Triangle Side Ratio: Comparing AB and DE in Similar Triangles

To solve the problem, we need to determine the ratio of lengths between sides $AB$ and $DE$ in triangles $\triangle ABC$ and $\triangle CDE$, using their similarity.

Given:

• Triangle $\triangle ABC$ and triangle $\triangle CDE$ are similar by AA criterion, having common angle $C$ and both right triangles.
• The vertical height from $B$ to $A$ is $3$, while the vertical height from $D$ to $C$ is $12$.
• The horizontal length from $A$ to $C$ is $2$, and from $C$ to $E$ is $8$.

To find the similarity ratio, we can compare corresponding segments:

• The vertical height ratio is $\frac{BD}{DE} = \frac{3}{12} = \frac{1}{4}$.
• The horizontal base ratio is $\frac{AC}{CE} = \frac{2}{8} = \frac{1}{4}$.

Thus, the triangles are similar with a ratio of $\frac{1}{4}$.

Since all corresponding dimensions of similar triangles are proportional by this ratio, it follows:

• $\frac{AB}{DE} = \frac{1}{4}$.

Therefore, the solution to the problem is: $\frac{AB}{DE} = \frac{1}{4}$.

The correct answer choice is: $\frac{1}{4}$.