# 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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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

**Answer** 1:1

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Semicircles with centre B have radius $2a$ and $3a$:

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C moves distance $2a\pi+a\pi = 3a\pi$