic Games. And Pythagoras was a philosopher. Many of his aphorisms are relevant today for us. Now you will perform an independent job. Each task is given several options for answers, next to which fragments of Pytagora aphorisms are recorded. Your task is to decide all the tasks, make a statement from the resulting fragments and write it down.

Objectives lesson:

Educational: to formulate and prove the theorem of Pythagora and theorem, the reverse Pythagoreo theorem. Show their historical and practical importance.

Developing: develop attention, memory, logical thinking of students, the ability to reason, compare, draw conclusions.

Rising: to educate interest and love for the subject, accuracy, the ability to listen to comrades and teachers.

Equipment: Portrait of Pythagora, posters with tasks for consolidation, textbook "Geometry" 7-9 classes (I.F. Sharygin).

Lesson plan:

I. Organizational moment - 1 min.

II. Checking homework - 7 min.

III. Teacher's introductory word, historical reference - 4-5 minutes.

IV. The wording and proof of the Pythagore's theorem is 7 minutes.

V. The wording and proof of the theorem, the inverse theorem of Pythagora - 5 min.

Fastening a new material:

a) oral - 5-6 min.
b) writing - 7-10 minutes.

VII. Homework - 1 min.

VIII. Summing up the lesson - 3 min.

During the classes

I. Organizational moment.

II. Check your homework.

p.7.1, No. 3 (at the boards on the finished drawing).

Condition: The height of the rectangular triangle divides the hypotenuse on the segments of length 1 and 2. Find the cathets of this triangle.

Bc \u003d a; Ca \u003d B; Ba \u003d C; Bd \u003d a 1; Da \u003d B 1; CD \u003d H C

Additional question: Write relations in a rectangular triangle.

p.7.1, No. 5. Cut the rectangular triangle to three similar triangles.

Explain.

ASN ~ ABC ~ SN

(draw the attention of students to the correctness of the recording of respective vertices of such triangles)

III. The introductory word of the teacher, historical reference.

Permanent truth will be, as soon as a weak person knows her!

And now the Pythagora theorem is true, as in his distant age.

It was not by chance that I began my lesson from the words of the German writer-novelist Shamisso. Our lesson today is devoted to the Pythagora theorem. We write the topic of the lesson.

Before you, the portrait of the Great Pythagorean. Born in 576 BC. Having lived 80 years, died in 496 to our era. Known as an ancient Greek philosopher and teacher. He was the son of a Menarch merchant who took him often on his trips, thanks to which the boy had inquisite and the desire to know the new one. Pythagoras is a nickname given to him for the eloquence ("Pythagoras" means "I am convincing speech"). He himself did not write anything. All his thoughts recorded his disciples. As a result of the first lecture, Pythagora acquired 2000 students who, together with their wives and children, have formed a huge school and created a state called "Great Greece", which is based on the laws and rules of Pythagora, revered as Divine Commandments. He was the first one who called his reasoning about the meaning of the life of philosophy (Lyubomatriy). It was inclined to mystification and demonstration in behavior. Once, Pythagoras hid underground, and everything was happening from the mother. Then, withered as a skeleton, he stated in the People's Assembly, which was in Aida, and showed an amazing awareness of earthly events. For this touched residents recognized him by God. Pythagoras never cried and was generally unavailable by passions and excitement. It believed that it comes from the seed, the best comparatively with human. The whole life of Pythagora is a legend, which came to our time and told us about the talented man of the ancient world.

IV. The wording and proof of the Pythagoreo Theorem.

The formulation of the Pythagore Theorem is known to you from the course of algebra. Let's remember it.

In a rectangular triangle, the square of the hypotenuse is equal to the sum of the squares of the cathets.

However, this theorem knew many years before Pythagora. For 1500 years before Pythagora, the ancient Egyptians knew that the triangle with the parties 3, 4 and 5 is rectangular and used this property to build direct angles when planning land plots and building buildings. In the most ancient times to us, the Chinese mathematic-astronomical essay of "Zhiu-bi", written in 600 years before Pythagora, among other proposals relating to the rectangular triangle, contains the Pytagora theorem. Earlier, this theorem was known to the Hindu. Thus, Pythagoras did not open this property of a rectangular triangle, he probably first managed to summarize it and prove, translate it from the practice of practicing science.

With deep antiquity of mathematics, more and more evidence of the Pythagoreo Theorem are found. They are known more than one and a half hundred. Let's remember the algebraic proof of the Pythagora theorem, known to us from the course of algebra. ("Mathematics. Algebra. Functions. Data Analysis" G.V. Dorofeev, M., "Drop", 2000 g).

Suggest students to remember proof to the drawing and write it on the board.

(a + b) 2 \u003d 4 · 1/2 A * B + C 2 B a

a 2 + 2A * B + B 2 \u003d 2A * B + C 2

a 2 + B 2 \u003d C 2 A A B

Ancient Indians who own this reasoning were usually not recorded, and accompanied the drawing with only one word: "Look".

Consider in modern presentation one of the evidence belonging to Pythagora. At the beginning of the lesson we remembered the theorem about the ratios in a rectangular triangle:

h 2 \u003d a 1 * b 1 a 2 \u003d a 1 * with b 2 \u003d b 1 *

Moving the recent recent two equality:

b 2 + A 2 \u003d B 1 * C + A 1 * C \u003d (B 1 + A 1) * C 1 \u003d C * C \u003d C 2; A 2 + B 2 \u003d C 2

Despite the seeming simplicity of this evidence, it is far from the simplest. After all, for this it was necessary to spend height in a rectangular triangle and consider such triangles. Write down, please, this is proof in the notebook.

V. The wording and proof of the theorem, the Pythagorean reverse theorem.

And what theorem is called the reverse to this? (... if the condition and conclusion change places.)

Let's now try to formulate theorem, the reverse Pythagoreo theorem.

If the triangle with the sides A, B and C is performed with the equality C 2 \u003d A 2 + B 2, then this triangle is rectangular, and the straight angle is opposed to the side with.

(Proof of the reverse theorem on the poster)

ABC, Sun \u003d A,

Ac \u003d b, va \u003d s.

a 2 + B 2 \u003d C 2

Prove

ABC - rectangular,

Evidence:

Consider a rectangular triangle A 1 in 1 C 1,

where from 1 \u003d 90 °, and 1 s 1 \u003d a, and 1 s 1 \u003d b.

Then, according to the Pytagora theorem in 1 A 1 2 \u003d a 2 + b 2 \u003d C 2.

That is, in 1 A 1 \u003d C A 1 in 1 C 1 \u003d ABC for three parties ABC - rectangular

C \u003d 90 °, which was required to prove.

Vi. Fixing the material studied (orally).

1. On a poster with ready-made drawings.

Fig.1: Find AD if CD \u003d 8, VA \u003d 30 °.

Fig.2: Locate the CD if we \u003d 5, waway \u003d 45 °.

Fig.3: Find the VD if Sun \u003d 17, AD \u003d 16.

2. Is the triangle rectangular if its parties are expressed by numbers:

5 2 + 6 2? 7 2 (no)	9 2 + 12 2 \u003d 15 2 (yes)	15 2 + 20 2 \u003d 25 2 (yes)

What are the top three numbers in the last two cases? (Pythagoras).

Vi. Solving tasks (writing).

№ 9. The side of the equilateral triangle is equal to a. Find the height of this triangle, the radius of the circle described, the radius of the inscribed circle.

№ 14. Prove that in a rectangular triangle, the radius of the circumference described is equal to the median conducted to the hypotenuse, and is equal to half the hypotenuse.

VII. Homework.

Paragraph 7.1, pp. 175-177, disassemble theorem 7.4 (generalized Pythagora theorem), No. 1 (orally), No. 2, No. 4.

VIII. The results of the lesson.

What new did you know today at the lesson? ............

Pythagoras was primarily a philosopher. Now I want to read a few of his checks, relevant and in our time for us with you.

• Do not raise dust on life path.
• Do just that later does not upset you and will not fit repent.
• Do not do what you do not know, but learn what you should know, and then you will lead a quiet life.
• Do not close your eyes when I want to sleep, do not raise all your actions last day.
• Take up to live just and without luxury.

Consideration of theme school program With the help of video tutorials is a convenient way to study and assimilate material. The video helps to concentrate the attention of students on the main theoretical provisions and not to miss important details. If necessary, schoolchildren can always listen to the video tutorial repeated or return to a few topics.

This video tutorial for the 8th grade will help students explore new topic By geometry.

In the previous topic, we studied Pythagore's theorem and disassembled its proof.

There is also a theorem that is known as the reverse theorem of Pythagora. Consider it in more detail.

Theorem. The triangle is rectangular if equality is performed in it: the value of one side of the triangle, erected into the square, is the same as the amount of the two other parties elevated to the square.

Evidence. Suppose the ABC triangle is given to us, in which the equality AB 2 \u003d Ca 2 + Cb 2 is performed. It is necessary to prove that the angle C is 90 degrees. Consider a triangle A 1 B 1 C 1, in which angle C 1 is 90 degrees, the side C 1 A 1 is Ca and the side B 1 C 1 is equal to the BS.

Using the Pythagora theorem, write down the ratio of the parties in the triangle A 1 C 1 B 1: A 1 B 1 2 \u003d C 1 A 1 2 + C 1 B 1 2. By replacing in the expression on the equal side, we obtain A 1 B 1 2 \u003d Ca 2 + CB 2.

From the conditions of the theorem, we know that AB 2 \u003d Ca 2 + CB 2. Then we can write a 1 b 1 2 \u003d AB 2, from which it follows that A 1 B 1 \u003d AB.

We found that in the triangles of ABC and A 1 B 1 C 1 are the three sides: A 1 C 1 \u003d AC, B 1 C 1 \u003d Bc, A 1 B 1 \u003d AB. So these triangles are equal. From the equality of triangles it follows that the angle with equal to the corner C 1 and, accordingly, 90 degrees. We determined that the ABC triangle rectangular and its angle C is 90 degrees. We have proven this theorem.

The author further gives an example. Suppose this is an arbitrary triangle. Known sizes of its parties: 5, 4 and 3 units. We verify the assertion from the theorem, the Pythagora reverse theorem: 5 2 \u003d 3 2 + 4 2. The statement is true, then this triangle is rectangular.

In the following examples, triangles will also be rectangular if their parties are equal:

5, 12, 13 units; Equality 13 2 \u003d 5 2 + 12 2 is faithful;

8, 15, 17 units; Equality 17 2 \u003d 8 2 + 15 2 is true;

7, 24, 25 units; Equality 25 2 \u003d 7 2 + 24 2 is true.

The concept of a Pythagora triangle is known. This is a rectangular triangle, in which the values \u200b\u200bof the sides are equal to the integers. If the Pythagorean Triangle Kartites indicate via a and c, and the hypothenus B, then the values \u200b\u200bof the sides of this triangle can be written using the following formulas:

b \u003d k x (m 2 - n 2)

c \u003d k X (M 2 + N 2)

where m, n, k- any integers, and the value M is greater than N values.

An interesting fact: the triangle with the parties 5, 4 and 3 is also called an Egyptian triangle, such a triangle has been known in ancient Egypt.

In this video, we got acquainted with the theorem, the Pythagorean reverse theorem. Details reviewed proof. Students also learned what triangles are called Pythagorov.

Students can easily familiarize themselves with the theorem theorem, reverse theorem of Pythagorean, independently with this video tutorial.