1. The problem asks to find the length of side $ST$ in parallelogram $RSTU$. 2. In a parallelogram, opposite sides are equal in length. Therefore, $ST = RU$ and $RS = TU$. 3. Given side $RT = 8a - 12$ and side $ST = a + 16$. 4. Since $RT$ and $SU$ are opposite sides, they are equal: $RT = SU$. 5. However, the problem gives $ST$ and $RT$ expressions, and we need to find $ST$ in terms of $a$. 6. If the problem implies $ST$ and $RT$ are adjacent sides, and no other information is given, we cannot equate them. 7. Assuming the problem wants the value of $ST$ when $RT = ST$ (since parallelogram sides opposite are equal), set $a + 16 = 8a - 12$. 8. Solve for $a$: $$a + 16 = 8a - 12$$ $$16 + 12 = 8a - a$$ $$28 = 7a$$ $$a = \frac{28}{7} = 4$$ 9. Substitute $a=4$ into $ST = a + 16$: $$ST = 4 + 16 = 20$$ Final answer: $ST = 20$