Introduction: Pythagorean theorem is one of the most important theorems in the set, which is very useful in production and life, and is also widely used in other natural sciences. When I didn't study Pythagorean theorem deeply, I thought it was amazing and profound. But after learning Pythagorean theorem, I know that "the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two right angles", which is the so-called Pythagorean theorem. I can easily solve many problems with this knowledge. So, how was this important theorem discovered, where did it originate, and what is its specific use in life? In order to understand Pythagorean theorem more deeply, I wrote this paper under the guidance of my math teacher.

Keywords: discovery, proof, application and extension

First, understand the discovery process of Pythagorean theorem

"The square of the hypotenuse of a right triangle is equal to the sum of the squares of its two right-angled sides" seems so simple, but the process of its discovery is not so simple: people's understanding of Pythagorean theorem manages the process from special to general. Looking back at history, almost all ancient civilizations have discovered this theorem, including Greece, China, Egypt, Babylon, India and so on. As early as 3,000 years ago, China had reached the conclusion that "the hook length is three, the head is four and the radius is five", that is, in a right triangle, if the hook length is three and the head is four, then the chord length is five. More than a thousand years before Pythagoras, the Babylonians knew this theorem. Pythagoras was the first person in the west to discover this theorem.

Pythagorean theorem is a theorem with a long history, which has a history of 5000 years from discovery to prominence. Throughout the ages, countless mathematicians have put forward the proof of this theorem, and even an American president (Garfield) once put forward a proof when he was a member of parliament. In addition, this theorem has been given many different names, such as Hundred Cows Theorem, Pythagorean Theorem, Quotient Theorem, Pythagorean Theorem and so on.

Second, prove Pythagorean theorem

You know, so far, there are more than 400 proofs of Pythagorean theorem! Then can we prove it ourselves and see if we can prove it with puzzles? Try it and you'll know:

Prove 1 As shown in the figure, the area of square ABCD.

= area of 4 right-angled triangles+area of square PQRS

∴ ( a + b )2 = 1/2 ab × 4 + c2

a2 + 2ab + b2 = 2ab + c2

Therefore, A2+B2 = C2. In figure 1, the area of a = (the area of a big square)-(the area of four right triangles).

In Figure 2, the sum of the areas of B and C = (the area of a big square)-(the areas of four right-angled triangles).

Since the area of the graph 1 is equal to that of the graph 2,

So the area of a = the area of b+the area of c.

c2 = a2 + b2

Are there any other puzzles to prove besides one or two methods? Let's see:

It is proved that the area of three trapezoids = the sum of the areas of three right triangles.

1/2×(a+b)×(a+b)= 2× 1/2×a×b+ 1/2×c×c

(a + b )2 = 2ab + c2

a 2 + 2ab + b2 = 2ab + c2

So a2+b2 = c2.

Wow! It turns out that it is also very interesting to prove a theorem by yourself!

Thirdly, from Pythagorean theorem to the expansion of graphic area.

As we know, Pythagorean theorem reflects the relationship between three sides of a right triangle: a2 +b2=c2. A2, b2, c2 and C2 can be regarded as square areas with side lengths A, B and C, so the Pythagorean theorem can also be expressed as: the sum of the areas with side lengths of two right-angled triangles is equal to the area with side lengths of hypotenuse being square. As shown in figure 1, S 1+S2=S3.

D

If the three sides A, B and C of a right triangle are taken as sides and the outward shape is made into regular triangles (as shown in Figure 2), then if the three sides A, B and C of the teaching triangle are taken as sides, there is also S 1+S2=S3.

c

b

a

B

F

A

E

C

S3

c

b

a

Figure 1

S2

S 1

B

A

C

Figure ii

Fourth, the application of Pythagorean theorem in life

What is the specific use of such a wonderful Pythagorean theorem in life and production? In fact, Pythagorean theorem has a very close relationship with people since ancient times.

In ancient times, Chen Zi, an outstanding mathematician in ancient China (6th-7th century BC), measured the height and distance of the sun, which people like to praise. In order to control the flood, Dayu decided the direction of the water flow according to the height of the terrain, followed the trend and made the flood flow into the sea, so that there would be no more flood disasters, which was also the result of applying Pythagorean theorem.

Today, many scientists in the world are trying to find "people" on other planets. Because of this, many signals are sent to the universe, such as human language, music, various graphics and so on. It is said that Hua, a famous mathematician in China, once suggested that the graph of Pythagorean theorem should be deduced. If cosmic people are "civilized people", then they will certainly understand this "language".

These facts can illustrate the great significance of Pythagorean theorem.

Verb (abbreviation of verb) summary and feeling

Feeling 1: Roentgen said: "The first is mathematics, the second is mathematics, and the third is mathematics." Mathematics is all around us. As long as we have a pair of eyes for discovery, we can gain a lot of knowledge about mathematics.

Perception 2: Although the Greeks called Pythagorean Theorem or Hundred Cows Theorem, France and Belgium also called this theorem "Donkey Bridge Theorem", it is estimated that they discovered Pythagorean Theorem later than China. China is the first country in the world to discover the geometric treasure of Pythagorean theorem! Pythagorean theorem is the crystallization of China people's wisdom and the essence of ancient China culture. So, besides being proud of it, how can it develop? This needs our consideration.