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Problem 290

Geometry Difficulty 2.2 Multiple choice CEMC Fermat · Canada · 2019

In the diagram, $PQRS$ is a rectangle. Also, $\triangle STU$ , $\triangle UVW$ and $\triangle WXR$ are congruent.

What fraction of the area of rectangle $PQRS$ is shaded?

Pick one

Official solution

Suppose that $SU=UW=WR=b$ and $PS=h$ .

Since the width of rectangle $PQRS$ is $3b$ and its height is $h$ , then its area is $3bh$ .

Since $SU = b$ and the distance between parallel lines $PQ$ and $SR$ is $h$ , then the area of $\triangle STU$ is $\frac{1}{2}bh$ . Similarly, the area of each of $\triangle UVW$ and $\triangle WXR$ is $\frac{1}{2}bh$ .

Therefore, the fraction of the rectangle that is shaded is $\dfrac{3 \times \frac{1}{2}bh}{3bh}$ which equals $\dfrac{1}{2}$ .

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