Chapter 6

Right Triangles

The Pythagorean Theorem

The Pythagorean Theorem is one of the most useful theorems of geometry.The theorem as we study it in modern geometry is often written as an algebraic statement, accompanying a diagram:

The theorem can also be written in words: "In a right triangle, the hypotenuse squared is equal to the sum of the squares of the legs."

But to the Greeks, the theorem was not algebra, it was geometry. Algebra had not yet been created, and so Euclid thought of the theorem in terms of area:

"In a right triangle, the square on the hypotenuse is equal to the sums of the squares on the other two sides."

Pythagorus was a mathematician who made important contributions to geometry. "He was a Greek philosopher and religious leader who was responsible for important developments in the areas of mathematics, astronomy, and music theory. He was also a healer, a wrestler, and was politically active. He founded a philosophical and religious school which has come to be known as the Pythagorean Society.

The Pythagoreans saw that many things in the universe were related in ways that could be stated in numbers. They reasoned that numbers must be the 'stuff' philosophers were looking for. The universe including man is a closed system. Both can be understood by the relation of the parts. These relations can be expressed in terms of numbers. These ideas led them to believe that if one could penetrate the secrets of numbers, he would penetrate the secrets of the universe and the destiny of man. This led to the careful study of geometry, the highest form of mathematics." This information, and more about Pythagorus and his followers, is from the website linked below:

http://sdcc14.ucsd.edu/~fillmore/blurbs/Pythagoras/pyth.html

Another website that delves further into the mystical and religious aspects of the Pythagorean Society can be found by clicking on the link below:

http://members.aol.com/areoasis/Reviews/pythagoras.html

There is a section about The Pythagorean Theorem in Euclid's 'Elements'. "The Elements, written about 300 B.C., a comprehensive treatise on geometry, proportions, and the theory of numbers, is the most long-lived of all mathematical works. This manuscript preserves an early version of the text. Shown here is Book I Proposition 47, the Pythagorean Theorem: the square on the hypotenuse of a right triangle is equal to the sum of the squares on the sides. This is a famous and important theorem that receives many notes in the manuscript." (from The Perseus Project referenced below)

This ancient manuscript and other historical mathematical mathematical information can be found at this website:

http://www.perseus.tufts.edu/

The Pythagorean Theorem is a fascinating subject, and one that has interested people for centuries. There are over 300 different proofs that have been written for this theorem! The link below will take you to a site where you can see a few of these proofs.

http://www.perseus.tufts.edu/GreekScience/Students/Tim/Pythag'sTheorem.html

Some other Pythagorean links are listed below:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

http://www.mcn.net/~jimloy/pythag.html

The Pythagorean Theorem, and much of the original Greek geometry depended on Geometric Constructions. Constructions are a way of creating diagrams of geometric figures using a method that incorporates the properties of the geometric figures into the creation of the diagram. The link below will explain Constructions further, and give some interesting examples:

http://www.perseus.tufts.edu/GreekScience/Students/Tim/Construction.html

If you were to construct, or very carefully draw, an isosceles right triangle with legs that were each one unit long, then you could use the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse would be the square root of 2, as shown below:

If you then construct another right triangle using the hypotenuse of the first right triangle as a leg of the second right triangle, and make the other new leg 1 unit, the new hypotenuse would be the square root of 3, by using the Pythagorean Theorem on the new triangle:

If we continue with this process, a beautiful geometric spiral results:

This right triangle spiral has an interesting property: the lengths of the hypotenuses of the right triangles form a sequence of special numbers - the square roots of the consecutive numbers 1, 2, 3, 4, 5, etc.

Other geometric spirals can be constructed using mathematical sequences such as The Golden Spiral on our Connecting Geometry welcome page, and The Fibonnacci sequence: 1,1,2,3,5,8,13,21,34 . . . (each number in this sequence is the sum of the previous two numbers . . . can you figure out which number would come next?)

The Fibonacci sequence has some fascinating applications ranging from the arrangement of leaves in the structure of plants to rabbit populations in the study of genetics. The link below will take you to a fascinating Fibonacci web site. While you are there, you can actually hear the music of the Fibonacci sequence if you click on the link in "The Rabbit Sequence", but that will take a while to download. To go to the general Fibonacci site, click on the second link below.To go directly to the music of the Fibonacci sequence, click on the first link below:

Fibonacci Rabbit Sequence

http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

The link below relates the Pythagorean Theorem, Golden Section, and the Fibonacci Sequence:

http://www.perseus.tufts.edu/GreekScience/Students/Tim/Golden.html

If you find the topic of spirals to be interesting, you should take a look at the website below:

http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiral.html

Project

Write a 2-3 page report on any one of the Right Triangle, Sequences, or Spirals topics above. Your report can be written using information you found on the internet and/or books. Keep in mind that this report must be your own work, and not just copied and pasted from the web. If you do use a sentence or two of someone else's work, be sure you include their name and the name of their book or web page to give them credit.

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