### 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?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

# Trigonometric Protractor

##### Stage: 4 Challenge Level:

Draw a right-angled triangle.

Click and drag near the centre of the protractor to move its centre over a vertex of a triangle (not at the right angle).
Click and drag on the inner circle to rotate it until the blue dot aligns with one of the adjacent sides.
Click and drag the blue dot until it aligns with the hypotenuse.
• What do the numbers refer to?
Click and drag the edge of the protractor to change its radius.
• Why don't the numbers change?
• How might this protractor be useful for finding the lengths of sides of different triangles?
• What do you notice about the numbers when the protractor is placed over each of the two acute angles? Can you explain what you see?
• What is the connection between the "lengths" of the two adjacent sides and the "length" of the hypotenuse?
• Do these things remain the same for any triangle? Can you explain why?

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