Prove the converse of Pythagoras' theorem. Mathematics lesson project "theorem inverse to the Pythagorean theorem"

It is remarkable that the property specified in the Pythagorean theorem is a characteristic property of a right triangle. This follows from the 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 right-angled.

Heron's formula

Let us 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 equal to a, b, c is calculated by the formula S=, where p is the semi-perimeter of the triangle.

Proof.

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

Prove:

Proof:

Consider 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 altitude CH of the triangle lies on side AB. Let us introduce the following notation: CH = h, AH=y, HB=x. by the Pythagorean theorem a 2 - x 2 = h 2 =b 2 -y 2, whence

Y 2 - x 2 = b 2 - a 2, or (y - x) (y + x) = b 2 - a 2, and since y + x = c, then y- x = (b2 - a2).

Adding the last two equalities, we get:

2y = +c, whence

y=, and, therefore, h 2 = b 2 -y 2 =(b - y)(b+y)=

Lesson objectives:

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

Developmental: develop attention, memory, logical thinking of students, the ability to reason, compare, and 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” for grades 7-9 (I.F. Sharygin).

Lesson plan:

I. Organizational moment – ​​1 min.

II. Checking homework – 7 min.

III. Introductory speech by the teacher, 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.

Consolidating new material:

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

VII. Homework- 1 min.

VIII. Summing up the lesson – 3 min.

During the classes

I. Organizational moment.

II. Checking homework.

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

Condition: The altitude 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 the ratios in a right triangle.

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

Explain.

ASN ~ ABC ~ SVN

(draw students’ attention to the correctness of writing the corresponding vertices of similar triangles)

The truth will remain eternal as soon as a weak person recognizes 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 down the topic of the lesson.

Before you is a portrait of the great Pythagoras. Born in 576 BC. Having lived 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 by 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 2000 students, who, together with their wives and children, formed a huge school and created a state called “Greater Greece,” which was 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. One day Pythagoras hid underground, and learned about everything that was happening from his mother. Then, withered like a skeleton, he declared in a public meeting that he had been to Hades and showed an amazing knowledge of earthly events. For this, the touched residents recognized him as God. Pythagoras never cried and was generally inaccessible to passions and excitement. He believed that he came from a seed that was better than a human one. 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.

You know the formulation of the Pythagorean theorem from your algebra course. 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 rectangular and used this property to construct right angles when planning land plots and constructing buildings. In the oldest Chinese mathematical and astronomical work that has come down to us, “Zhiu-bi,” written 600 years before Pythagoras, among other proposals relating to the right triangle, the Pythagorean theorem is contained. Even earlier this theorem was known to the Hindus. Thus, Pythagoras did not discover this property of a right 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. More than one and a half hundred of them are known. Let's remember the algebraic proof of the Pythagorean theorem, known to us from the algebra course. (“Mathematics. Algebra. Functions. Data analysis” G.V. Dorofeev, M., “Drofa”, 2000).

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

(a + b) 2 = 4 1/2 a * b + c 2 b a

a 2 + 2a * b + b 2 = 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 about relations in a right triangle:

h 2 = a 1* b 1 a 2 = a 1* c b 2 = b 1* c

Let's add the last two equalities term by term:

b 2 + a 2 = b 1* c + a 1* c = (b 1 + a 1) * c 1 = c * c = c 2 ; a 2 + b 2 = c 2

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

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

Let's now try to formulate the theorem converse to the Pythagorean theorem.

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

(Proof of the converse theorem on the 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 = 90°, A 1 C 1 = a, A 1 C 1 = b.

Then, by the Pythagorean theorem, B 1 A 1 2 = a 2 + b 2 = c 2.

That is, B 1 A 1 = c A 1 B 1 C 1 = ABC on three sides ABC is rectangular

C = 90°, which is what needed to be proven.

1. Based on a poster with ready-made drawings.

Fig. 1: find AD if ВD = 8, ВDA = 30°.

Fig.2: find CD if BE = 5, BAE = 45°.

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

2. Is a triangle rectangular if its sides are expressed by numbers:

What are the names of triplets of numbers in the last two cases? (Pythagorean).

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

No. 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 the hypotenuse.

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

What new did you learn in class today? …………

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

• Don't raise dust on life's path.
• Do only what will not upset you later and will not force you to repent.
• Never do what you don’t 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 having sorted out all your actions of the past day.
• Learn to live simply and without luxury.

The Pythagorean theorem states:

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

a 2 + b 2 = c 2,

• a And b– legs forming a right angle.
• With– hypotenuse of the triangle.

Formulas of the Pythagorean theorem

• a = \sqrt(c^(2) - b^(2))
• b = \sqrt (c^(2) - a^(2))
• c = \sqrt (a^(2) + b^(2))

Proof of the Pythagorean Theorem

The area of ​​a right triangle is calculated by the formula:

S = \frac(1)(2)ab

To calculate the area of ​​an arbitrary triangle, the area formula is:

• p– semi-perimeter. p=\frac(1)(2)(a+b+c) ,
• r– radius of the inscribed circle. For a rectangle r=\frac(1)(2)(a+b-c).

Then we equate the right sides of both formulas for the area of ​​the triangle:

\frac(1)(2) ab = \frac(1)(2)(a+b+c) \frac(1)(2)(a+b-c)

2 ab = (a+b+c) (a+b-c)

2 ab = \left((a+b)^(2) -c^(2) \right)

2 ab = a^(2)+2ab+b^(2)-c^(2)

0=a^(2)+b^(2)-c^(2)

c^(2) = a^(2)+b^(2)

Converse Pythagorean theorem:

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. That is, for any triple of positive numbers a, b And c, such that

a 2 + b 2 = c 2,

there is a right triangle with legs a And b and hypotenuse c.

Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It was proven by the learned mathematician and philosopher Pythagoras.

The meaning of the theorem the fact that with its help you can prove other theorems and solve problems.

Additional material:

Reviewing school curriculum topics using video lessons is a convenient way to study and master the material. The video helps to focus students' attention on the main theoretical concepts and not miss important details. If necessary, students can always listen to the video lesson again or go back several topics.

This video lesson for 8th grade will help students learn a new topic in geometry.

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

There is also a theorem that is known as the inverse Pythagorean theorem. Let's take a closer look at it.

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

Proof. Let's say we are given a triangle ABC, in which the equality AB 2 = CA 2 + CB 2 holds. It is necessary to prove that angle C is equal to 90 degrees. Consider a triangle A 1 B 1 C 1 in which angle C 1 is equal to 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. Replacing the expression with equal sides, we get A 1 B 1 2 = CA 2 + CB 2 .

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

We 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 equal. From the equality of triangles it follows that angle C is equal to angle C 1 and, accordingly, equal to 90 degrees. We have determined that triangle ABC is right-angled and its angle C is 90 degrees. We have proven this theorem.

Next, the author gives an example. Suppose we are given an arbitrary triangle. The sizes of its sides are known: 5, 4 and 3 units. Let's check the statement from the theorem inverse to the Pythagorean theorem: 5 2 = 3 2 + 4 2. The statement is true, which means this triangle is right-angled.

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

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

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

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

The concept of a Pythagorean triangle is known. This is a right triangle whose sides are equal to whole numbers. If the legs of the Pythagorean triangle are denoted by a and c, and the hypotenuse by b, then the values ​​of the sides of this triangle can be written using the following formulas:

b = k x (m 2 - n 2)

c = k x (m 2 + n 2)

where m, n, k are any natural numbers, and the value of m is greater than the value of n.

Interesting fact: a triangle with sides 5, 4 and 3 is also called an Egyptian triangle; such a triangle was known in Ancient Egypt.

In this video lesson we learned the theorem converse to the Pythagorean theorem. We examined the evidence in detail. Students also learned which triangles are called Pythagorean triangles.

Students can easily familiarize themselves with the topic “Theorem, converse of the theorem Pythagoras" yourself using this video tutorial.