# Geometry and Algebra - October 2006, All Stages

## Problems

### Tessellating Capitals

##### Stage: 1 Challenge Level:

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

### Four-triangle Arrangements

##### Stage: 1 Challenge Level:

How many different shapes can you make by putting four right- angled isosceles triangles together?

### Coordinate Challenge

##### Stage: 2 Challenge Level:

Use the clues about the symmetrical properties of these letters to place them on the grid.

### Tubular Path

##### Stage: 2 Challenge Level:

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

### A Maze of Directions

##### Stage: 2 Challenge Level:

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

### Escher Tessellations

##### Stage: 2 Challenge Level:

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

### Decoding Transformations

##### Stage: 3 Challenge Level:

See the effects of some combined transformations on a shape. Can you describe what the individual transformations do?

### Combining Transformations

##### Stage: 3 Challenge Level:

Does changing the order of transformations always/sometimes/never produce the same transformation?

### Simplifying Transformations

##### Stage: 3 Challenge Level:

How many different transformations can you find made up from combinations of R, S and their inverses? Can you be sure that you have found them all?

### 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?

##### Stage: 5 Challenge Level:

Find a quadratic formula which generalises Pick's Theorem.

### Proof of Pick's Theorem

##### Stage: 5 Challenge Level:

Follow the hints and prove Pick's Theorem.