Hello. This blog is for a project in my geometry class called the Triangle Blog Project. We have to do a series of investigations about right triangles and the Pythagorean Theorem (a2+b2=c2) and record our results here. Right now, I am investigating to answer the following question:
“Pythagorean triples are right triangles that have integer side lengths. How many of these are there? Can you predict when they will occur?”
To investigate, I followed the following procedure. First, I looked at a list of Pythagorean triples. I took out the ones with whose lowest number was odd and looked only at them:
(3-4-5)
(5-12-13)
(7-24-25)
(9-40-41)
(11-60-61)
(13-84-85)
(15-112-113)
(17-144-145)
(19-180-181)
…
This is a picture of Pascal’s Triangle from Wikipedia.org. The green highlighted region shows the sequence explained below↓.
All the numbers in the middle are divisible by four. If the numbers are divided by four, then the following sequence is discovered: 1;3;6;10;15;21;28;36;45…. These are the numbers in the second set of numbers parallel to the set with all 1s in Pascal’s Triangle. To get a Pythagorean triple, you have to multiply a number in that set (the one with 1;3;6;10;15;21;28;36;45…) by four to get the second number in the triple, and add 1 to that number to get the third number in the triple. Then do the following operation to get the first number in the triple: 2n-3 where n is the row of Pascal’s Triangle from which you got the number that you multiplied by four to get the second number in the triple. There are more Pythagorean triples. To find those, you just have to multiply all the numbers in a Pythagorean triples that you can find with this method by the same number.
