# Geometry and Algebra - October 2006, Stage 4&5

## Problems

### Points in Pairs

##### Stage: 4 Challenge Level:

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

### The Line and Its Strange Pair

##### Stage: 4 Challenge Level:

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

##### Stage: 4 Challenge Level:

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?

### Intersections

##### Stage: 4 and 5 Challenge Level:

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

### Lattice Points

##### Stage: 5 Challenge Level:

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?