We have discovered that there are patterns that appear in the lengths of right triangles with the angles 30-60-90 and with the angles 45-45-90. Are there any other angle combinations that show similar patterns?
The angle combination that shows a similar pattern is the 80-60-40 Triangle. There is a relationship between the Short Leg and the Hypotenuse Leg and between the Short Leg and the Longer Leg. The relationship between the Short Leg and the Hypotenuse Leg is the Short Leg x 1.532 = Hypotenuse Leg. This relationship is shown in the table below. There is another relationship between the Short Leg and the Longer Leg. The relationship is Short Leg x 1.347 = Longer Leg. This relationship is shown in the table below.
| Short Leg | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Hypotenuse Leg | 1.532 | 3.065 | 4.597 | 6.129 | 7.661 | 9.193 | 10.725 | 12.257 | 13.789 | 15.321 |
| Longer Leg | 1.347 | 2.694 | 4.041 | 5.389 | 6.736 | 8.083 | 9.431 | 10.778 | 12.125 | 13.472 |
I had to take many steps to investigate this triangle. I used the website: http://www.saltire.com/applets/triangles/tri1s2a.htm to find my information. It also took me some complicated calculations to get this pattern.
It is really neat that you found a pattern in this data. Why do you think that this pattern exists? Do you think you could find patterns for any set of angles or will only certain ones work? You may want to focus on right triangles in the future, it will help you to limit an otherwise very large investigation.
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