1. **Problem statement:** We have a right triangle ABC with a right angle at C. The hypotenuse AB is 13, side BC is 12, and side AC is 5. We need to find $\tan(\beta)$ where $\beta$ is the angle at vertex B. 2. **Recall the definition of tangent:** For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Identify sides relative to angle $\beta$:** - Opposite side to $\beta$ is AC (length 5). - Adjacent side to $\beta$ is BC (length 12). 4. **Calculate $\tan(\beta)$:** $$\tan(\beta) = \frac{AC}{BC} = \frac{5}{12}$$ 5. **Conclusion:** The value of $\tan(\beta)$ is $\frac{5}{12}$, which corresponds to answer choice A.