1. **State the problem:** We have points $P(a,b)$ and midpoint $M(c,d)$ of segment $PQ$. We want to find expressions for the distances $PM$ and $PQ$. 2. **Recall the distance formula:** The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Expression for $PM$:** Using the distance formula between $P(a,b)$ and $M(c,d)$, $$PM = \sqrt{(a - c)^2 + (b - d)^2}$$ 4. **Relationship between $P$, $Q$, and $M$:** Since $M$ is the midpoint of $PQ$, the coordinates of $M$ are $$c = \frac{a + x}{2}, \quad d = \frac{b + y}{2}$$ where $Q$ has coordinates $(x,y)$. 5. **Express $Q$ in terms of $P$ and $M$:** Solving for $x$ and $y$, $$x = 2c - a, \quad y = 2d - b$$ 6. **Expression for $PQ$:** Using the distance formula between $P(a,b)$ and $Q(x,y)$, $$PQ = \sqrt{(x - a)^2 + (y - b)^2} = \sqrt{(2c - a - a)^2 + (2d - b - b)^2} = \sqrt{(2c - 2a)^2 + (2d - 2b)^2}$$ 7. **Simplify $PQ$:** $$PQ = \sqrt{4(c - a)^2 + 4(d - b)^2} = \sqrt{4[(c - a)^2 + (d - b)^2]} = 2\sqrt{(c - a)^2 + (d - b)^2}$$ 8. **Relate $PQ$ to $PM$:** Note that $$PM = \sqrt{(a - c)^2 + (b - d)^2} = \sqrt{(c - a)^2 + (d - b)^2}$$ so $$PQ = 2PM$$ **Final answers:** $$PM = \sqrt{(a - c)^2 + (b - d)^2}$$ $$PQ = 2\sqrt{(a - c)^2 + (b - d)^2}$$