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Problem 174

Geometry Difficulty 1.4 Multiple choice CEMC Gauss (Grade 7) · Canada · 2012

On the spinner shown, $PQ$ passes through centre $O$ .

If areas labelled $R$ and $S$ are equal, then what percentage of the time will a spin stop on the shaded region?

Pick one

Official solution

Since $PQ$ passes through centre $O$ , then it is a diameter of the circle.

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Since $\angle QOT=90\degree$ , then $\angle POT=180\degree-90\degree=90\degree$ .

Thus, the area of sector $POT$ is $\frac{90\degree}{360\degree}=\frac{1}{4}$ or 25% of the area of the circle.

Since the areas labelled $R$ and $S$ are equal, then each is

$25\%\div 2=12.5\%$ of the area of the circle.
Therefore, a spin will stop on the shaded region $12.5\%$ of the time.

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