No.
The few right triangles that correspond to Pythagorean Triples (3, 4, 5, for example) will have integer-length sides. The vast majority will have at least one side that is irrational. (1, 1, √2, for example)
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Of course, there are an infinite number of Pythagorean Triples, and they can each be scaled by an infinite number of integer scale factors. However, the infinite number of real and irrational numbers is far greater.
