So far...

When I went back to one of the websites I had, I decided to put in side lengths over 10. When I did this, I noticed that when you get to the length of the hypotenuse being 17, the second longest side turned out to be two less than the side of the hypotenuse. For example, if the hypotenuse was 17, the second longest side would be 15. This only starts when I got to the hypotenuse being 17. 16 and down were all one apart. Also, for the shortest side, for three numbers in a row (for the hypotenuse side) the shortest side is the same number. for example, if the hypotenuse of three triangles was 12, 13, and 14, the shortest side would be 4. When I got to side lengths 33 and 34 for the hypotenuse, the longest side for both was 31 and the shortest side for both was 11. From then on, the second longest side is three numbers smaller instead of the starting 1. When I got to 49 and 50 being the length of the hypotenuse, the longest side were the same (46 and 46), but the shortest side were different (16 and 17).


A question that I have been wondering about is, do all triangle angle combinations (like 30-60-90 or 20-70-90) all have the same side lengths if you change the length of the hypotenuse?

What I've found out so far

I used this website to help me further my investigation. http://www.saltire.com/applets/triangles/tri1s2a.htm On this website, I can put in any side lengths and any 2 angles (I chose 20 degrees and 70 degrees.) when I made the side either 3, 6, or 9, the third angle wasnt exactly 90 degrees, (it was actually 89 degrees) as any other side length is 90 degrees. So far, the pattern I found is that the hypotenuse of the triangle is one more than the second longest side.

What I will be investigating

Question: We have discovered that there are patterns that appear in the lengths of right triangles with the angles 30-60-90 and 45-45-90. Are there any other any other angle combinations that show similar patterns?

I am going to see if a triangle with the angles, 20-70-90, is another one of these combinations. So far, I have found out that a triangle with angles, 20-70-90's shortest side doubles as its hypotenuse increases by one. To continue my investigation, I will be going to some given sites and try out some triangles with angles measures, 20-70-90 and change side lengths to see how it will affect the triangle.




DATA:

Line AC: 1 2 3 4 5 6 7 8 9 10
Line AB: .34 .68 1.02 1.36 1.71 2.05 2.39 2.73 3.07 3.42
Line BC: .93 1.87 2.81 3.75 4.69 5.63 6.57 7.51 8.45 9.39