Lesson Objectives:

Educational: formulate and prove the Pythagorean theorem and the converse of the Pythagorean theorem. Show their historical and practical significance.

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

Educational: to cultivate interest and love for the subject, accuracy, the ability to listen to comrades and teachers.

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

Lesson plan:

I. Organizing time- 1 minute.

II. Checking homework - 7 min.

III. opening speech teachers, historical background - 4-5 min.

IV. Formulation and proof of the Pythagorean theorem - 7 min.

V. Formulation and proof of the theorem converse to the Pythagorean theorem - 5 min.

Fixing new material:

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

VII. Homework - 1 min.

VIII. Summing up the lesson - 3 min.

During the classes

I. Organizational moment.

II. Checking homework.

p.7.1, No. 3 (at the board according to the finished drawing).

Condition: The height of a right triangle divides the hypotenuse into segments of length 1 and 2. Find the legs of this triangle.

BC = a; CA=b; BA=c; BD = a 1 ; DA = b 1 ; CD = hC

Additional question: write down the ratios in a right triangle.

item 7.1, No. 5. Cut right triangle into three similar triangles.

Explain.

ASN ~ ABC ~ SVN

(draw students' attention to the correct recording of the corresponding vertices of similar triangles)

III. Introductory speech of the teacher, historical background.

The truth will remain eternal, as soon as a weak person knows it!

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

It is no coincidence that I began my lesson with the words of the German novelist Chamisso. Our lesson today is about the Pythagorean theorem. Let's write the topic of the lesson.

Before you is a portrait of the great Pythagoras. Born in 576 BC. After living 80 years, he died in 496 BC. Known as an ancient Greek philosopher and teacher. He was the son of the merchant Mnesarchus, who often took him on his trips, thanks to which the boy developed curiosity and a desire to learn new things. Pythagoras is a nickname given to him for his eloquence (“Pythagoras” means “persuasive speech”). He himself did not write anything. All his thoughts were recorded by his students. As a result of the first lecture he gave, Pythagoras acquired 2,000 students who, together with their wives and children, formed a huge school and created a state called “Great Greece”, which is based on the laws and rules of Pythagoras, revered as divine commandments. He was the first to call his reasoning about the meaning of life philosophy (philosophy). He was prone to mystification and demonstrative behavior. Once Pythagoras hid underground, and learned about everything that was happening from his mother. Then, withered like a skeleton, he declared in the public assembly that he had been in Hades, and showed amazing awareness of earthly events. For this, the touched inhabitants recognized him as God. Pythagoras never cried and was generally inaccessible to passions and excitement. He believed that he comes from a seed that is better compared to human. The whole life of Pythagoras is a legend that has come down to our time and told us about the most talented man of the ancient world.

IV. Formulation and proof of the Pythagorean theorem.

The formulation of the Pythagorean theorem is known to you from the course of algebra. Let's remember her.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

However, this theorem was known many years before Pythagoras. 1500 years before Pythagoras, the ancient Egyptians knew that a triangle with sides 3, 4 and 5 is right-angled and used this property to build right angles when planning land plots and construction of buildings. In the oldest Chinese mathematical and astronomical work that has come down to us, “Zhiu-bi”, written 600 years before Pythagoras, among other sentences related to a right triangle, the Pythagorean theorem is also contained. Even earlier, this theorem was known to the Hindus. Thus, Pythagoras did not discover this property of a right-angled triangle; he was probably the first to generalize and prove it, to transfer it from the field of practice to the field of science.

Since ancient times, mathematicians have been finding more and more proofs of the Pythagorean theorem. There are over a hundred and fifty known. Let's recall the algebraic proof of the Pythagorean theorem, known to us from the course of algebra. (“Mathematics. Algebra. Functions. Data analysis” G.V. Dorofeev, M., “Bubblehead”, 2000).

Invite students to remember the proof for 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 = c 2 a a b

The ancient Hindus, to whom this reasoning belongs, usually did not write it down, but accompanied the drawing with only one word: “Look.”

Let us consider in a modern presentation one of the proofs belonging to Pythagoras. At the beginning of the lesson, we remembered the theorem on ratios in a right triangle:

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

We add the last two equalities term by term:

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 = c 2

Despite the apparent simplicity of this proof, it is far from being the simplest one. After all, for this it was necessary to draw a height in a right-angled triangle and consider similar triangles. Please write down this proof in your notebook.

V. Statement and proof of the theorem converse to the Pythagorean theorem.

What is the inverse of this theorem? (... if the condition and conclusion are reversed.)

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

If in a triangle with sides a, b and c the equality with 2 \u003d a 2 + b 2 is true, then this triangle is right-angled, and the right angle is opposite to side c.

(Proof of the inverse theorem on a poster)

ABC, BC = a,

AC = b, BA = c.

a 2 + b 2 = c 2

Prove:

ABC - rectangular,

Proof:

Consider a right triangle A 1 B 1 C 1,

where C 1 \u003d 90 °, A 1 C 1 \u003d a, A 1 C 1 \u003d b.

Then, according to the Pythagorean theorem, B 1 A 1 2 \u003d a 2 + b 2 \u003d c 2.

That is, B 1 A 1 \u003d c A 1 B 1 C 1 \u003d ABC on three sides of ABC - rectangular

C = 90°, which was to be proved.

VI. Consolidation of the studied material (orally).

1. According to the poster with ready-made drawings.

Fig.1: find AD if BD = 8, BDA = 30°.

Fig. 2: find CD if BE = 5, BAE = 45°.

Fig. 3: find BD if BC = 17, AD = 16.

2. Is a triangle right-angled if its sides are expressed by numbers:

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

What are the triples of numbers in the last two cases called? (Pythagorean).

VI. Problem solving (in writing).

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

№ 14. Prove that in a right triangle the radius of the circumscribed circle is equal to the median drawn to the hypotenuse and equal to half of the hypotenuse.

VII. Homework.

Item 7.1, pp. 175-177, analyze Theorem 7.4 (generalized Pythagorean theorem), No. 1 (oral), No. 2, No. 4.

VIII. Lesson results.

What new did you learn at the lesson today? …………

Pythagoras was first and foremost a philosopher. Now I want to read you a few of his sayings, which are relevant in our time for you and me.

• Do not raise dust on the path of life.
• Do only what in the future will not upset you and will not force you to repent.
• Never do what you do not know, but learn everything you need to know, and then you will lead a quiet life.
• Don't close your eyes when you want to sleep without understanding all your actions on the previous day.
• Learn to live simply and without luxury.

It is remarkable that the property indicated in the Pythagorean theorem is a characteristic property of a right triangle. This follows from a theorem converse to the Pythagorean theorem.

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

Heron's formula

We derive a formula expressing the plane of a triangle in terms of the lengths of its sides. This formula is associated with the name of Heron of Alexandria, an ancient Greek mathematician and mechanic who probably lived in the 1st century AD. Heron paid much attention to the practical applications of geometry.

Theorem. The area S of a triangle whose sides are a, b, c is calculated by the formula S=, where p is the half-perimeter of the triangle.

Proof.

Given: ?ABC, AB=c, BC=a, AC=b. Angles A and B are acute. CH - height.

Prove:

Proof:

Consider a triangle ABC in which AB=c , BC=a, AC=b. Every triangle has at least two acute angles. Let A and B be acute angles of triangle ABC. Then the base H of height CH of the triangle lies on side AB. Let's introduce the notation: CH = h, AH=y, HB=x. according to the Pythagorean theorem a 2 - x 2 \u003d h 2 \u003d b 2 -y 2, whence

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).

Adding the last two equalities, we get:

2y = +c, whence

y \u003d, and, therefore, h 2 \u003d b 2 -y 2 \u003d (b - y) (b + y) \u003d

The solution of the problem:

252 \u003d 242 + 72, then the triangle is right-angled and its area is equal to half the product of its legs, i.e. S \u003d hc * s: 2, where c is the hypotenuse, hc is the height drawn to the hypotenuse, then hc = = = 6.72 (cm)

Answer: 6.72 cm.

Purpose of the stage:

slide number 4

"4" - 1 wrong answer

"3" - the answers are incorrect.

I suggest doing:

slide number 5

Purpose of the stage:

At the end of the lesson:

The phrases are written on the board:

The lesson is useful, everything is clear.

Still have to work hard.

Yes, it's hard to learn!

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"The project of a lesson in mathematics "Theorem, the inverse of the Pythagorean theorem""

Lesson project "Theorem, converse theorem Pythagoras"

Lesson of "discovery" of new knowledge

Lesson Objectives:

activity: the formation of students' abilities to independently build new modes of action based on the method of reflexive self-organization;

educational: expansion of the conceptual base by including new elements in it.

Stage of motivation learning activities(5 minutes)

Mutual greeting of the teacher and students, checking preparedness for the lesson, organizing attention and internal readiness, quickly involving students in a business rhythm by solving problems according to ready-made drawings:

Find BC if ABCD is a rhombus.

ABCD is a rectangle. AB:AD = 3:4. Find AD.

Find AD.

Find AB.

Find sun.

Answers to tasks according to ready-made drawings:

1.BC = 3; 2.AD=4cm; 3.AB = 3√2cm.

Stage of "discovery" of new knowledge and ways of action (15 min)

Purpose of the stage: formulation of the topic and objectives of the lesson with the help of a leading dialogue (reception "problem situation").

Formulate statements that are inverse to the data and find out if they are true:slide number 1

In the latter case, students can formulate a statement opposite to this one.

Instruction for work in pairs on the study of the proof of the theorem, the converse of the Pythagorean theorem.

I instruct students about the method of activity, about the location of the material.

Assignment to couples: slide number 2

Independent work in pairs to study the proof of the theorem, the converse of the Pythagorean theorem. Public defense of evidence.

One of the pairs begins their presentation with the formulation of a theorem. There is an active discussion of the evidence, during which one or another option is substantiated with the help of questions from the teacher and students.

Comparing the proof of the theorem with the teacher's proof

The teacher works at the blackboard, addressing the students who are working in a notebook.

Given: ABC - triangle, AB 2 \u003d AC 2 + BC 2

Find out if ABC is rectangular. Proof:

Consider A 1 B 1 C 1 such that ˂C = 90 0 , A 1 C 1 = AC, B 1 C 1 = BC. Then, according to the Pythagorean theorem, A 1 B 1 2 \u003d A 1 C 1 2 + B 1 C 1 2.

Since A 1 C 1 \u003d AC, B 1 C 1 \u003d BC, then: A 1 C 1 2 + B 1 C 1 2 \u003d AC 2 + BC 2 \u003d AB 2, therefore, AB 2 \u003d A 1 B 1 2 and AB \u003d A 1 B 1.

A 1 B 1 C 1 = ABC on three sides, whence ˂C = ˂C 1 = 90 0, that is, ABC is rectangular. So, if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

This statement is called a theorem converse to the Pythagorean theorem.

Public presentation of one of the students about the Pythagorean triangles (pre-prepared information).

slide number 3

After the information, I ask the students a few questions.

Are the following triangles Pythagorean triangles?

with hypotenuse 25 and leg 15;

with legs 5 and 4?

The stage of primary consolidation with pronunciation in external speech (10 min)

Purpose of the stage: demonstrate the application of the theorem, the reverse of the Pythagorean theorem in the process of solving problems.

I propose to solve problem No. 499 a) from the textbook. One of the students is invited to the board, solves the problem with the help of the teacher and students, pronouncing the solution in external speech. During the presentation of the invited student, I ask a few questions:

How to check if a triangle is a right triangle?

To which side will the smallest height of the triangle be drawn?

What method of calculating the height of a triangle is often used in geometry?

Using the formula for calculating the area of ​​a triangle, find the desired height.

The solution of the problem:

25 2 \u003d 24 2 + 7 2, then the triangle is right-angled and its area is equal to half the product of its legs, i.e. S = h s * s: 2, where s is the hypotenuse, h s is the height drawn to the hypotenuse, then h s = = = 6.72 (cm)

Answer: 6.72 cm.

Stage of independent work with self-test according to the standard (10 min)

Purpose of the stage: improve independent activity in the classroom, carrying out self-examination, teach to evaluate activities, analyze, draw conclusions.

Offered independent work with a proposal to adequately evaluate their work and put an appropriate assessment.

slide number 4

Evaluation criteria: "5" - all answers are correct

"4" - 1 wrong answer

"3" - the answers are incorrect.

The stage of informing students about homework, briefing on its implementation (3 min).

I inform the students about the homework, explain the methodology for its implementation, check the understanding of the content of the work.

I suggest doing:

slide number 5

The stage of reflection of educational activity in the lesson (2 min)

Purpose of the stage: to teach students to assess their readiness to discover ignorance, to find the causes of difficulties, to determine the result of their activities.

At this stage, I suggest that each student choose only one of the guys who wants to say thank you for their cooperation and explain what exactly this cooperation manifested itself in.

The teacher's word of thanks is the final one. At the same time, I choose those who got the least amount of compliments.

At the end of the lesson:

The phrases are written on the board:

The lesson is useful, everything is clear.

Only a few things are a little unclear.

Still have to work hard.

Yes, it's hard to learn!

Children come up and put a sign (tick) next to those words that are most suitable for them at the end of the lesson.

Consideration of topics school curriculum with the help of video lessons is a convenient way to study and assimilate the material. Video helps to focus students' attention on the main theoretical points and not to miss important details. If necessary, students can always listen to the video lesson again or go back a few topics.

This 8th grade video lesson will help students learn new theme by geometry.

In the previous topic, we studied the Pythagorean theorem and analyzed its proof.

There is also a theorem which is known as the inverse Pythagorean theorem. Let's consider it in more detail.

Theorem. A triangle is right-angled if it satisfies the equality: the value of one side of the triangle squared is the same as the sum of the other two sides squared.

Proof. Suppose we are given a triangle ABC, in which the equality AB 2 = CA 2 + CB 2 is true. We need to prove that angle C is 90 degrees. Consider a triangle A 1 B 1 C 1 in which angle C 1 is 90 degrees, side C 1 A 1 is equal to CA and side B 1 C 1 is equal to BC.

Applying the Pythagorean theorem, we write the ratio of the sides in the triangle A 1 C 1 B 1: A 1 B 1 2 = C 1 A 1 2 + C 1 B 1 2 . By replacing the expression with equal sides, we get A 1 B 1 2 = CA 2 + CB 2.

We know from the conditions of the theorem that AB 2 = CA 2 + CB 2 . Then we can write A 1 B 1 2 = AB 2 , which implies that A 1 B 1 = AB.

We have found that in triangles ABC and A 1 B 1 C 1 three sides are equal: A 1 C 1 = AC, B 1 C 1 = BC, A 1 B 1 = AB. So these triangles are congruent. From the equality of triangles it follows that the angle C equal to the angle With 1 and respectively equal to 90 degrees. We have determined that triangle ABC is a right triangle and its angle C is 90 degrees. We have proved this theorem.

The author then gives an example. Suppose we are given an arbitrary triangle. The dimensions of its sides are known: 5, 4 and 3 units. Let's check the statement from the theorem converse to the Pythagorean theorem: 5 2 = 3 2 + 4 2 . If the statement is correct, then the given triangle is a right triangle.

In the following examples, the triangles will also be right-angled if their sides are equal:

5, 12, 13 units; the equality 13 2 = 5 2 + 12 2 is true;

8, 15, 17 units; the equation 17 2 = 8 2 + 15 2 is true;

7, 24, 25 units; the equation 25 2 = 7 2 + 24 2 is true.

The concept of the Pythagorean triangle is known. It is a right triangle whose side values ​​are integers. If the legs of the Pythagorean triangle are denoted by a and c, and the hypotenuse b, then the values ​​of 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 are any integers, and the value of m is greater than the value of n.

An interesting fact: a triangle with sides 5, 4 and 3 is also called the Egyptian triangle, such a triangle was known in ancient Egypt.

In this video tutorial, we got acquainted with the theorem, the converse of the Pythagorean theorem. Consider the proof in detail. Students also learned which triangles are called Pythagorean triangles.

Students can easily get acquainted with the topic "Theorem, the inverse of the Pythagorean theorem" on their own with the help of this video lesson.

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.

On the this moment in scientific literature 367 proofs of this theorem were recorded. Probably the theorem

Pythagoras is the only theorem with such an impressive number of proofs. Such diversity

can only be explained by the fundamental significance of the theorem for geometry.

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

proof of 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 sharp corners 90°, a

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 given equation and using the initial conditions, we get:

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 leg b). Then for the integration constant we get: