1. **State the problem:** We have two similar quadrilaterals. We know the lengths of some sides and need to find the missing length $q$ in the smaller figure. 2. **Identify given lengths:** - Smaller figure sides: $q$ (unknown), 81 yards - Larger figure sides: 11 yards, 99 yards 3. **Recall the property of similar figures:** Corresponding sides of similar figures are proportional. This means: $$\frac{q}{11} = \frac{81}{99}$$ 4. **Set up the proportion:** $$\frac{q}{11} = \frac{81}{99}$$ 5. **Simplify the right side fraction:** $$\frac{81}{99} = \frac{\cancel{9} \times 9}{\cancel{9} \times 11} = \frac{9}{11}$$ 6. **Rewrite the proportion:** $$\frac{q}{11} = \frac{9}{11}$$ 7. **Cross multiply to solve for $q$:** $$q \times 11 = 9 \times 11$$ 8. **Simplify both sides:** $$q \times \cancel{11} = 9 \times \cancel{11}$$ 9. **Result:** $$q = 9$$ **Final answer:** The missing length $q$ is 9 yards.