Fun Puzzle with the Pythagorean Theorem

Normally, I spend the third week of the month discussing a practical application of a cool mathematical concept. However, my post from last week fit that standard perfectly. So, I chose to give you a practical use of the Pythagorean Theorem, but less on a applicable level and more on a personal level. This is an identity that can be turned into a puzzle for your friends. It can also be used as a proof of the Pythagorean Theorem. Enjoy!

Math in the News: The Pythagorean Theorem in Football

I mentioned last week that I wanted to focus this month on the famous Pythagorean Theorem. Since I usually reserve the second week for a "Math in the News" segment, I was looking for an example of the Pythagorean Theorem in the news.

Though I couldn't find a controversial article about it, I did find a fantastic video that showed its applications to football. And it was done by the National Science Foundation in their news section, so I think it counts. Click here to see the video.

My friends were just joking with me a few weeks ago about how in a sports game, players wouldn't actually be thinking about the math behind what they are doing. It's true that they probably aren't solving for the hypotenuse of a triangle in this dynamic moment. However, they actually do need to take an educated guess as to what angle and speed they should be running at. This does require number sense, which is definitely mathematics. The actual math might not be applied directly, but it is definitely present.

When I talked about connecting game theory with penalty kicks in soccer, this example actually uses the mathematics. Teams do hire statisticians to analyze the skills of the players on both sides, and will give their players advice on what to do. I'm sure coaches of football and other sports do this as well. Why else would they watch the opposing team's previous games to prepare themselves?

I had never thought the Pythagorean Theorem could be applied that way, and it is another reminder of the influence that mathematics has on our lives.

Proof of the Pythagorean Theorem

This month, I decided I would focus on one of the most famous theorems in all of mathematics: The Pythagorean Theorem. Though it sounds kind of dull on the face, it has a lot of extremely interesting things within it.

First, let me go over what it is. If you take a right triangle, and label the measure of its shortest side a, its middle side b, and its longest side c, then you can count on the fact that:

a² + b² = c² 

This can be used in geometry problems, like to find the longest side, or the hypotenuse, of a triangle with shorter sides, or legs, 3 and 4. You would simply plug 3 in for a, 4 in for b, and solve for c.

(3)² + (4)² = c² 

9 + 16 = c² 

25 = c² 

5 = c

The first thing I wondered when seeing this theorem is why it is true. It seems odd that any right triangle’s measurements must fit this criteria. And I never found a proof I liked until extremely recently. So, I would like to share it.

Take the following diagram:

We are going to try to figure out the area of this square. The simplest way to do it would probably be to square the square’s side. This length would be a + b.

(a + b)²

a² + 2ab + b²

Another way we could get this result is to find the area of the inside square, and then the area of the four triangles surrounding it. The inside square has length c, and the outer triangles have a base of b and height of a. So, we would get:

c² + 4(ab/2)

c² + 2ab

Since these two quantities are both the area of the same square, we can set them equal to each other. After simplification, we would get:

a² + 2ab + b² = c² + 2ab

a² + b² = c²

And we end up with our Pythagorean Theorem. I was so intrigued to see that this proof was so quick and simple, as well as pretty interesting.

Eratosthenes: The First Estimate of the Earth's Circumference

You might have thought that the Earth was proven round by Christopher Columbus in 1492. However, it was actually proven almost 2000 years earlier by a mathematician named Eratosthenes.

Eratosthenes was born in 276 BC in Cyrene, which was then a Greek colony. It would now be considered part of northern Libya. He was educated at Plato's school in Athens and went on to become chief librarian at the University of Alexandria.

Aside from his other contributions, he came up with a very close estimate of the circumference of the earth. First, he found out that at noon on the summer solstice, people in the Egyptian city of Syene noticed that there were no shadows from the sun. This is because Syene is very close to the Tropic of Cancer. This was confirmed by some tests he did by looking for brightness contrast inside of deep wells.

In his hometown of Alexandria, thought to be on the same meridian as Syene, he measured the angle of the sun's shadow on a pillar at noon of the next summer solstice. He found a 7° angle. This meant that when a circle is drawn to represent the circumference of the earth, Alexandria would be 7° around the circle.

Syene was known to be about 500 miles from Alexandria, and 7° is about one fiftieth of a circle. So, multiplying 500 x 50 gave Eratosthenes his estimate of 25,000 miles. This is a very impressive estimate, considering the modern distance is 24,901 miles.

It is extremely cool that you can determine the circumference of such a big thing without much technology at all. Just with a few mathematical principles, Eratosthenes could solve this daunting problem.

Euler Paths and Circuits

This week, I wanted to talk about a topic that I have not yet mentioned on this blog. I wanted to introduce something called Graph Theory. This is a branch of Discrete Mathematics, along with Number Theory and Combinatorics.

Though I don't know much about graph theory, I was recently introduced to a graph theory property that I wanted to share. Since it is a visual area, I found a YouTube video that explains the concept pretty well.

I found it really interesting that you can always predict the possibility of an Euler Path and Euler Circuit. If anyone knows of a proof of what was mentioned in the video, please let me know. I would love to post it.