1. **Stating the problem:** We have a triangle with points A, B, C and a segment DE parallel to side AB. Given lengths are AB = 12, CB = 18, and DE is parallel to AB. We want to find the length DE using Thales' theorem. 2. **Formula and theorem:** Thales' theorem states that if a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally. Mathematically, if DE \parallel AB, then: $$\frac{CD}{CB} = \frac{CE}{CA} = \frac{DE}{AB}$$ 3. **Given values:** - AB = 12 - CB = 18 - DE is parallel to AB - From the problem, DE is calculated as: $$DE = \frac{19.5 \times 5}{7.5}$$ 4. **Calculation:** Calculate DE: $$DE = \frac{19.5 \times 5}{7.5} = \frac{97.5}{7.5} = 13$$ 5. **Explanation:** We used the proportionality of segments created by the parallel line DE to AB. By substituting the known lengths into the proportion, we solved for DE. **Final answer:** $$DE = 13$$