1. **State the problem:** We are given three vertices of a parallelogram DEFG: $D(5, 2)$, $E(2, 6)$, and $F(-8, -3)$. We need to find the coordinates of the fourth vertex $G$. 2. **Formula and rule:** In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal $DF$ is the same as the midpoint of diagonal $EG$. 3. **Calculate midpoint of diagonal $DF$:** $$\text{Midpoint}_{DF} = \left(\frac{5 + (-8)}{2}, \frac{2 + (-3)}{2}\right) = \left(\frac{-3}{2}, \frac{-1}{2}\right) = \left(-1.5, -0.5\right)$$ 4. **Let the coordinates of $G$ be $(x, y)$. Calculate midpoint of diagonal $EG$:** $$\text{Midpoint}_{EG} = \left(\frac{2 + x}{2}, \frac{6 + y}{2}\right)$$ 5. **Set midpoints equal and solve for $x$ and $y$:** $$\frac{2 + x}{2} = -1.5 \implies 2 + x = -3 \implies x = -3 - 2 = -5$$ $$\frac{6 + y}{2} = -0.5 \implies 6 + y = -1 \implies y = -1 - 6 = -7$$ 6. **Final answer:** The coordinates of vertex $G$ are $(-5, -7)$.