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Geometry Difficulty 3.5 AMC 10/12 Find the answer Canada

The diameters $PQ$ , $RS$ and $TU$ of three identical semi-circles are placed along line $PU$ , as shown.If $PR=SU=8$ and $QT=5$ , then the length of $PQ$ is

Pick one

Solution

Suppose the diameter of each semi-circle is $x$ . That is, $PQ=RS=TU=x$ , and so $PU=PQ+QT+TU=x+5+x=2x+5$ . Also, $PU=PR+RS+SU=8+x+8=x+16$ . Thus, $2x+5=x+16$ or $x=11$ , and so $PQ=11$ .

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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.