given c2 how do you sove for both a2 and b2

Question

given c2 of a right triangle how do you determine the length of both the a2 and the b2 segments

Answer 1

I guess by a2, b2 and c2 you mean a², b² and c², because a²+b²=c² (Pythagoras).

To find any of the sides a, b or c, you need to know the other two. For example, if you know a and b, c=√(a²+b²) and may not be a rational number. You can also find a if you know b and c: a=√(c²-b²). Some triangles have rational length sides, the most popular being a=3, b=4, c=5, because 3²+4²=5², that is: 9+16=25. There are an infinitude of triangles with such rational length sides. 

If you are only given c or c², there are many possible segment lengths for a and b because there are many numbers whose sum is c², but most of these will be the squares of irrational numbers, that is, not integers or fractions.