Let's start by assigning labels to the sides of triangle $ABC$ where $\angle C = 90^\circ$. Let $AC = a$, $BC = b$, and $AB = c$. A semicircle is constructed along side $\overline{AC}$ such that it is tangent to $\overline{BC}$ and $\overline{AB}$. Given that the radius of the semicircle is equal to $\frac{b}{4}$, we need to find the ratio $\frac{a}{b}$.

Step 1: Analyze the geometric setup

The semicircle is tangent to both $\overline{BC}$ and $\overline{AB}$. Let the center of the semicircle be $O$, and the radius be $r = \frac{b}{4}$. Since the semicircle is along $\overline{AC}$, its diameter equals $AC = 2r$. Therefore: $a = 2r = 2 \times \frac{b}{4} = \frac{b}{2}$

Step 2: Calculate the ratio $\frac{a}{b}$

We now calculate the desired ratio $\frac{a}{b}$: $\frac{a}{b} = \frac{\frac{b}{2}}{b} = \frac{1}{2}$

Thus, the required ratio $\frac{AC}{BC}$ is $\boxed{\frac{1}{2}}$.

Conclusion

The solution has determined that the ratio $\frac{AC}{BC}$ is $\frac{1}{2}$.

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

Here are some related questions:

1. How would the solution change if the semicircle was along side $\overline{BC}$ instead of $\overline{AC}$?
2. What is the length of the hypotenuse $AB$ in terms of $b$?
3. How does the Pythagorean theorem apply to this problem?
4. What happens if the radius of the semicircle was instead $\frac{BC}{3}$?
5. How would the construction be affected if $\angle C$ was not $90^\circ$?

Tip: When dealing with problems involving tangents and circles, visualize the geometric configuration carefully and remember the properties of tangents to a circle from a point outside the circle.