P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by
replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
We believed that the triangle took up a quarter of the square,
and that a total of four triangles could fit around the square. We
created a moving example:
We started by rotating a square inside the four triangles as
this has the same effect as rotatiing the triangle (editors note:I
"liked this bit of lateral thinking").
From our "moving" representation (Fig. 1) we could see that it
is always possible to fit four right angled triangles around the
centre of the square. This is because the centre of the square
allows a 360° rotation and, as the traingles are right
angled, they have angles of 90° (360 / 90 = 4).
As can be seen the square in the
middle of the four triangles is 2 units by 2 units. This means that
the overlap of each of the four triangles is congruent and makes up
a quarter of the square.