Solution: The triangle with sides 9, 12, and 15 is a right triangle since $9^2 + 12^2 = 81 + 144 = 225 = 15^2$. The area is $\frac12 \times 9 \times 12 = 54$. The longest altitude corresponds to the shortest side (9 units). Using the formula for altitude $h = \frac2 \times \textArea\textbase$, we calculate $h = \frac2 \times 549 = 12$. \boxed12 - Nelissen Grade advocaten