In triangle $ABC,$ $\angle C = 90^\circ.$ A semicircle is constructed along side $\overline{AC}$ that is tangent to $\overline{BC}$ and $\overline{AB}.$ If the radius of the semicircle is equal to $\frac{BC}{4},$ then find $\frac{AC}{BC}.$

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.

Explore a geometry problem involving a semicircle tangent to side AC in triangle ABC, where angle C is 90 degrees. Learn how to find the ratio AC/BC using circle tangency and geometric properties. Suitable for high school students.

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