Sorry, I Was Wrong In My First and Third entries. Bye.

Hello. This is my last blog entry. I have bad news.

This is the blog entry where I make up my own question and try to answer it. My question is:

If you have a set of integer side lengths for a triangle, how can you figure out if it is a right triangle without actually creating the triangle?

This is what I did to try to answer my question. I made a step of procedures that you can follow to find out if a set of triangle integer side lengths:

DO THIS BEFORE THE REST OF THE PROCESS:

Take the set of side lengths and find their greatest common factor. Divide by the common factor.

Criteria:

Check for these conditions in your set of side lengths.

• The third number in the number must be the second number plus one in order for this set to be a Pythagorean triple.
• Take the middle number. If it is not divisible by 4, then it can’t be a Pythagorean triple. If it is divisible by 4, divide it by 4. Subtract 1. Subtract 2. Subtract 3. Subtract 4. Subtract 5. Keep doing this until you reach zero. If you can’t reach zero this way, then it can’t be a Pythagorean triple.
• Count up how many numbers you subtracted in order to get zero from the second number. This must be the first number in order for it to be a Pythagorean triple.

If any of these conditions are not met, then it is not a Pythagorean triple.

Sound good so far? Well, I was looking at my classmates' blogs today in class, and I found some sets of side lengths listed as Pythagorean triples on a blog called brianna² by a²+brianna²=c². I tested them on Triangles with Specified Side Lengths. They were real Pythagorean triples, and they disproved everything I said in entries 1 and 2.

I couldn't find a pattern in this new data.

Here are the new triples:

8	15	17
12	35	37
16	63	65
20	21	29
28	45	53
33	56	65
36	77	85
39	80	89
48	55	73
65	72	97

I graphed this data. Everything is color coded. There was no pattern.



Even though the graphs are blurry, you can see that they contain no pattern.

So, don't listen to the content of my first and third blog entry unless you don't need to find the triples listed here.

I have thought of what I can do next. I can put the other triples into the graph that I know, but I don't think that that will make much of a difference. I don't think there will be a pattern even if I do that. So, if I were to go on with my investigation, the thing I would do next would be kind of starting my investigation over, only with triples that are relatively prime.

Good-bye.
--anonymous PKOW!ian.

P.S. Thank you to Ms. Sheppard-Brick and Sammy for commenting on my blog.

Pythagorean Triples Again

Just to recap the first two entries, in the first entry, I found a way to find Pythagorean triples. My second entry was not very successful.

I am back on Pythagorean triples. If you saw the comments on my second blog entry, Ms. Sheppard-Brick gave me two questions to answer. This entry answers one of them.

To get the second number of a Pythagorean Triple, take a series of consecutive integers. The series must start with 1. Add all the numbers in the series. Take that sum and multiply it by 4. Add 1 to that number to get the third number in the triple. To get the first number in the triple, count how many integers you added to get the second number in the triple. Add 2 to that. You can also find more Pythagorean triples by multiplying all the numbers in a Pythagorean triple you found with the method described before.

I used the first blog entry to find these results. It took some time, but after some fruitless attempts, I finally broke through. The next entry or last entry will answer my own question. That is part of the project. Also, I have to put two comments on two of my classmate's blogs. The project assignment and my classmates' blogs are available on Ms. Sheppard-Brick's blog, Triangle Investigations.

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.