In the diagram, the points $(2,1),(4,1)$ and $(2,5)$ are three vertices of a rectangle. What are the coordinates of the fourth vertex of the rectangle?
(A) $(5,2)$
(B) $(4,4)$
(C) $(1,5)$
(D) $(4,5)$
(E) $(2,4)$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-003.jpg?height=341&width=377&top_left_y=1331&top_left_x=1313)
Suppose we call the unknown vertex $V$.

The side of the rectangle joining the points $(2,1)$ and $(2,5)$ is vertical, and so the opposite side of the rectangle (the side joining $(4,1)$ to $V$ ) must also be vertical.

This means that $V$ has the same $x$-coordinate as $(4,1)$, which is 4 .

Similarly, the side of the rectangle joining the points $(2,1)$ and $(4,1)$ is horizontal, and so the opposite side of the rectangle (the side joining $(2,5)$ to $V$ ) must also be horizontal.

This means that $V$ has the same $y$-coordinate as $(2,5)$, which is 5 .

Therefore, the coordinates of the fourth vertex of the rectangle are $(4,5)$.

ANSWER: (D)