Library / CEMC Cayley /442 of 470

CEMC Cayley, 2026

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 $PQ=11$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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