Tag Archives: Pythagorean theorem

Proof of the Pythagorean theorem

History of the Pythagorean theorem

The Pythagorean theorem is one of the greatest scientific discovery of the human, and it is also one of the basic elementary geometry theorems. There are also many other names to call this theorem, like Shang-Gao theorem, Bai-Niu theorem and so on. Someone maybe will ask that what is the Pythagorean theorem. According to Wikipedia, the Pythagorean theorem “is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.[1]” This theorem has a very long history. Almost all ancient civilizations (Greece, China, Egypt, Babylon, India, etc.) have studied this theorem. In the West, this theorem was called Pythagorean theorem. According to legend, Pythagoras, an ancient Greek mathematician and philosopher, was the first person to discover this theorem in 550 BC. Unfortunately Pythagoras’ method of proving this theorem had been lost and we could not see how he proved now. But another famous Greek mathematician, Euclid (330 BC – 275 BC), gave us a good proof in his book called Euclid’s Elements. But Pythagoras was not the first person who discovered this theorem around the world. Ancient China discovered this theorem much earlier than him. So there is another name for the Pythagorean theorem in China, the Gou-Gu theorem. Zhong Jing is the first book about mathematics in China. And in the beginning of this book, there was a conversation between Zhong Gong and Shang Gao. They were talking about the way to solve the triangle problem. From this conversation, we could know that they already found out the Pythagorean theorem around 1100 BC. They found this theorem 500 years earlier than Pythagorean.

Proof of the Pythagorean theorem

Usually in a right triangle, we need to find the length of the third side when we already know the length of other two sides. For such problems, we can directly use the formula to calculate. In many problems, we need this theorem to solve many complex questions. And then, I will introduce two basic method to prove the Pythagorean theorem.

1) Proof by Zhao Shuang

In China, Zhang Shuang was the first person who gave us the earliest proof of the Pythagorean theorem. Zhao Shuang created a picture of “Pythagorean Round Square”, and used method of symbolic-graphic combination gave us a detailed proof of the Pythagorean theorem.

Assume a, b are two Right-angle side (b > a) and c is Hypotenuse. Then each area of a right triangle is equal to ab/2.

ZhaoShuangRtTriangleGraph

[Fig.1] Proof by Zhao Shuang

∵ RtΔDAH ≌ RtΔABE,

∴ ∠HDA = ∠EAB.

∵ ∠HAD + ∠HAD = 90º,

∴ ∠EAB + ∠HAD = 90º,

∴ ABCD is a square with side c, and the area of ABCD is equal to c2.

∵ EF = FG =GH =HE = DG―DH , ∠HEF = 90º.

∴ EFGH is also a square, and the area of ABCD is equal to (b-a)2.

∴ 4 *(1/2)(DG*DH)+(DG-DH)2=AD2

∴ DH2+DG2=AD2

2) Proof by Euclid

Just like we said before, Euclid gave us a good proof in his Euclid’s Elements. He also used method of symbolic-graphic combination.

In the first, we draw three squares and the side of each square are a, b, c. And then, let points H、C、B in a straight line. Next we draw two lines between F、B and C、D and draw a line parallel to BD and CE from A. This line will perpendicularly intersect BC and DE at K and L.

[Fig.2] Proof by Euclid

[Fig.2] Proof by Euclid

∵ BF = BA,BC = BD,

∠FBC = ∠ABD,

∴ ΔFBC ≌ ΔABD

∵ The area of ΔFBC is equal to (1/2)*FG2 and the area of ΔABD is half of the area of BDLK.

∴ The area of BDLK is equal to FG2. And then we can find the area of KLCE is equal to AH2 with the same method.

∵ The area of BDEC = The area of BDLK + The area of KLCE.

∴ FG2+AH2=BD2

Conclusion

The Pythagorean theorem’s development has exerted a significant impact on mathematics. And this theorem gave us an idea to solve geometric problems with Algebraic thinking. It is also a great example about symbolic-graphic combination. This idea is very important for solving mathematical problems. By the Pythagorean theorem, we can derive a number of other true propositions and theorems, which will greatly facilitate our understanding of geometry problems, but it also has driven the development of mathematics.

Reference

[1] http://en.wikipedia.org/wiki/Pythagorean_theorem

[Fig.1] https://upload.wikimedia.org/wikipedia/commons/e/e0/ZhaoShuangRtTriangleGraph.PNG

[Fig.2]https://upload.wikimedia.org/wikipedia/commons/5/59/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem2.svg

The Pythagorean Theorem

Introduction

Have you ever pondered where mathematical equations come from or how they were derived? If the answer is yes, I want you to think if you have ever wondered where the Pythagorean theorem came from. Whether you’re in geometry, trigonometry, algebra or calculus you have to admit that we see this theorem often in each one of these math classes. Wow, it seems like it’s stalking us! Well we can thank a few cultures for that!

The History

The Babylonians are known for their discovery of Pythagorean triples. How do Pythagorean triples relate to the Pythagorean theorem, you ask? Well, they actually do play a big role and I’ll explain why later. Similarly, Pythagorean triples were also discovered by the Chinese during the Han Dynasty. However, the Chinese mentioned one thing in their proof that the Babylonians left out. This was the relationship between Pythagorean triples and a right triangle. This Discovery, made by the Chinese, is very similar to the Pythagorean theorem that we know and use today.

There was also a gentleman by the name of Pythagoras who was made famous for his discovery of the Pythagorean theorem. Though he had a prior knowledge about Pythagorean triples he was still able to find a relationship between the Pythagorean triples and right triangles. The Pythagorean theorem is mostly attributed to Pythagoras because authors like Plutarch and Cicero gave him the credit. So, I want to provide you readers with a little bit of background information on Pythagoras.

Pythagoras is from Samos Island. From a young age he was very well educated but, at the time, his passion was poetry not mathematics. However, later on Pythagoras stated to become much more interested in math and science because of the influence of Thales. Pythagoras even traveled to Egypt and he attended math related lectures there. As he gained more interest in mathematics he decided to move to the island of Croton fulltime, and specialize in Geometry. It wasn’t until later in his career that he derived the Pythagorean Theorem.

What is the Pythagorean theorem?

The cool thing about the Pythagorean theorem is that it is known to be one of the earliest geometry related theorems! The theorem states that in right triangles the square of the hypotenuse equals the sum of the squares of the other two sides. This may be a little bit confusing written out in word so I have provided a picture below!

In this particular picture, c2 = a2+b2. Hopefully that makes more sense! Now lets break down the Pythagorean theorem just a little bit more. Imagine that you have two square of two different sizes and you used them to construct multiple right triangles. Now I know that this sounds a bit confusing and you may be thinking how can I get multiple triangles from just two squares? Well, What if we put the smaller square in the center of the larger square, but we rotated the small square slightly so that it resembled a diamond. It should look something like this!

Image: Yuval Madar, via Wikimedia Commons.

Now you are able to divide the drawing up in different lengths by using different variables. From the drawing you can see that the letter “c” labels each side of the diamond or the Hypotenuse (the longest side) of the triangle.   The letter “a” labels the shortest side of the triangle and “b” labels the medium size leg of the triangle. Also notice that side “a” and side “b” both create a right angle within the triangle. I’m sure that this is making sense visually but not mathematically. Well then, I will explain in mathematical terms how these two squares and this picture relates to the Pythagorean theorem.

Proof

  • The area of a square can be written like this: (a + b)^2 = a^2 + b^2 + 2ab
  • The area of a square can also be written in term of the four triangles that we created, with the variable “c”, in the diagram above: c^2 + 4(ab/2)
    • So this means that (a + b)^2 +2ab = c^2 + 4(ab/2)
  • Now all we have to do is simplify the equation!
    • If we subtract 2ab from the left side and we are able completely cancel it out.
      • So we end up with (a + b)^2 = c^2
    • Lastly if we distribute the ^2 (on the left side of the equation) to both the “a” and “b” variables we end up with: a^2 + b^2 = c^2 and that’s the how we derive the Pythagorean theorem!!!

What can we do with the Pythagorean theorem?

Like I said before the Pythagorean theorem is used on right triangles. More particularly we use the theorem when we know the value of two sides of the triangle and we want to find the value of the remaining side of that particular triangle. We can also find the distance between points with this theorem. The Pythagorean Theorem is often used in higher-level math classes like calculus. For example in calculus three, we use this theorem to find the distance between two points on a plane, finding the surface area and volume of different shapes and etc.

Conclusion

Thanks to Pythagoras, the Babylonians and the Chinese we have the Pythagorean theorem. The famous theorem is a^2 + b^2 = c^2. We are able to derive this formula by taking the area of a square. And lastly the theorem is used a lot in finding the side lengths of a triangle and is also helpful in higher-level math courses.

Sources

http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html

http://www.thefamouspeople.com/profiles/pythagoras-504.php

http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html

http://mathforum.org/dr.math/faq/faq.pythagorean.html

http://www.purplemath.com/modules/distform.htm

http://en.wikipedia.org/wiki/Pythagorean_theorem

Who Was Zhao Shuang?

Zhao Shuang was an ancient Chinese mathematician who lived in the 3rd century CE. He spent his life on the study of the book called “Ling Xian” from famous Chinese scientist Zhang Heng and the ancient arithmetic book called “Zhou bi”. Zhao Shuang’s main achievement was summarizing the ancient mathematical achievements about “Gou gu mathematics” from the Han dynasty. Meanwhile he rediscovered the “Gou gu therorem”, which is actually the Chinese version of the “Pythagorean theorem”. He wrote hundreds of articles about “Gou gu geometric figures”, which are the earliest records of using several propositions to prove the sum-difference relations between the three sides of a triangle. In addition, he explored the roots of the quadratic equation, and he even put forward some conclusions, which are similar to Vieta’s formula (see the last link).

In Zhao Shuang’s articles about “Gou gu geometric figures”, Zhao Shuang used a method called the “cutting and compensation principle” sophisticatedly. And the meaning of “cutting and compensation principle” is actually: from 2ab+(b − a)2=c2, we can get a2 +b2 =c2. Its basic idea is that after appropriate “cutting” and “compensating” the area of the figure will not change. He also gave several propositions about the relations of the three edges of a triangle, such as:
(2(c − a)(c − b))1/2+(c-b)=a, (2(c − a)(c − b))1/2+(c-a)=b, (2(c − a)(c − b))1/2+(c-a)+(c-b)=c. Here c is the length of hypotenuse.

The most famous achievement of Zhao Shuang is an article with five hundreds words, which gives a brief proof of the “Gou gu theorem”. In that, he used figures to prove this theorem. And here I give two ways to show that.

6a600c338744ebf801757c25daf9d72a6059a7a7

A figure used to illustrate the proof of the “Gou gu theorem.”

Proof 1: Set a right triangle. Its two legs have lengths a and b respectively (assume b is larger than a) and the hypotenuse has length c. (In ancient Chinese we call the leg with length a as “gou”, leg with length b as “gu”, and the hypotenuse as “xian”, that’s why we call this theorem as “Gou gu theorem.”) Then four of such right triangles combine with a small square whose edge length is (b-a) to make a big square whose edge length is c. The small square is the white inner part of the picture below. Then apparently the area of the big square is c2, which can also be seen as the sum of the areas of the triangles and the small square. So we can get that 4ba/2 + (b − a)2=c2, from this we can get that c2=a2+b2. Thus it shows that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Proof 2: When a square with the edge length c (see the picture in the below, the large white square is just this square) is surrounded by 4 right triangles (see the 4 black right triangles in the picture), which are the same as the right triangles in the last proof, they form a bigger square with the edge length b+a. Then the area of the bigger square can be seen in two ways, the first way is (b − a)2. The second way is the sum of the area of the small square and the 4 right triangles. Thus we can get that (b − a)2= c2 +4ba/2, then we can get that c2=a2+b2.

These two proofs use the figures to show the “Gou gu theorem”, and both of them are very brief. They reflect the wisdom of ancient Chinese people. Also they are one of the great achievements of Zhao Shuang.
Links:
http://books.google.com/books?id=_q3DTqHfSFAC&pg=PA259&lpg=PA259&dq=zhao+shuang+math&source=bl&ots=P43wfC0zD_&sig=ReVMYnPYQfvNrmCM8xc2MdO9siw&hl=zh-CN&sa=X&ei=gDwVVJPJCZPkoASlkYH4Bg&ved=0CD4Q6AEwAw#v=onepage&q=zhao%20shuang%20math&f=false
http://baike.baidu.com/subview/77313/6501498.htm
http://en.wikipedia.org/wiki/Pythagorean_theorem
http://en.wikipedia.org/wiki/Vieta’s_formulas