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Just Rolling Round

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 of P?

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Coke Machine

Stage: 4 Challenge Level: Challenge Level:1

This is another tough nut and perhaps the diagram of the 50p piece will help.

Fifty P
A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing the straight line DE with an arc centred at A and radius AE; replacing the straight line EF with an arc centred at B radius BF ...etc..

The 50p piece can roll in the same chute as a disc of radius $r$. Suppose the seven arcs forming the edge of the 50p piece (the arcs AB, BC etc. ) all have radius $R$ (where $R$=AD=AE=BE=BF...) then you need to find $R$ in terms of $r$. These seven arcs subtend angles of $2\pi /7$ at the centre of the disc and $2\pi /14$ at the opposite edge.