To solve the question:

The proportion provided is:

$\frac{AB}{AC} = \frac{AD}{?}$

Key observations:

1. The triangle $\triangle ABC$ is a right triangle.
2. $D$ is a point on $AC$, making $\triangle ABD$ and $\triangle ADC$ smaller right triangles inside $\triangle ABC$.
3. The question involves similarity of triangles. Since $\triangle ABD \sim \triangle ABC$ (by AA similarity due to shared angle $\angle A$ and right angles), the corresponding sides will be proportional.

Step-by-step reasoning:

1. Using the similarity between $\triangle ABD$ and $\triangle ABC$, the side ratios are: $\frac{AB}{AC} = \frac{AD}{BC}.$
2. Comparing this with the given proportion: $\frac{AB}{AC} = \frac{AD}{?},$ it follows that the missing side must be $BC$.

Final Answer:

$BC$.

Would you like a detailed explanation of similar triangles, or do you have any questions?

Related Questions:

1. What are the criteria for triangle similarity?
2. How can proportions be used to solve geometric problems involving similar triangles?
3. How does the Pythagorean theorem relate to this triangle?
4. Why does the proportionality hold for corresponding sides of similar triangles?
5. How would the answer change if $D$ were not on $AC$?

Tip:

Always check if triangles in a diagram are similar. Similarity helps establish proportional relationships between corresponding sides, simplifying many problems.