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 I 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)

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 min.

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: Height rectangular triangle Delivers the hypotenuse on the segments of length 1 and 2. Locate 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)

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.

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

In a rectangular triangle Square hypotenuse equal to sum Squares of 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.

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.

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:

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

№ 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.

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

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.

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 is equal to the sum of the squares of the carriage 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

there is a rectangular triangle with customs 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:

It is remarkable that the property specified in the Pythagora theorem is the characteristic property of a rectangular triangle. This follows from the theorem, the Pythagorean reverse theorem.

Theorem: If the square of one side of the triangle is equal to the sum of the squares of the other other sides, then the triangle is rectangular.

Formula Gerona

We derive the formula expressing the plane of the triangle through the lengths of its sides. This formula is associated with the name of Gerona Alexandrian - ancient Greek mathematics and mechanics who lived, probably in 1 V.N. Geron paid a lot of attention to practical geometry applications.

Theorem. The area S is a triangle, the sides of which are equal to A, B, C, is calculated by the formula S \u003d, where P is a half-versioner of the triangle.

Evidence.

Danched :? ABC, AB \u003d C, Sun \u003d A, AC \u003d B.Glons A and B, sharp. CH - height.

Prove

Proof:

Consider the ABC triangle, in which AB \u003d C, BC \u003d A, AC \u003d B. In every triangle, at least two angles are sharp. Let A and B be the sharp corners of the ABC triangle. The base of H height CH triangle lies on the AB side. We introduce the notation: ch \u003d h, ah \u003d y, hb \u003d x. According to Pythagorea theorem A 2 - x 2 \u003d H 2 \u003d B 2 -Y 2, from where

Y 2 - x 2 \u003d b 2 - a 2, or (y - x) (y + x) \u003d b 2 - a 2, and since y + x \u003d c, then y- x \u003d (B2 - A2).

Folding the last two equality, n Olchache:

2y \u003d + C, from where

y \u003d, and, therefore, H 2 \u003d B 2 -Y 2 \u003d (b - y) (B + Y) \u003d