### 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

##### Age 14 to 16 Challenge Level:

Lukaz Smith, from Parkside School, sent in the following correct approach to finding the ratio of the Earth-Sun and Venus-Sun distances (diagram added to help clarify the solution)

If you said the Earth to sun distance was 1 astronomical unit ($a$ in diagram), then you could simply measure the angle ($\theta$ in diagram) between Sun, Earth and Venus, when you knew that there was a 90 degree angle between Earth, Sun and Venus. Then you just find the tangent of the angle, so it's 1 : $\tan \theta$ as the ratio of Earth to Sun distance and Venus to Sun distance.