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Problem 330

Geometry Difficulty 2.4 Multiple choice CEMC Fermat · Canada · 2013

$PQRS$ is a square. The midpoint of $PQ$ is $M$ and the midpoint of $RS$ is $N$ . If the perimeter of rectangle $PMNS$ is 36, the area of square $PQRS$ is

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Official solution

Suppose that the side length of square $PQRS$ is $x$ .

Then $PQ=QR=RS=SP=x$ .

Since $M$ is the midpoint of $PQ$ , then $PM = \frac{1}{2}x$ .

In terms of $x$ , the perimeter of rectangle $PMNS$ is $2(PM + PS) = 2(\tfrac{1}{2}x+x)=3x$

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(Note that $SN = PM = \frac{1}{2}x$ since $N$ is the midpoint of $RS$ . Also, $MN=PS = x$ , since $MN$ is parallel to $PS$ and joins two parallel line segments.)

Since we are told that the perimeter of $PMNS$ is 36, then $3x=36$ or $x=12$ .

Therefore, the area of square $PQRS$ is $x^2=144$ .

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