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CEMC Gauss (Grade 7), 2026

Geometry Difficulty 3.7 AMC 10/12 Find the answer Canada

In the diagram, $PQRS$ is a
square with $RS=30~\text{cm}$ . $TUVW$ is a square with $U$ and $V$ on $PS$ so that $PU=UV=VS$ .

Point $X$ is on $QR$ so that $TX$ is parallel to $WR$ . The area of parallelogram $TXRW$ is

Pick one

Solution

$PQRS$ is a square with $PS=RS=30\text{ cm}$ .

Since $PU+UV+VS=PS=30\text{ cm}$ and
$PU=UV=VS$ , then $UV=\dfrac{30\text{ cm}}{3}=10\text{ cm}$ .

$TUVW$ is a square with $UT=TW=VW=UV=10\text{ cm}$ .

$TXRW$ is a parallelogram, and so
$XR=TW=10\text{ cm}$ .

The height of $TXRW$ is the vertical
distance between its two parallel and horizontal sides $TW$ and $XR$ . This vertical height is equal to
$RS+VW=30\text{ cm}+10\text{ cm}=40\text{ cm}$ .

The area of parallelogram $TXRW$ is
the product of its base and its height, which is equal to $10\text{ cm}\times40\text{ cm}=400\text{ cm}^2$ .

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