The Pythagorean Theorem

Introduction

There are an uncountable number of topics that students are expected to cover each year in school. For example, they are expected to learn about right triangles, similar triangles, and polygons. We expect them to learn about angles, lines, and graphs. One of the topics that almost every high school geometry student learns about is the Pythagorean Theorem.

When asked what the Pythagorean Theorem is, students will often state that a2+b2=c2 where a, b, and c are sides of a right triangle. However, students often don't know why this is true. Most have never proved it. On the pages that follow, there is history, several proofs, and ways for high school students to use the proof in real life situations.

To understand the following information, a general knowledge of geometry and algebra should be sufficient. The proofs are not difficult and with some thought it is clear that the Pythagorean theorem works and is important.

I. The Pythagorean Theorem

To begin, the Pythagorean theorem states that the square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides. (Gardner, 152) One will find the converse of this statement to also be true.

The Pythagorean theorem was a mathematical fact that the Babylonians knew and used. However 1000 years later, between the years of 580-500 BC, Pythagoras of Samos was the first to prove the theorem. It is possible that someone proved the theorem before Pythagoras, but no proof has been found. Because of this, Pythagoras is given credit for the first proof. (MacTutor History of Mathematics Archive)

Before a proof was ever given, besides the Babylonians it was thought that "Egyptian temple builders used ropes in laying foundations, suggested that perhaps the obtained accurate right angles by using marked ropes that could be stretched around stakes to form a 3,4,5 right triangle." There is no documented evidence of this but Cantor, a historian of mathematics agreed this could be true. (Gardner, 155)

Following the first proof, many proofs followed. Proofs have been found by Euclid, Socrates, and even President Garfield. The Pythagorean Proposition, by Elisha S. Loomis gives 367 different proofs of this theorem. With the Pythagorean theorem being such a popular topic, it is no wonder high school students study the theorem. (Gardner)

Before the proofs, it is important for students to see the theorem as it is worded. "The square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides" can be drawn to get a better idea about what this proof is stating.

I+II=III where I, II, and III represent areas of the squares.

This definition gives rise to the definition that is used in today's classroom textbooks. "In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse." (Algebra, 414)

a2`+b2=c2

Two proofs of the Pythagorean theorem are available from this page and both of them are suitable to teach in the classroom. One proof may predate Pythagoras, and the second proof President Garfield devised.

Proofs of the Pythagorean Theorem

In order to fully understand the Pythagorean theorem, it is important to know that it is only a generalization of the law of cosines.

Lastly, here are a few exercises to play with.

Exercises

Bibliography

Martin Gardner, "The Pythagorean Theorem." Sixth Book of Mathematical Games from Scientific American. Charles Scribner’s Sons: New York, 1971. Chapter 16.

MacTutor History of Mathematics Archive. University of St. Andrews: Scotland. http://www.bulletproof.org/math.

Bellman, Allan., et al. Algebra, Tools for a Changing World. Massachusetts and New Jersey: Prentice Hall, 1998.

Larson, Roland and Hostetler, Robert. Precalculus. Boston and NewYork: Houghton Mifflin Co., 1997. 522-524.