1. **Problem statement:** We have a rectangle $ABCD$ with diagonal $\overline{AC} = 25$ cm. Point $O$ is the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$. We need to find the length of $\overline{AO}$. 2. **Key property:** In a rectangle, the diagonals are equal in length and they bisect each other. This means that the point of intersection $O$ divides each diagonal into two equal parts. 3. **Formula and explanation:** Since $O$ is the midpoint of diagonal $AC$, we have: $$\overline{AO} = \overline{OC} = \frac{1}{2} \overline{AC}$$ 4. **Calculation:** Substitute the given length: $$\overline{AO} = \frac{1}{2} \times 25 = 12.5$$ 5. **Answer:** The length of $\overline{AO}$ is $12.5$ cm. This result follows from the fundamental property of rectangles that their diagonals bisect each other, making $O$ the midpoint of $AC$.