Prove the reverse theorem of Pythagora. The lesson "Theorem - Pythagore's theorem"
Pythagore's theorem says:
In a rectangular triangle, the sum of the squares of the cathets is equal to the square of the hypotenuse:
a 2 + B 2 \u003d C 2,
- a. and b. - Roots that form a straight corner.
- from - triangle hypotenuse.
Pythagora theorem formulas
- a \u003d \\ sqrt (C ^ (2) - b ^ (2))
- b \u003d \\ SQRT (C ^ (2) - A ^ (2))
- c \u003d \\ SQRT (A ^ (2) + B ^ (2))
Proof of Pythagora theorem
The area of \u200b\u200bthe rectangular triangle is calculated by the formula:
S \u003d \\ FRAC (1) (2) AB
To calculate the area of \u200b\u200ban arbitrary triangle Formula Square:
- p. - half-meter. P \u003d \\ FRAC (1) (2) (A + B + C),
- r. - radius inscribed circle. For rectangleR \u003d \\ FRAC (1) (2) (A + B-C).
Then we equate the right parts of both formulas for the triangle area:
\\ FRAC (1) (2) AB \u003d \\ FRAC (1) (2) (A + B + C) \\ FRAC (1) (2) (A + B-C)
2 AB \u003d (A + B + C) (A + B-C)
2 ab \u003d \\ left ((a + b) ^ (2) -c ^ (2) \\ RIGHT)
2 ab \u003d a ^ (2) + 2ab + b ^ (2) -c ^ (2)
0 \u003d a ^ (2) + b ^ (2) -c ^ (2)
c ^ (2) \u003d a ^ (2) + b ^ (2)
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. That is, for all the three of positive numbers a, B. and c., such that
a 2 + B 2 \u003d C 2,
there is a rectangular triangle with customs a. and b. and hypotenuse c..
Pythagorean theorem - One of the fundamental theorems of Euclidean geometry, which establishes the ratio between the sides of the rectangular triangle. She proven by a scientist mathematician and philosopher Pythagore.
Theorem value In that, with its help, you can prove other theorems and solve problems.
Additional material:
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 three 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, 367 evidence of this theorem was recorded in the scientific literature. 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:
A more general expression for changing the hypotenuse in the event of increments of both cathets:
Integrating this equation and using the initial conditions, we obtain:
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:
According to Van der Varden, it is very likely that the ratio in general was known in Babylon near the XVIII century BC. e.
Approximately 400 BC. E. According to the probe, Plato gave the method of finding Pythagora Trok, combining algebra and geometry. About 300 BC. e. In the "beginning of" Euclidea, the oldest axiomatic proof of the Pythagoreo Theorem appeared.
Formulation
The main formulation contains algebraic actions - in a rectangular triangle, whose cathettes are equal A (\\ DisplayStyle A) and B (\\ DisplayStyle B), and the length of hypotenuses - C (\\ DisplayStyle C)The ratio is completed:
.Equivalent geometric formulation is possible, resorting to the concept of an area of \u200b\u200bthe figure: 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 the categories. In this form the theorem is formulated in the beginning of the Euclidea.
Pythagora reverse theorem - approval of the rectangles of any triangle, the length of the sides of which are related by the relation a 2 + b 2 \u003d C 2 (\\ displaystyle a ^ (2) + b ^ (2) \u003d c ^ (2)). As a result, for all three of positive numbers A (\\ DisplayStyle A), B (\\ DisplayStyle B) and C (\\ DisplayStyle C), such that a 2 + b 2 \u003d C 2 (\\ displaystyle a ^ (2) + b ^ (2) \u003d c ^ (2)), There is a rectangular triangle with customs A (\\ DisplayStyle A) and B (\\ DisplayStyle B) and hypotenuse C (\\ DisplayStyle C).
Proof of
The scientific literature recorded at least 400 evidence of the Pythagora theorem, which is explained as a fundamental value for geometry and the elementaryness of the result. The main directions of evidence: the algebraic use of the relationship of the triangle elements (such, for example, the popular method of similarity), the method of space, there are also various exotic evidence (for example, using differential equations).
Through such triangles
The classical evidence of Euclidea is aimed at establishing equality of the area between rectangles formed from the migration of the square above the hypotenurium height of direct angle with squares above the customs.
The design used for the proof is as follows: for a rectangular triangle with direct angle C (\\ DisplayStyle C), squares over customs and squares over hypotenuse A B I K (\\ DisplayStyle Abik) Built height C H (\\ DisplayStyle CH) and continuing her beam S (\\ DisplayStyle S), breaking the square over hypotenur with two rectangles and. Evidence is aimed at establishing equality of rectangle area A h j k (\\ DisplayStyle Ahjk) Square over cathet A C (\\ DisplayStyle AC); Equality of the area of \u200b\u200bthe second rectangle constituting the square above the hypotenuse, and the rectangle over the other cathe is set in the same way.
Equality of rectangle squares A h j k (\\ DisplayStyle Ahjk) and A C E D (\\ DisplayStyle Aced) Installed through the congruence of triangles △ A C K \u200b\u200b(\\ DisplayStyle \\ Triangle ACK) and △ A B D (\\ DisplayStyle \\ Triangle ABD), the area of \u200b\u200beach of which is equal to half the square square A h j k (\\ DisplayStyle Ahjk) and A C E D (\\ DisplayStyle Aced) Accordingly, due to the following property: The triangle area is equal to half the rectangle area, if the figures have a common party, and the height of the triangle to the general side is the other side of the rectangle. Congrunce of triangles follows from the equality of both sides (sides of the squares) and the corner between them (composed of a straight corner and angle at A (\\ DisplayStyle a).
Thus, the proof establishes that the square of the square above the hypotenuse composed of rectangles A h j k (\\ DisplayStyle Ahjk) and B h j i (\\ displaystyle bhji)is equal to the sum of squares of squares over customs.
Proof Leonardo da Vinci
The proof of Leonardo da Vinci found to the area of \u200b\u200bthe square. Let a rectangular triangle △ A B C (\\ DisplayStyle \\ Triangle ABC) With direct angle C (\\ DisplayStyle C) and squares A C E D (\\ DisplayStyle Aced), B C F G (\\ DisplayStyle BCFG) and A B H J (\\ DisplayStyle ABHJ) (See Figure). In this proof on the side H j (\\ DisplayStyle HJ) The latter in the outside is a triangle, congruent △ A B C (\\ DisplayStyle \\ Triangle ABC), moreover, reflected both relative to hypotenuses and relatively height to it (that is, J i \u003d b c (\\ displaystyle ji \u003d bc) and H i \u003d a C (\\ DisplayStyle Hi \u003d AC)). Straight C i (\\ DisplayStyle Ci) breaks the square built on hypotenuse into two equal parts, since triangles △ A B C (\\ DisplayStyle \\ Triangle ABC) and △ J H i (\\ DisplayStyle \\ Triangle JHI) equal to construction. The proof establishes the congruence of the quadricles C A J i (\\ DisplayStyle Caji) and D A B G (\\ DisplayStyle DABG)The area of \u200b\u200beach of which turns out to be on the one hand equal to the sum of the half of the squares of squares on the cateches and the area of \u200b\u200bthe original triangle, on the other hand, half the square of the square on the hypotenuse plus the area of \u200b\u200bthe original triangle. Total, half the sum of squares of squares over the categories is equal to half the square of the square over the hypotenuse, which is equivalent to the geometric formulation of the Pythagores theorem.
Proof by the method of infinitely small
There are several evidence that resort to the technique of differential equations. In particular, Hardy is attributed to the proof, using infinitely small increments of cathets A (\\ DisplayStyle A) and B (\\ DisplayStyle B) and hypotenuses C (\\ DisplayStyle C), and preserving the similarity with the original rectangle, that is, providing the following differential relations:
d a d c \u003d c a (\\ displaystyle (\\ FRAC (DA) (DC)) \u003d (\\ FRAC (C) (A))), d b d c \u003d c b (\\ displaystyle (\\ FRAC (DB) (DC)) \u003d (\\ FRAC (C) (B))).Differential equation is derived by separating the variables c d c \u003d a d a + b d b (\\ displaystyle c \\ dc \u003d a \\, da + b \\, db)whose integration gives the ratio C 2 \u003d A 2 + B 2 + C O N S T (\\ DisplayStyle C ^ (2) \u003d A ^ (2) + B ^ (2) + \\ Mathrm (Const)). Application of initial conditions a \u003d b \u003d c \u003d 0 (\\ displaystyle a \u003d b \u003d c \u003d 0) Determines the constant as 0, which results in the statement of the theorem.
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.
Variations and generalizations
Similar geometric shapes on the three sides
An important geometric generalization of the Pythagorean theorem gave Euclium in the "Beginning", crossing the squares of the squares on the sides of arbitrary similar geometric figures: the sum of the areas of such figures built on catetes will be equal to the area of \u200b\u200bthe figure similar to the hypotenuse.
The main idea of \u200b\u200bthis generalization is that the area of \u200b\u200bsuch a geometric shape is proportional to the square of any linear size and in particular the square of the length of any side. Consequently, for similar shapes with squares A (\\ DisplayStyle a), B (\\ DisplayStyle B) and C (\\ DisplayStyle C)built on customations with lengths A (\\ DisplayStyle A) and B (\\ DisplayStyle B) and hypotenuse C (\\ DisplayStyle C) Accordingly, the ratio is:
A A A 2 \u003d B B 2 \u003d C C 2 ⇒ A + B \u003d A 2 C 2 C + B 2 C 2 C (\\ DisplayStyle (\\ FRAC (A) (A ^ (2))) \u003d (\\ FRAC (B ) (B ^ (2))) \u003d (\\ FRAC (C) (C ^ (2))) \\, \\ rightarrow \\, a + b \u003d (\\ FRAC (A ^ (2)) (C ^ (2) )) C + (\\ FRAC (B ^ (2)) (C ^ (2))) c).Since Pythagora theorem a 2 + b 2 \u003d C 2 (\\ displaystyle a ^ (2) + b ^ (2) \u003d c ^ (2)), then performed.
In addition, if it is possible to prove without attracting the Pythagora theorem, that for the areas of three similar geometric figures on the sides of the rectangular triangle, the ratio was performed A + B \u003d C (\\ DISPLAYSTYLE A + B \u003d C), Using the reverse stroke of proof of the generalization of Euclidea, the proof of the Pythagora theorem can be derived. For example, if on the hypotenuse to build a congruent initial rectangular triangle area C (\\ DisplayStyle C), and on categories - two similar rectangular triangles with squares A (\\ DisplayStyle a) and B (\\ DisplayStyle B), it turns out that triangles on catetes are formed as a result of dividing the initial triangle of its height, that is, the sum of two smaller areas of triangles is equal to the area of \u200b\u200bthe third, so A + B \u003d C (\\ DISPLAYSTYLE A + B \u003d C) And, applying the ratio for such figures, the Pythagora theorem is displayed.
Kosinus theorem
The Pythagoreo Theorem is a special case of a more general cosine theorem, which binds the lengths of the parties in an arbitrary triangle:
a 2 + b 2 - 2 a b cos \u2061 θ \u003d C 2 (\\ displaystyle a ^ (2) + b ^ (2) -2ab \\ cos (\\eta) \u003d c ^ (2)),where - the angle between the parties A (\\ DisplayStyle A) and B (\\ DisplayStyle B). If the angle is 90 °, then cos \u2061 θ \u003d 0 (\\ displaystyle \\ cos \\eta \u003d 0)And the formula is simplified to the usual Pythagoreo theorem.
Arbitrary triangle
There is a generalization of the Pythagora theorem on an arbitrary triangle, operating exclusively by the ratio of the lengths of the parties, it is believed that it was first established by Sabi astronomer Sabit Ibn Kury. In it for an arbitrary triangle with the sides, an equifiable triangle fits into it with the base on the side C (\\ DisplayStyle C), vertex that coincides with the top of the original triangle, the opposite side C (\\ DisplayStyle C) and angles at the base equal to the corner θ (\\ DisplayStyle \\ Theta), opposite side C (\\ DisplayStyle C). As a result, two triangles are formed, similar to the original: the first - with the parties A (\\ DisplayStyle A), the long-sided side of the sideways inscribed by an elevated triangle, and R (\\ DisplayStyle R) - Parts Parts C (\\ DisplayStyle C); The second is symmetrically to him from side B (\\ DisplayStyle B) from the side S (\\ DisplayStyle S) - the corresponding part of the part C (\\ DisplayStyle C). As a result, the relation: Relation:
a 2 + b 2 \u003d C (R + S) (\\ displaystyle a ^ (2) + b ^ (2) \u003d c (R + S)),degenerate into the Pythagora theorem with θ \u003d π / 2 (\\ DISPLAYSTYLE \\ THATA \u003d \\ PI / 2). The ratio is a consequence of the similarity of the formed triangles:
CA \u003d AR, CB \u003d BS ⇒ CR + CS \u003d A 2 + B 2 (\\ DisplayStyle (\\ FRAC (C) (A)) \u003d (\\ FRAC (A) (R)), \\, (\\ FRAC (C) (b)) \u003d (\\ FRAC (B) (S)) \\, \\ rightarrow \\, cr + cs \u003d a ^ (2) + b ^ (2)).Pappa theorem on squares
Neevklidova Geometry
The Pythagoreo Theorem is derived from an axiom of Euclidean geometry and is invalid for non-child geometry - the implementation of the Pythagorean theorem is equivalent to the postulate of Euclidea of \u200b\u200bparallelism.
In non-child geometry, the ratio between the sides of the rectangular triangle will necessarily be in the form other than the Pythagorean theorem. For example, in spherical geometry, all three sides of a rectangular triangle, which limit the aircraft of a single sphere, have a length π / 2 (\\ DisplayStyle \\ PI / 2)which contradicts the Pythagorean theorem.
In this case, the Pythagora theorem is valid in hyperbolic and elliptical geometry, if the requirement of the rectangle of the triangle is replaced by the condition that the sum of two triangle angles should be equal to the third.
Spherical geometry
For any rectangular triangle on the sphere of radius R (\\ DisplayStyle R) (for example, if the angle in the triangle is straight) with the parties A, B, C (\\ DisplayStyle A, B, C) The ratio between the parties has the form:
COS \u2061 (C R) \u003d COS \u2061 (A R) ⋅ COS \u2061 (B R) (\\ DisplayStyle \\ COS \\ LEFT ((\\ FRAC (C) (R)) \\ Right) \u003d \\ cos \\ left ((\\ FRAC (a) (R)) \\ Right) \\ CDOT \\ COS \\ LEFT ((\\ FRAC (B) (R)) \\ Right)).This equality can be derived as a special case of the spherical cosine theorem, which is valid for all spherical triangles:
COS \u2061 (C R) \u003d COS \u2061 (A R) ⋅ COS \u2061 (B R) + SIN \u2061 (A R) ⋅ SIN \u2061 (B R) ⋅ Cos \u2061 Γ (\\ DisplayStyle \\ COS \\ LEFT ((\\ FRAC ( c) (R)) \\ Right) \u003d \\ cos \\ left ((\\ FRAC (A) (R)) \\ RIGHT) \\ CDOT \\ COS \\ left ((\\ FRAC (B) (R)) \\ Right) + \\ CH \u2061 C \u003d CH \u2061 A ⋅ CH \u2061 B (\\ DISPLAYSTYLE \\ OPERATORNAME (CH) C \u003d \\ Operatorname (CH) A \\ CDot \\ Operatorname (CH) B). Where,cH (\\ DisplayStyle \\ Operatorname (CH)) - hyperbolic cosine. This formula is a special case of a hyperbolic cosine theorem, which is valid for all triangles: CH \u2061 C \u003d CH \u2061 A ⋅ CH \u2061 B - SH \u2061 A ⋅ SH \u2061 B ⋅ COS \u2061 γ (\\ DisplayStyle \\ Operatorname (CH) C \u003d \\ Operatorname (CH) A \\ CDot \\ Operatorname (CH) B- \\ Operatorname (SH) A \\ CDOT \\ Operatorname (SH) B \\ CDOT \\ COS \\ GAMMA)
γ (\\ DisplayStyle \\ Gamma),cH (\\ DisplayStyle \\ Operatorname (CH)) - angle whose vertex is the opposite to the side Using a series of Taylor for a hyperbolic cosine ( C (\\ DisplayStyle C).
CH \u2061 X ≈ 1 + x 2/2 (\\ DisplayStyle \\ Operatorname (CH) X \\ APPROX 1 + X ^ (2) / 2) ) It can be shown that if the hyperbolic triangle decreases (that is, when A (\\ DisplayStyle A), B (\\ DisplayStyle B) and C (\\ DisplayStyle C) They strive for zero), then hyperbolic relations in a rectangular triangle are approaching the ratio of the classical Pythagore's theorem.
Application
Distance in two-dimensional rectangular systems
The most important use of the Pythagora theorem is the determination of the distance between two points in the rectangular coordinate system: distance S (\\ DisplayStyle S) between points with coordinates (A, B) (\\ DisplayStyle (A, B)) and (C, D) (\\ DisplayStyle (C, D)) equally:
S \u003d (A - C) 2 + (b - d) 2 (\\ displayStyle S \u003d (\\ SQRT ((A-C) ^ (2) + (B - D) ^ (2)))).For complex numbers, the Pythagorea Theorem gives a natural formula to find a complex integrated module - for z \u003d x + y i (\\ displaystyle z \u003d x + yi) It is equal to the length
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 from 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: they shared the knurles on 12 equal parts, the ends were associated, 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 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 minutes.
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 Pyphagora, the ancient Egyptians knew that the triangle with the parties 3, 4 and 5 is rectangular and used this property to build direct corners 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.
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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.
