Problem 144

Based on the diagram shown, the order of the vertices of the rectangle must be $PQRS$. [[IMAGE0]] (In any other configuration, either $\angle QRP$ or $\angle QPR$ would be an angle in the rectangle, which is not possible.) Since $P$ and $Q$ have the same $x$-coordinate, then side $PQ$ of the rectangle is vertical. This means that side $SR$ must also be vertical, and so the $x$-coordinate of $S$ is the same as the $x$-coordinate of $R$, which is 7. Since $Q$ and $R$ have the same $y$-coordinate, then side $QR$ of the rectangle is horizontal. This means that side $PS$ must also be horizontal, and so the $y$-coordinate of $S$ is the same as the $y$-coordinate of $P$, which is $-2$. Thus, the coordinates of $S$ are $(7,-2)$.