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Problem 338

Geometry Difficulty 2.4 Multiple choice CEMC Pascal · Canada · 2022

In the diagram, points $P$ ,
$Q$ , $R$ , and $S$ are at intersections of gridlines in a
$6 \times 6$ grid.

What is the perimeter of parallelogram $PQRS$ ?

Pick one

Official solution

Since the given grid is $6 $\times$
6, the size of each of the small squares is 1 $\times$ 1$.

This means that $QR = PS = 3$ .

Join $Q$ to $S$ .

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Since $QS$ is vertical, and $QR$ and $PS$ are both horizontal, then $\angle RQS = 90^\circ$ and $\angle PSQ = 90^\circ$ .

We note further that $QS = 4$ .

Since $\triangle RQS$ is right-angled
at $Q$ , by the Pythagorean Theorem,
$RS^2 = QR^2 + QS^2 = 3^2 + 4^2 = 25$ Since $RS>0$ , then
$RS = 5$ .

Similarly, $PQ = 5$ .

Thus, the perimeter of $PQRS$ is
$PQ + QR + RS + PS = 5 + 3 + 5 + 3 =
16$.

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