1. **State the problem:** We are given two triangles ABC and DEC where AB is parallel to DE, and the triangles are similar. We need to find the length of AB. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{AB}{DE} = \frac{BC}{EC} = \frac{AC}{DC}$$ 3. **Identify known lengths:** - $DE = 10.5$ cm - $BC = 8$ cm - $AC = 10$ cm - $DC = 15$ cm - $AB = ?$ - $EC = ?$ (unknown but not needed directly) 4. **Use the ratio involving $AC$ and $DC$ to find the scale factor:** $$\frac{AC}{DC} = \frac{10}{15} = \frac{2}{3}$$ 5. **Since the triangles are similar, the ratio $\frac{AB}{DE}$ must be the same:** $$\frac{AB}{10.5} = \frac{2}{3}$$ 6. **Solve for $AB$:** $$AB = 10.5 \times \frac{2}{3}$$ 7. **Calculate:** $$AB = 10.5 \times \frac{2}{3} = \cancel{10.5} \times \frac{2}{\cancel{3}} = 3.5 \times 2 = 7$$ 8. **Final answer:** $$\boxed{7 \text{ cm}}$$ The length of AB is 7 cm.