At the beginning of China's earliest mathematical work "Parallel Calculation of Classics in Weeks", there was a dialogue in which Duke Zhou asked Shang Gao for mathematical knowledge. Duke Zhou asked, "I heard that you are very proficient in mathematics. Excuse me: there is no ladder to go up in the sky, and you can't use a ruler to measure one by one on the ground. So how can we get data about heaven and earth? " Shang Gao replied: "The number comes from the physical hunger and thirst of the other party and the circle. There is a principle: when one right-angled side' hook' obtained from the moment of a right-angled triangle is equal to 3 and the other right-angled side' chord' is equal to 4, then its hypotenuse' chord' must be 5. This truth was summed up when Dayu was managing water. " As can be seen from the above conversation, people in ancient China discovered and applied Pythagorean Theorem thousands of years ago, which is an important principle to understand mathematics. Readers who know a little about plane geometry know that Pythagorean theorem means that in a right triangle, the sum of the squares of two right-angled sides is equal to the square of the hypotenuse. As shown in the figure, we use hook (a) and strand (b) to represent right-angled triangles respectively to get two right-angled sides, which are represented by chord (c). Then we can get Pythagorean theorem, that is, a2+b2=c2, which is called Pythagorean theorem in the west and is said to have been first discovered by Pythagoras, an ancient Greek mathematician and philosopher, in 550 BC. In fact, the ancient people of China discovered and applied this mathematical theorem much earlier than Pythagoras. If Dayu's flood control is too old to be accurately verified, then the dialogue between Duke Zhou and Shang can be determined in the Western Zhou Dynasty around 1 100 BC, more than 500 years earlier than Pythagoras. Among them, the hook of 3 strands, 4 chords and 5 is a special case of Pythagorean theorem (32+42=52). So it should be very appropriate to call it Pythagorean theorem in mathematics now. In a later book, "multiply the hook and chord separately, then add up their products and make a root, and you can get the string." Write this passage as an equation, that is, chord = (hook 2+ strand 2)( 1/2), that is, c=(a2+b2)( 1/2) in ancient China. Zhao Shuang, a mathematician in the Three Kingdoms Period, drew a pythagorean square diagram and proved the pythagorean theorem in detail by combining shape and number. In this Pythagorean square diagram, the square ABDE obtained by taking the chord as the side length is composed of four equal right triangles and a small square in the middle. The area of each right triangle is AB/2; If we know that the side length of a small square is b-a, the area is (b-a)2. Then we can get the following formula: 4*(ab/2)+(b-a)2=c2. After simplification, we can get: a2+b2=c2, that is, c=(a2+b2)( 1/2). It is rigorous and intuitive, which sets an example for China's unique style of proving the number of forms, unifying the number of forms, and closely combining algebra and geometry. Most mathematicians in the future inherited this style and developed it from generation to generation. For example, Liu Hui later used the method of proving Pythagorean theorem in form, but the division, combination and supplement of specific numbers were slightly different. Ancient mathematicians in China discovered and proved Pythagorean theorem. It has a unique contribution and position in the history of world mathematics, especially the thinking method of "unity of shape and number", which is of great significance to scientific innovation. In fact, the thinking method of "unity of form and number" is an extremely important condition for the development of mathematics. As Wu Wenjun, a contemporary mathematician in China, said, "In traditional mathematics in China, the relationship between quantity and spatial form often develops side by side. It is analytic geometry invented by Descartes in17th century.
Proof Methods of Pythagorean Theorem (10)
Prove that 1 (the proof of the textbook) makes eight congruent right triangles. Let their two right-angled sides be A and B, and their hypotenuse be C, and then make three squares with sides A, B and C, and make two squares as shown in the above figure. As can be seen from the figure, the sides of these two squares are both a+b, so the areas are equal. That is, in Proof 2 (Zou's proof), four congruent right-angled triangles are made with A and B as right-angled sides and C as hypotenuse, and the area of each right-angled triangle is equal to. Put these four right triangles into the shape shown in the figure, so that A, E and B are in a straight line, B, F and C are in a straight line, and C, G and D are in a straight line. ，∴ ∠AEH + ∠BEF = 90？ . ∴ ∠HEF = 180？ ―90? = 90? . ∴ quadrilateral EFGH is a square with a side length of c, and its area is equal to C2. ∫rtδgdh≌rtδhae，∴ HGD = ∠ EHA。 ∫∠HGD+∠GHD = 90？ ，∴ ∠EHA + ∠GHD = 90？ ∫∠GHE = 90 again? ，∴ ∠DHA = 90？ + 90? = 180? . ABCD is a square with a side length of a+b, and its area is equal to ...
On the Proof of Pythagorean Theorem
Proof of Pythagorean Theorem: Among these hundreds of proof methods, some are very wonderful, some are very concise, and some are very famous for their special identities.
Firstly, the two most wonderful proofs of Pythagorean theorem are introduced, which are said to come from China and Greece respectively. 1. China method: draw two squares with side length (a+b), as shown in the figure, where a and b are right-angled sides and c is hypotenuse.
The two squares are congruent, so the areas are equal. The left picture and the right picture each have four triangles that are the same as the original right triangle, and the sum of the areas of the left and right triangles must be equal.
If all four triangles in the left and right images are deleted, the areas of the rest of the image will be equal. There are two squares left in the picture on the left, with A and B as sides respectively.
On the right is a square with C as its side. So a 2+b 2 = c 2.
This is the method introduced in our geometry textbook. Intuitive and simple, everyone can understand.
2. Greek method: draw squares directly on three sides of a right triangle, as shown in the figure. It is easy to see △ ABA '△ AA 'c.
Draw a vertical line through C to A''B'', cross AB at C' and cross A''B'' at C'. △ABA' and square ACDA' have the same base height, and the area of the former is half that of the latter, while the base height of△ AA'' C and rectangle AA''C''C' are the same, and the area of the former is also half that of the latter.
From △ ABA '△ AA'' C, we can see that the area of square ACDA' is equal to that of rectangle AA''C''C'. Similarly, the area of square BB'EC is equal to the area of rectangle B''BC'C'
Therefore, s squared AA''B''B=S squared ACDA'+S squared BB'EC, that is, a2+b2=c2. As for the triangle area, it is half of the rectangular area with the same base and height, which can be obtained by digging and filling method (please prove it yourself).
Only the simple area relation is used here, and the area formulas of triangles and rectangles are not involved. This is the proof of the ancient Greek mathematician Euclid in the Elements of Geometry.
The above two proof methods are wonderful because they use few theorems and only use two basic concepts of area: (1) the area of congruence is equal; ⑵ Divide a graph into several parts, and the sum of the areas of each part is equal to the area of the original graph. This is a completely acceptable simple concept that anyone can understand.
Mathematicians in China have demonstrated Pythagorean Theorem in many ways, and illustrated Pythagorean Theorem in many ways. Among them, Zhao Shuang (Zhao) proved Pythagorean Theorem in his paper Pythagorean Diagrams, which was attached to Zhou Bi Shu Jing. Digging and filling method: as shown in the figure, the four right-angled triangles in the figure are colored with vermilion, the small square in the middle is colored with yellow, which is called the middle yellow solid, and the square with the chord as the edge is called the chord solid. Then, after patchwork and collocation, "the entrance and exit complement each other, each according to its type", and he affirmed that the relationship between Pythagorean triad accords with Pythagorean theorem.
That is, "Pythagoras shares multiply each other, and they are real strings, and they are divided, that is, strings." Zhao Shuang's proof of Pythagorean theorem shows that China mathematicians have superb ideas of proving problems, which are concise and intuitive.
Many western scholars have studied Pythagoras theorem and given many proof methods, among which Pythagoras gave the earliest proof in written records. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate.
Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.
The following is the proof of Pythagorean theorem by Garfield, the twentieth president of the United States. As shown in the figure, S trapezoidal ABCD= (a+b)2 = (a2+2ab+b2), ① S trapezoidal ABCD = S △ AED+S △ EBC+S △ CED = AB+BA+C2 = (2ab+C2).
② Comparing the above two formulas, we can get a2+b2=c2. This proof is quite concise because it uses trapezoidal area formula and triangular area formula.
On April 1876, Garfield published his proof of Pythagorean theorem in the New England Journal of Education. Five years later, Garfield became the twentieth president of the United States.
Later, in order to commemorate his intuitive, simple, easy-to-understand and clear proof of Pythagorean theorem, people called this proof "presidential proof" of Pythagorean theorem and it was passed down as a story in the history of mathematics. After studying similar triangles, we know that in a right triangle, the height on the hypotenuse divides the right triangle into two right triangles similar to the original triangle.
As shown in the figure, in Rt△ABC, ∠ ACB = 90. Let CD⊥BC and the vertical foot be D.
Then △BCD∽△BAC, △CAD∽△BAC. From △BCD∽△BAC, we can get BC2=BD? BA, ① AC2=AD can be obtained from △CAD∽△BAC? AB .
② We found that by adding ① and ②, we can get BC2+AC2=AB(AD+BD) and AD+BD=AB, so we have BC2+AC2=AB2, which means a2+b2=c2. This is also a method to prove Pythagorean theorem, and it is also very concise.
It makes use of similar triangles's knowledge. In the numerous proofs of Pythagorean theorem, people also make some mistakes.
For example, some people have given the following methods to prove Pythagorean theorem: let △ABC, ∠ C = 90, and cosC=0 follow the cosine theorem c2=a2+b2-2abcosC, because ∠ C = 90. So a2+b2=c2.
This seemingly correct and simple proof method actually makes a mistake in the theory of circular proof. The reason is that the proof of cosine theorem comes from Pythagorean theorem.
People are interested in Pythagorean theorem because it can be generalized. Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".
From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter". Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.
If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides. And so on.
In addition, the proof of Pythagorean theorem in eighth grade mathematics (introduction 16 proof method) (mathematics teaching plan) ydgz/.
Describe and prove Pythagorean theorem.
It is proved that the square on the left consists of 1 square with side length A and 1 square with side length B and four right-angled triangles with side lengths A and B and hypotenuse C. The square on the right consists of 1 square with side length C and four right-angled triangles with side lengths A and B and hypotenuse C. Because the areas of these two squares are equal, So we can list the equation A2+B2+4 *12ab = C2+4 *12ab and simplify it to a 2 +b 2 =c 2. The following is a false proof: Pythagorean Theorem: The property that the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse is called Pythagorean Theorem or Pythagorean Theorem. A), the length of the hypotenuse is C. Then make a square with the side length of C and make them into polygons as shown in the figure, so that E, A and C are in a straight line. The intersection Q is qp∑BC, the intersection AC is P, the intersection B is BM⊥PQ, and the vertical scale is M; Make FN⊥PQ after point F, and the vertical foot is n .∫∠BCA = 90, qp∨BC, ∴∠ MPC = 90, ∵BM⊥PQ, ∴∠ BMP. ∠ BCA = 90, BQ=BA=c, ∴Rt△BMQ≌Rt△BCA. Similarly, Rt△QNF≌Rt△AEF can also be proved, that is, a 2 +b 2 =c 2.
The proof method of Pythagorean theorem is graphic, which is difficult and diverse.
When Liu Hui proved Pythagorean theorem, he also used the method of proving numbers in form, but the specific division, combination and supplement were slightly different. Liu Hui's proof was originally a picture, but unfortunately the picture has been lost, leaving only a paragraph: "Hooking Zhu Fang, the shares are multiplied by the square, so that the entry and exit complement each other, and other things are the same, synthesizing the power of chords. In addition to roots, chords also. " The square with hook A as the edge is Zhu Fang, and the square with stock B as the edge is Fang Qing. The surplus made up for the deficiency, and Zhu Fang and Fang Qing combined into a chord box. According to their area relationship, there is a+b = C. Because Zhu Fang and Fang Qing each have a part in the string box, that part will not move. The square with the hook as the edge is Zhu Fang, and the square with the stock as the edge is Fang Qing. When III moves to III', a square (the square of C) with the chord as the side length is spelled out. It can be proved that the square of A+the square of B = the square of C, which was put forward by Liu Hui, a mathematician of Wei State in the Three Kingdoms period. In the fourth year of Wei Jingyuan (AD 263), Liu Hui annotated the ancient book Nine Chapters Arithmetic. In the annotation, he drew a diagram similar to Figure 5 (b) to prove the Pythagorean theorem. Because he used "green out" and "Zhu out" to represent yellow, purple and green in the figure, and explained how to fill the blank part of the hypotenuse square with "green in" and "Zhu in", later mathematicians called this figure "green in and out". Some people also use "in and out, complement each other."
What is Pythagorean Theorem? What methods can be used to prove the problem
In any right-angled triangle (RT△), the sum of squares of two right-angled sides is equal to the square of the hypotenuse, which is the Pythagorean theorem. That is, the square of the hook plus the square of the strand is equal to the square of the chord (6). The sum of squares of two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse. Pythagorean theorem is a special case of cosine theorem. This theorem is also called "quotient height theorem" in China (it is said that this theorem will be used in the calculation of water conservancy when Dayu is in charge of water conservancy), and it is called Pythagoras theorem or Hundred Cows theorem abroad. (Pythagoras discovered this theorem and beheaded a hundred cows to celebrate, so it is also called "Hundred Cows Theorem"), and France and Belgium also call this theorem "Donkey Bridge Theorem" (Donkey Bridge Theorem-Euclid's Elements of Geometry Part I) to find equilateral triangles. Proposition 2: Find a known point as the endpoint and make a line segment equal to the known line segment. Proposition 3: Find two line segments with known sizes, and find a line segment on the big line segment equal to the small line segment. Proposition 4: The two sides of two triangles and their included angles are equal. Then these two triangles are the same. Proposition 5: The two base angles of an isosceles triangle are equal. They discovered Pythagorean theorem later than China (China was the first country to discover this geometric treasure). At present, junior two students begin to learn, and the methods of proving textbooks mostly use Zhao Shuang's string diagram, which is a basic geometric theorem proved by Green-Zhu path diagram. Pythagorean theorem is one of the most important tools to solve geometric problems with algebraic ideas. It is also one of the ties that combine numbers and shapes. The sum of squares of two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse. If a, b, and c respectively represent two right-angled sides and hypotenuse of a right-angled triangle, then a 2; +b^2； =c^2； Pythagorean theorem points out that the sum of squares of sides of a right triangle (that is, the short side of a hook and the long side of a strand are hooks) is equal to the square of the side of the hypotenuse (that is, the chord). That is to say, if the two right sides of a right triangle are A and B respectively and the hypotenuse is C, then the square of A+the square of B = the square of C? +b？ =c？ There are about 500 ways to prove Pythagorean theorem, which is one of the most proven theorems in mathematics. Shang Gao, a famous mathematician in ancient China, said: "If you hook three, share four, then string five." It is recorded in Nine Chapters of Arithmetic. It is extended to 1. If the hypotenuse of a right triangle is regarded as a vector on a two-dimensional plane and the two right angles are regarded as projections on the coordinate axis of a plane rectangular coordinate system, then we can examine the significance of Pythagorean theorem from another angle, that is, the square of the vector length is equal to the sum of the squares of its projection lengths on a set of orthogonal bases in its space.
How to prove Pythagorean theorem in primary school? I know how to teach you. Thank you.
Pythagorean theorem was first proved by Zhao Shuang, a mathematician of the State of Wu in the Three Kingdoms period. The detailed proof of Pythagorean theorem is given by combining shape and number. In this Pythagorean theorem, strings are regarded as edges. The name BDE is composed of four equal right triangles and a small square in the middle. The area of each right triangle is AB/2. If we know that the side length of a small square is b-a, the area is (b-a)2. Then we can get the following formula: 4*(ab/2)+(b-a)2=c2. After simplification, we can get: a2+b2=c2, that is, c=(a2+b2)( 1/2). He cut out (cut out) some areas on the square with the Pythagorean line as the edge and moved them to the blank area with the chord as the edge of the square. The result was just filled in, and the problem was completely solved by graphic method. Then he gave two heights of 1, and then made them with similar triangle proportions. 2. The right triangle is inscribed in a circle and then expanded into a rectangle. Finally, he used Ptolemy theorem.