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Problem 423

Geometry Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2017

Two sheets of $11\text{ cm}\times 8\text{ cm}$ paper are placed on top of each other, forming an overlapping $8\text{ cm}\times 8\text{ cm}$ square in the centre, as shown.

The area of rectangle $WXYZ$ is

Pick one

Official solution

In the diagram, rectangles $WQRZ$ and $PXYS$ are the two sheets of 11 cm $\times$ 8 cm paper.

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The overlapping square $PQRS$ has sides of length 8 cm.

That is, $WQ=ZR=PX=SY=11$ cm and

$WZ=QR=PS=XY=PQ=SR=8$ cm.

In rectangle $WQRZ$ , $WQ=WP+PQ=11$ cm and so

$WP=11-PQ=11-8=3$ cm.

In rectangle $WXYZ$ , $WX=WP+PX=3+11=14$ cm.

Since $WX=14$ cm and $XY=8$ cm, the area of $WXYZ$ is

$14\times8=112$ cm $^2$ .

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