Rectangle Fourth Vertex 48Bafc

1. **State the problem:** We are given three vertices of a rectangle: $A(-1,4)$, $B(2,4)$, and $C(-1,-2)$. We need to find the coordinates of the fourth vertex $D$. 2. **Recall the properties of a rectangle:** Opposite sides are parallel and equal in length, and adjacent sides are perpendicular. 3. **Identify the given points:** - $A(-1,4)$ - $B(2,4)$ - $C(-1,-2)$ 4. **Find the vector from $A$ to $B$:** $$\overrightarrow{AB} = (2 - (-1), 4 - 4) = (3, 0)$$ 5. **Find the vector from $A$ to $C$:** $$\overrightarrow{AC} = (-1 - (-1), -2 - 4) = (0, -6)$$ 6. **Since $D$ is the fourth vertex, it can be found by adding vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ to point $A$:** $$D = A + \overrightarrow{AB} + \overrightarrow{AC} = (-1,4) + (3,0) + (0,-6) = (-1 + 3 + 0, 4 + 0 - 6) = (2, -2)$$ 7. **Check if $D$ matches one of the given options:** $(2, -2)$ is one of the options. **Final answer:** The coordinates of the fourth vertex are $\boxed{(2, -2)}$.