Different ways to prove the Pythagorean theorem: examples, description and reviews. Different ways to prove the Pythagorean theorem: examples, descriptions and reviews Algebraic formulation of the Pythagorean theorem
Class: 8
Lesson Objectives:
- Educational: achieve the assimilation of the Pythagorean theorem, instill the skills of calculating the unknown side of a right-angled triangle using two known ones, 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 ( Appendix 1)
Lesson plan:
- Organizing time
- oral exercises
- Research work, putting forward a hypothesis and testing it in 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
– Is there a 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
– An angle whose measure is 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. (Injection)
- 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 sides?
- 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.
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 of area method, axiomatic and exotic evidence(For example,
via 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 be 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:
Question - answer Angle whose measure is 90 ° DIRECT The side lying opposite the right angle of the triangle HYPOTENUSE Triangle, square, trapezoid, circle are geometric ... FIGURES The smaller side of a right triangle CATETH The figure formed by two rays emanating from one point ANGLE Perpendicular segment drawn from the vertex of a triangle to the line containing the opposite side HEIGHT A triangle whose two sides are equal isosceles
Pythagoras of Samos (c. 580 - c. 500 BC) Ancient Greek mathematician and philosopher. Born on the island of Samos. He organized his own school - the school of Pythagoras (Pythagorean Union), which was at the same time a philosophical school, a political party, and a religious brotherhood. He was the first to prove the relationship between the hypotenuse and the legs of a right triangle.
Problem of the Indian mathematician of the XII century Bhaskara On the bank of the river a lonely poplar grew. Suddenly a gust of wind broke its trunk. The poor poplar has fallen. And the angle of a straight line With the course of the river, its trunk was. Remember now that in this place the river B was only four feet wide. The head leaned at the edge of the river. There are only three feet left from the trunk, I beg you, tell me soon now: How high is the poplar tree?
