1. **State the problem:** We are given the midpoint $M(3,4)$ of points $A(2,a)$ and $B(b,12)$, and we need to find the values of $a$ and $b$. 2. **Formula for midpoint:** The midpoint $M$ of two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ 3. **Apply the formula:** Using the given points, $$3 = \frac{2 + b}{2} \quad \Rightarrow \quad 2 \times 3 = 2 + b \quad \Rightarrow \quad 6 = 2 + b \quad \Rightarrow \quad b = 4$$ $$4 = \frac{a + 12}{2} \quad \Rightarrow \quad 2 \times 4 = a + 12 \quad \Rightarrow \quad 8 = a + 12 \quad \Rightarrow \quad a = -4$$ 4. **Final answer:** The values are $a = -4$ and $b = 4$.