1. **Problem statement:** Find $\cos(\alpha)$ in the right triangle with sides $BC=7$, $AC=24$, and hypotenuse $AB=25$, where $\alpha$ is the angle at vertex $A$. 2. **Formula:** In a right triangle, $\cos(\alpha) = \frac{\text{adjacent side to } \alpha}{\text{hypotenuse}}$. 3. **Identify sides:** Angle $\alpha$ is at vertex $A$. The side adjacent to $\alpha$ is $AC=24$, and the hypotenuse is $AB=25$. 4. **Calculate:** $$\cos(\alpha) = \frac{24}{25}$$ 5. **Conclusion:** The value of $\cos(\alpha)$ is $\frac{24}{25}$. **Answer:** A 24/25