1. **Problem Statement:** We have two similar trapezoids. The larger trapezoid has a vertical side of length 8 and a bottom base of length 12. The smaller trapezoid has a vertical side of length $n$ and a bottom base of length 9. We need to find the length of $n$ using proportions. 2. **Formula and Concept:** Since the trapezoids are similar, their corresponding sides are proportional. This means: $$\frac{8}{n} = \frac{12}{9}$$ 3. **Set up the proportion:** $$\frac{8}{n} = \frac{12}{9}$$ 4. **Solve for $n$:** Cross-multiply: $$8 \times 9 = 12 \times n$$ $$72 = 12n$$ Divide both sides by 12: $$n = \frac{72}{12} = 6$$ 5. **Answer:** The length of $n$ is 6. This means the smaller trapezoid's vertical side is 6 units long, maintaining the similarity ratio with the larger trapezoid.