In the diagram, we have two triangles: triangle $ABC$ and triangle $ADE$. These triangles are similar, and the task is to find the length of $AC$.

Step-by-Step Explanation:

1. Similarity of Triangles: Since the triangles $ABC$ and $ADE$ are similar, the corresponding sides of these triangles are proportional.

2. Given Data:

   • $AB = 24$
   • $AD = 9$ (since the distance from $A$ to $D$ is 9 units)
   • $DC = 15$ (as shown on the diagram)

3. Setting Up Proportions: Using the property of similar triangles, we can set up the following ratio between the corresponding sides of triangles $ABC$ and $ADE$:

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

   $AC = AD + DC = 9 + 15 = 24$. So $AC$ is the entire length from $A$ to $C$.

4. Conclusion: Therefore, the length of $AC$ is 24 units.

Would you like any further explanation or additional details?

Here are five related questions to explore further:

1. How can you prove that two triangles are similar?
2. What other properties can be derived from similar triangles?
3. How can the Pythagorean theorem be applied to right triangles like $ABC$?
4. How do you calculate the area of triangle $ABC$ using known side lengths?
5. How can we find unknown angles using the properties of similar triangles?

Tip: Always check for corresponding sides and angles when dealing with similar triangles; this helps in setting up correct proportions!