1. **Problem statement:** In parallelogram ABCD, point M is the midpoint of side BC. Given vectors $\overline{AB} = \vec{a}$ and $\overline{AD} = \vec{b}$, express vectors $\overline{AM}$ and $\overline{MD}$ through $\vec{a}$ and $\vec{b}$. 2. **Recall vector properties in a parallelogram:** - $\overline{AB} = \vec{a}$ - $\overline{AD} = \vec{b}$ - $\overline{BC} = \overline{AD} = \vec{b}$ (opposite sides are equal and parallel) - $\overline{AC} = \vec{a} + \vec{b}$ 3. **Find vector $\overline{AM}$:** - Point M is midpoint of BC, so $\overline{BM} = \frac{1}{2} \overline{BC} = \frac{1}{2} \vec{b}$. - Vector $\overline{AM} = \overline{AB} + \overline{BM} = \vec{a} + \frac{1}{2} \vec{b}$. 4. **Find vector $\overline{MD}$:** - Vector $\overline{MD} = \overline{MD} = \overline{MB} + \overline{BD}$. - Since $\overline{MB} = -\overline{BM} = -\frac{1}{2} \vec{b}$. - Vector $\overline{BD} = \overline{BA} + \overline{AD} = -\vec{a} + \vec{b}$. - So, $\overline{MD} = -\frac{1}{2} \vec{b} + (-\vec{a} + \vec{b}) = -\vec{a} + \frac{1}{2} \vec{b}$. **Final answers:** $$\overline{AM} = \vec{a} + \frac{1}{2} \vec{b}$$ $$\overline{MD} = -\vec{a} + \frac{1}{2} \vec{b}$$