1. **Problem statement:** Given three vertices of a parallelogram $R(-2,-1)$, $S(2,1)$, and $T(0,-3)$, find the coordinates of the fourth vertex. 2. **Formula and rule:** In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal $RT$ is the same as the midpoint of diagonal $S$ and the unknown vertex $U$. We can use the midpoint formula: $$\text{Midpoint} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ 3. **Step 1: Calculate midpoint of diagonal $RT$:** $$\left(\frac{-2+0}{2}, \frac{-1+(-3)}{2}\right) = \left(\frac{-2}{2}, \frac{-4}{2}\right) = (-1, -2)$$ 4. **Step 2: Let the fourth vertex be $U(x,y)$. The midpoint of diagonal $SU$ must be $(-1,-2)$:** $$\left(\frac{2+x}{2}, \frac{1+y}{2}\right) = (-1, -2)$$ 5. **Step 3: Set up equations and solve for $x$ and $y$:** $$\frac{2+x}{2} = -1 \implies 2+x = -2 \implies x = -4$$ $$\frac{1+y}{2} = -2 \implies 1+y = -4 \implies y = -5$$ 6. **Answer:** The coordinates of the fourth vertex are $U(-4, -5)$. This is the only possible fourth vertex that completes the parallelogram with the given three vertices.