Geometry consists of different theorems connected to figures, structures, and math in general. The Pythagorean Theorem is one of Euclidean geometry’s most prevalent and fundamental theorems. It remains one of the primary subjects to study in geometry class in every country. This rule is a cornerstone for math, other subjects like physics, and every other natural science. However, the Pythagorean Theorem is far more complex and widely used than people think. It can become more exciting and innovative using different points of view.
To be precise, Pythagoras of Samos was an Ancient Greek scientist, philosopher, mystic, and mathematician. None of his works has survived, but his heritage is based on the works of his followers, which is why all humanity knows about him. Like other scientists of the Ancient world, Pythagoras was contradictory and versatile. Many ideas he believed in were ahead of time. With that, he thought he had already lived four lives and remembered all of them. The Pythagorean Theorem means that the square built on the hypotenuse equals the sum of the squares on the legs. As Chambers stated, the “Pythagorean Theorem is one of the traditional bits of mathematics that many people remember from their schooldays” (22). It has a significant impact on the application in math and practical use. Builders create stable constructions using this fundamental rule as long as it is used by GPS coordinates systems (TED-ed). The theorem is named after Pythagoras, who lived in the sixth century B.C., but there are many examples of people using this rule much earlier.
One of them is a Babylonian tablet that was found to have fifteen sets of numbers that confirm the theorem, and other cases came from Ancient Egypt and India. Historians state that ancient Egyptians used numbers “three, four, and five” to make square corners using a robe with knots and twelve equal lengths between them. This way, they made right triangles and subsequently square corners. In the earliest mathematical works of India, it is stated that to produce a square twice as big, and the original square needs to be divided across the diagonal of it by the rope. One of the ways of using the Pythagorean Theorem in life is laying the foundation of houses or other buildings. Nowadays, when someone needs to build a foundation, they understand the angles must be right. To mathematically count it up and not eyeball, all they need is this theorem.
The Pythagorean theorem can be found using in many different areas and ways. The theorem is constantly used in navigation, and “all GPS measurements are possible because of this theorem” (Kiger). The Pythagorean Theory gives pilots and navigators of ships the way to calculate the distance to a certain point. Cartographers use it to calculate the steep mountain or hill they plan to climb. This process is called surveying, and they usually use a telescope to guide it toward a measuring stick. Another way the theorem is used nowadays is through electronic data. Programmers use the Pythagorean Theorem to calculate if there is a minor mistake in it or a missing bit. They use data as points and track if these points are in their places. Everyone uses this theorem “from church builders to video-game designers” (Zamboni). The Pythagorean Theorem is one of the most influential and grounding rules in the history of humanity.
Even though the Pythagorean Theorem is thousands of years old, it still is a theorem that needs to be proven. The concept of the theorem itself is a mathematical statement, the truth of which is established by proof, leaning on axioms and previously proven theorems. It leads this essay to a question: how many ways are there to prove the Pythagorean Theorem? According to a TED-ed video about this theorem, there are “more than three hundred and fifty proofs and counting” (TED-ed). One of the interesting ways to prove the theorem is shown by the YouTuber Steve Chow. He demonstrates how this theorem can exist in three-dimensional space. He uses “a” and “b” as the width and length of a room and “c” as the height (blackpenredpen). In order to find the diagonal from the bottom corner to another upper corner, first of all, he uses the two-dimensional formula. He calculates the hypotenuse and uses it as one of the legs, with another leg as “c.” Now with all the numbers, he estimates the three-dimensional diagonal and proves that a2+b2+c2=d2. In this equation, “c” means height, and “d” is a three-dimensional diagonal.
Another interesting approach to the Pythagorean Theorem is the Inverse Pythagoras’ Theorem. It demonstrates the same triangle with the right angle, but this time the hypotenuse is used as a basis, and “h” is perpendicular from the right angle to the hypotenuse. In this situation, “h” is the height. The inverse Pythagorean Theorem says that 1/a+1/b2=1/h2, and it is proven. To prove this theorem, it is argued that ab=hc, which means c=ab/h. This way a2+b2=(ab/h)2 and if it is rearranged 1/a2+1/b2 equals 1/h2 or another way a-2+b-2=h-2. This statement is called Inverse Pythagorean Theorem and serves other needs using the Pythagorean Theorem as a basis.
The Pythagorean Theorem is simple and understandable enough for people of different ages and statuses. However, one of the biggest problems is how to convey this information in schools. There are three ways of teaching this theorem, and the first one is to use only language to explain. The teacher can state that “the square on the hypotenuse is equal to the sum of the squares on the other two sides” and not demonstrate anything (Akeshkevich 2). The second variant is the most common one, and it is being explained by using the diagram with the right-angled triangle, with “a” and “b” as legs and “c” as a hypotenuse. However, there are many different options to state the formula. The third way of expressing the theorem is to areas, not lines. This method is appropriate because of its practical use and the examples in life. At the same time, it can cause misunderstanding in some students’ minds about whether the Pythagorean Theorem is about lines or areas.
To conclude, geometry is the section of mathematical science that studies spatial structures and relations and their generalizations. The Pythagorean Theorem is one of the ground rules of geometry that has served humanity for thousands of years. Using this theorem, people can produce even details for every area that can be imagined. It drastically affects the whole world and people’s lives, and it will have a considerable impact on the world’s future. In fact, the Pythagorean Theorem is a diamond of geometry.
Works Cited
Aleshkevich, Natalia. “Geometric and Algebraic Interpretation of Primitive Pythagorean Triples Parameters”. Peter the Great St. Petersburg Polytechnic University, 2019, pp. 2-3.
“How Many Ways Are There to Prove the Pythagorean Theorem”. Youtube, uploaded by TED-ed, 2017.
Kiger, Patrick. The Pythagorean Theorem Makes Construction and GPS Possible. HowStuffWorks, 2019.
“Pythagorean Theorem, But in Three Dimensions”. Youtube, uploaded by blackpenredpen, 2018.
Zamboni, Jon. Real Life Uses of the Pythagorean Theorem. Sciencing, 2018.