### The Line and Its Strange Pair

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

### Mapping the Wandering Circle

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

### Like a Circle in a Spiral

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

# Symmetric Trace

##### Stage: 4 Challenge Level:
Pranav from Vardhana School had a good suggestion : copy the image from the NRICH webpage into Paint and then make a copy you can rotate and compare.

Also if you copy into Word, and rotate a copy, you could use one of the drawing tool to trace over one of the curves you want to compare and then move that new line around to see if it fits over the rotated copy.

Trace 2 does not line up with itself upside down. This can be spotted by looking at the lowest point on the line, the curve is sharper than at the highest point on the line, and so when rotated these two can not line up.

Trace 3 on the other hand does line up with itself upside down. This can be spotted by the fact that when you rotate the graph and slide it along it lines up. Another method is to draw a line at the top:

You can now notice that when the wheel has turned 180 degrees, then the rotated diagram is the same as the one above, and so the rotated graph does line up with itself.