Pencil turning
Rotating a pencil twice about two different points gives surprising results...
Problem
A pencil AB lying on a table is given a half turn about the end B (so that A moves to A*) and then a half turn about A* (so that B moves to B*). The point C on the pencil is one third of the way from A to B.
What is the ratio of the total distances moved by A and by C?
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![Pencil Turning Pencil Turning](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob7-pencil%252520turning%252520problem.png?itok=wgiAWRMM)
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer 1:1
Image
![Pencil Turning Pencil Turning](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob7-pencil%252520turning%252520sol%2525201.png?itok=vwBXTtFa)
Semicircles with centre B have radius $2a$ and $3a$:
Image
![Pencil Turning Pencil Turning](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob7-pencil%252520turning%252520sol%2525202.png?itok=xmHTXC5A)
Image
![Pencil Turning Pencil Turning](/sites/default/files/styles/large/public/thumbnails/content-04-weekly-prob7-pencil%252520turning%252520sol%2525203.png?itok=MEaYNnMn)
C moves distance $2a\pi+a\pi = 3a\pi$