Math Is Fun Forum

In trying to fully grok the concept of irrational vs. rational, I try to visualize the difference between them represented by two Pythagorean Triangles.  One triangle, like the 3,4,5, will have commensurate, rational values between the legs and hypotenuse.  Change the aspect slightly, so as to make it 3,1, and the sqrt of 10, and the legs and hypotenuse are now incommensurate and irrationality introduced.  My question is: by what mechanism does this occur?  Why should a slight shift in aspect of the legs take us from one realm of numbers to another? 
I am a hobbyist, this is my first post.  Perhaps the answer is obvious, but it eludes me and I hope someone can help me.  Perhaps it is purely a numerical property that is detached from the physical representations of geometry and therefore lacks any mechanism.