### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# A Scale for the Solar System

##### Stage: 4 Challenge Level:

This accessible problem will hopefully lead students into the whole area of solar and astronomical scales. The internet is full of sites offering information. Google anything and you'll start a trail of enquiry.

The methods often take some thinking about and nearly all of them require the imagination of different viewpoints. Grasping the methods and especially the required adjustment of perspective will not be immediate for most students and serves as a useful reminder that mathematical understanding isn't an "instant" or "not-at-all" experience.