Pencil Turning
Rotating a pencil twice about two different points gives surprising results...
Age
14 to 16
Challenge level
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?
Image
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Answer 1:1
Image
Semicircles with centre B have radius $2a$ and $3a$:
Image
Image
C moves distance $2a\pi+a\pi = 3a\pi$