1. **Problem statement:** We have triangle ABC with M as the midpoint of line segment AC. Given vectors: $$\overrightarrow{AB} = 8a - 6b$$ $$\overrightarrow{BC} = 20b$$ We need to find vector $$\overrightarrow{AM}$$ in terms of $$a$$ and $$b$$, fully simplified. 2. **Recall vector addition rule:** $$\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}$$ 3. **Calculate $$\overrightarrow{AC}$$:** $$\overrightarrow{AC} = (8a - 6b) + 20b = 8a + ( -6b + 20b ) = 8a + 14b$$ 4. **Since M is midpoint of AC,** $$\overrightarrow{AM} = \frac{1}{2} \overrightarrow{AC}$$ 5. **Calculate $$\overrightarrow{AM}$$:** $$\overrightarrow{AM} = \frac{1}{2} (8a + 14b) = 4a + 7b$$ **Final answer:** $$\overrightarrow{AM} = 4a + 7b$$