1. **Problem statement:** We have triangle ACD with a smaller triangle ABE inside it. Line BE is parallel to line DC. Given lengths: AB = 12 cm, AC = 15 cm, BE = 8 cm. We need to find the length of line CD. 2. **Key concept:** When a line is drawn parallel to one side of a triangle, it creates similar triangles. Here, triangle ABE is similar to triangle ACD because BE \parallel DC. 3. **Similarity ratio:** The sides of similar triangles are proportional. So, $$\frac{AB}{AC} = \frac{BE}{CD}$$ 4. **Substitute known values:** $$\frac{12}{15} = \frac{8}{CD}$$ 5. **Solve for CD:** $$\frac{12}{15} = \frac{8}{CD} \implies 12 \times CD = 15 \times 8$$ $$12 \times CD = 120$$ $$CD = \frac{120}{12}$$ 6. **Simplify fraction:** $$CD = \cancel{\frac{120}{12}} = 10$$ 7. **Answer:** The length of line CD is 10 cm.