### 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?

### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

### 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?

# Tilting Triangles

##### Age 14 to 16Challenge Level

A right-angled isosceles triangle whose two equal sides are 2 units in length is attached at its right-angled vertex to the centre of a square of side 2 units and rotated about this centre point.

What can you say about the area of the part of the square covered by the triangle as it rotates?

What happens to this area if the triangle is reduced in size so that its two equal sides are $\sqrt 2$ units?

What happens if the triangle is further reduced in size?