Second Entry

Hello again. This is my second blog entry. Last time, I had some good results, but this time, I wasn’t so fortunate and I do not have results that are as good, accurate, or relevant. I have found something different than I was looking for.

This time, the question I was investigating the question:

If you know the side lengths of a right triangle, can you predict what the angles will be?

I did not find the answer to that question. Instead, I found a semi-pattern that is not really a pattern. It seems like a pattern, but I didn’t find a pattern. Maybe you can find one. This is what I did.

My teacher us to find patterns, but I was stuck. So, my teacher gave me a clue. She said that the relationship was in the ratio of the legs. After getting that clue, I started experimenting with the legs. I divided the longer leg by the shorter leg in some of the Pythagorean triples. These are my results:

4 ÷ 3 = 1.33333333

12 ÷ 5 = 2.4

24 ÷ 7 = 3.42857143

40 ÷ 9 = 4.44444444

60 ÷ 11 = 5.45454545

84 ÷ 13 = 6.46153846

112 ÷ 15 = 7.46666667

…

The quotient increases by about 1 each time, but not exactly. In fact, I calculated the difference of every two numbers. These are the results for that:

4 ÷ 3 = 1.33333333

+1.06666667

12 ÷ 5 = 2.4

+1.02857143

24 ÷ 7 = 3.42857143

+1.01587301

40 ÷ 9 = 4.44444444

+1.01010101

60 ÷ 11 = 5.45454545

+1.00699301

84 ÷ 13 = 6.46153846

+1.00512821

112 ÷ 15 = 7.46666667

…

This time, the numbers decrease each time. I found the difference between those, too, but they don’t have a pattern them either, so I won’t bother posting them here. That is what I have done since my last entry. Thank you.

Pythagorean Triples

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.