Let $\#$ be the relation defined by $A \# B = A^2 + B^2$. If $A \# 5 = 169$, what is the positive value of $A$?
Using the definition given in the problem, we have $A^2+5^2=169=13^2$. Recognizing this as Pythagorean Theorem for a 5-12-13 right triangle, $A=\boxed{12}$.