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

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