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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.

Pythagoras’ Comma

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Hint : What fraction is 2/3 of 2/3 of 2/3 . . . twelve times ?

And you've had to double 7 times, so what's the fraction now?

Maybe work in decimals and give the error as a percentage.