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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.

Circuit Training

Stage: 4 Challenge Level: Challenge Level:1

It is worth spending some time discussing the strategies pupils have adopted.

Another question might be:
What is the ratio of the speeds of the two runners if they are to meet at the start after $1$ lap, $2$ laps, $3$ laps... $n$ laps?.

What is the relationship betweent he ratio of the speeds of the runners and where they meet?