1. **State the problem:** We are given point T with coordinates $(0,6)$ and the midpoint $M$ of segment $ST$ as $(3,-4)$. We need to find the coordinates of point $S$. 2. **Recall the midpoint formula:** The midpoint $M$ of a segment with endpoints $S(x_1,y_1)$ and $T(x_2,y_2)$ is given by: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ 3. **Set up equations using the midpoint coordinates:** Given $M = (3,-4)$ and $T = (0,6)$, we have: $$3 = \frac{x_1 + 0}{2}$$ $$-4 = \frac{y_1 + 6}{2}$$ 4. **Solve for $x_1$ and $y_1$:** Multiply both sides by 2: $$2 \times 3 = x_1 + 0 \Rightarrow 6 = x_1$$ $$2 \times (-4) = y_1 + 6 \Rightarrow -8 = y_1 + 6$$ 5. **Isolate $y_1$:** $$y_1 = -8 - 6 = -14$$ 6. **Final answer:** The coordinates of point $S$ are: $$S = (6, -14)$$