For any right triangle, the side lengths of the triangle can be put in the equation a2 + b2 = c2 where a, b, and c are the side lengths. a triangle with the side lengths 3 inches, 4 inches, and 5 inches is a right triangle. which way(s) can you substitute the values into the equation to make it true? which variable has to match the longest side length? why?

The short answer is trial and error.  The side lengths "3" and "4" can both be substituted for a² and b² but not c² because their value squared is not high enough since 5² is 25. "c²" as to match the longest side because the smaller numbers will cause the equation to not be true. See Below. 

a² + b² = c²
3² + 4² = 5²
9 + 16 =  25
25 = 25


a² + b² = c²
4² + 3² = 5²
16 + 9 = 25
25 = 25


a² + b² = c²
5² + 4² = 3²
25 + 16 = 9
41 ≠ 9



a² + b² = c²
3² + 5² = 4²
9 + 25 = 16
34 ≠ 16

Hope this helped!