Let's solve this problem step-by-step.

Step 1: Setting Up Known Information

We have:

• $AX = 12$
• $AX = 4 \cdot CX$

From $AX = 4 \cdot CX$, we can solve for $CX$: $AX = 4 \cdot CX \Rightarrow 12 = 4 \cdot CX \Rightarrow CX = \frac{12}{4} = 3$

Since $AX = 12$ and $CX = 3$, we can conclude that: $AC = AX + CX = 12 + 3 = 15$

Step 2: Using Right Triangle Properties

We know:

• $\angle ABC = \angle BXA = 90^\circ$

This implies that triangles $\triangle ABC$ and $\triangle BXA$ are both right triangles.

Step 3: Applying the Pythagorean Theorem in $\triangle BXA$

In $\triangle BXA$:

• $AX = 12$
• $\angle BXA = 90^\circ$

Let $BX = b$. By the Pythagorean Theorem: $AX^2 + BX^2 = AB^2$ However, we need additional information or relations between $BX$ and the other sides to fully determine its length without assumptions. Please confirm if you would like to proceed with additional clarifications or if there’s an element overlooked.

Would you like more details, or do you have any questions?

Here are five related questions that might extend this topic:

1. How would we solve for $BX$ if $\triangle ABC$ had a different configuration?
2. What properties can be derived if point $X$ was in a different location on $\overline{AC}$?
3. How would this problem change if $\angle BXA \neq 90^\circ$?
4. How can similar triangles help in solving for unknown lengths in geometric problems?
5. What other methods can confirm the solution, like trigonometric identities?

Tip: Always check if there are similar triangles in geometry problems, as they can simplify the solution by establishing proportional relationships.