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Problem 502

Geometry Difficulty 2.5 Find the answer CEMC Cayley

The points $Q(1,-1), R(-1,0)$ and $S(0,1)$ are three vertices of a parallelogram. What could be the coordinates of the fourth vertex of the parallelogram?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

We plot the first three vertices on a graph. We see that one possible location for the fourth vertex, $V$ , is in the second quadrant. If $V S Q R$ is a parallelogram, then $S V$ is parallel and equal to $Q R$ . To get from $Q$ to $R$ , we go left 2 units and up 1 unit. Therefore, to get from $S$ to $V$ , we also go left 2 units and up 1 unit. Since the coordinates of $S$ are $(0,1)$ , then the coordinates of $V$ are $(0-2,1+1)=(-2,2)$ . This is choice (A).

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.