All geometry students study the
Pythagorean Theorem, named after Pythagoras, the Greek mathematician who led a
mathematical society in the 6th century BC.
The society was somewhat secretive, and close-knit, and any work done by a member
of the school was identified by the name 'Pythagorean,' rather than the individual’s
name.
The Pythagorean theorem is the most famous of mathematical
theorems, and states the following: in a right triangle (a triangle with a 90 degree
angle,) the sum of the squares of the two legs equals the square of the third side, called
the hypotenuse.
The Greeks are widely known for their work in geometry, but
properties of right triangles were also discovered by the Egyptians and the Hindus.
Any three numbers that satisfy the Pythagorean Theorem are
called Pythagorean triples. Some of
these include: (3, 4, 5); (5, 12, 13);
(8, 15, 17); (12, 16, 20); (12, 35, 37).
Most high school geometry students have met these numbers, and
verified them by squaring each, adding the squares of the first two to see that they equal
the square of the third. But few know how to
generate new triples. Here’s a neat
method, developed by the Greek mathematician Diophantus.
Pick any two numbers, a and b.
To get the first number of the Pythagorean triple, find a²
- b².
To get the second number of the Pythagorean triple, find 2ab.
To get the third number of the triple, (the hypotenuse), find a²+b².
For example, take a = 6 and b = 5.
Then a² - b² = 11.
2ab = 60.
a² + b² = 61.
Now check to see if the Pythagorean Theorem
works.
11²+60² = 61² ?
121 + 3600 = 3721. Yes!
It works!
isn't math lovely? |
I knew that.
I didn't know that.
Of course.
Oh.
Yes.
3² + 4² = 5²
(9 + 16 = 25)
etcetera
Hmm.
What?...
WOW.
WOW.
Well, now don't get carried away. |
