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 analyze the problem step by step. Given a right triangle $ ABC $ with $ \angle C = 90^\circ $, we are to find the ratio $ \frac{AC}{BC} $ under the condition that a semicircle with radius $ r = \frac{BC}{4} $ is constructed on side $ \overline{AC} $, and this semicircle is tangent to both $ \overline{BC} $ and $ \overline{AB} $.

### Step 1: Establish Relations in the Triangle
We are given that $ r = \frac{BC}{4} $. Let:
- $ AB = c $
- $ BC = a $
- $ AC = b $

Since the semicircle is tangent to both $ BC $ and $ AB $, and the radius is $ \frac{a}{4} $, the distance from $ C $ to the point of tangency on $ AB $ is equal to the radius of the semicircle.

### Step 2: Express the Length of $ AC $ in Terms of $ a $
The radius of the semicircle $ r $ is equal to $ \frac{a}{4} $. The semicircle is tangent to both sides $ BC $ and $ AB $, so the center of the semicircle is at a distance $ \frac{a}{4} $ from $ BC $ and from $ AB $.

This setup means that $ b - \frac{a}{4} $ is the distance from the point $ C $ (along side $ \overline{AC} $) to the center of the semicircle, making the length of $ b $ equal to the diameter of the semicircle plus the extra length beyond the semicircle's radius:

\[
b = \frac{a}{2} + \frac{a}{4} = \frac{3a}{4}
\]

### Step 3: Compute $ \frac{AC}{BC} $
We are asked to find $ \frac{AC}{BC} = \frac{b}{a} $:

\[
\frac{AC}{BC} = \frac{b}{a} = \frac{\frac{3a}{4}}{a} = \frac{3}{4}
\]

Thus, the ratio $ \frac{AC}{BC} $ is $ \boxed{\frac{3}{4}} $.

Would you like more details, or do you have any questions? Here are some additional questions to deepen your understanding:

1. How do you find the relationship between $ AC $ and $ BC $ in a right triangle?
2. How can you use the properties of tangents to determine the position of the semicircle?
3. What would change in the problem if the semicircle was constructed along side $ \overline{BC} $?
4. How does the Pythagorean Theorem relate to this problem?
5. What other methods could be used to solve this problem?

**Tip:** When dealing with geometric problems involving tangents, always consider the radii of the circles and their relation to the sides of the triangle for a clearer analysis.

Explore how to find the ratio AC/BC in a right triangle with a semicircle constructed on side AC. The semicircle is tangent to sides AB and BC with a radius equal to BC/4. Detailed step-by-step solution and geometric analysis provided.

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