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Problem 154

Geometry Difficulty 1.2 Multiple choice CEMC Fermat · Canada · 2015

The points $P(3,-2)$ , $Q(3,1)$ , $R(7,1)$ , and $S$ form a rectangle. What are the coordinates of $S$ ?

Pick one

Official solution

Based on the diagram shown, the order of the vertices of the rectangle must be $PQRS$ .

(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 $(7,-2)$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.