1. **Stating the problem:** We have two similar triangles, $\triangle ABC \sim \triangle DBE$. Given side lengths are $AC=10$, $DE=4$, and $BE=3$. We need to find the length of $AB$. 2. **Recall similarity properties:** Similar triangles have corresponding sides proportional. That means: $$\frac{AB}{DB} = \frac{BC}{BE} = \frac{AC}{DE}$$ 3. **Identify corresponding sides:** Since $\triangle ABC \sim \triangle DBE$, the vertices correspond as $A \leftrightarrow D$, $B \leftrightarrow B$, and $C \leftrightarrow E$. 4. **Use the ratio involving $AC$ and $DE$:** $$\frac{AC}{DE} = \frac{10}{4} = 2.5$$ 5. **Use the ratio involving $BE$ and $BC$:** Since $BE$ corresponds to $BC$, and $BE=3$, then $$BC = 3 \times 2.5 = 7.5$$ 6. **Use the Pythagorean theorem in $\triangle ABC$ to find $AB$:** $$AB = \sqrt{AC^2 + BC^2} = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25}$$ 7. **Calculate the square root:** $$AB = 12.5$$ **Final answer:** $$\boxed{12.5}$$ units