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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.

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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.

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Star Gazing

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

Slippage

Age 14 to 16 Challenge Level:

There are a number of routes towards the solution of this problem. Sharing different strategies, such as scale drawing, use of spreadsheets or dynamic geometry software and the use of iterative methods, seem a useful opportunity for discussion about the merits of different approaches.