1. **Stating the problem:** We have a rectangle ABCD with points E and I outside it. Given lengths are $BE=4$ cm and $AI=8$ cm. We want to analyze the figure and find relationships or lengths involving points E and I. 2. **Understanding the setup:** Since ABCD is a rectangle, opposite sides are equal and all angles are right angles. Points E and I are outside the rectangle, with E above segment AB and I on segment AB. 3. **Using given lengths:** Since $AI=8$ cm and $BE=4$ cm, and I lies on AB, we can deduce the length of AB. Because $AI + IB = AB$ and $I$ lies on AB, if $AI=8$ cm and $BE=4$ cm, then $AB = AI + IB = 8 + 4 = 12$ cm. 4. **Summary:** The length of segment AB is $12$ cm based on the given points and lengths. Final answer: $AB = 12$ cm.