1. **State the problem:** Find the missing length in the first triangle proportionality problem where the given lengths are 10, 4, 6, and the missing length is ?. The proportion is set as $\frac{6}{?} = \frac{6}{9}$. 2. **Formula and rules:** The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. This means the ratios of the corresponding segments are equal: $$\frac{segment_1}{segment_2} = \frac{segment_3}{segment_4}$$ 3. **Set up the proportion:** Given $\frac{6}{?} = \frac{6}{9}$, we want to solve for ?. 4. **Cross multiply:** $$6 \times 9 = 6 \times ?$$ $$54 = 6?$$ 5. **Solve for ?:** $$? = \frac{54}{6}$$ 6. **Simplify the fraction:** $$? = \cancel{\frac{54}{6}} = 9$$ 7. **Answer:** The missing length is $9$.