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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.

Rhombus in Rectangle

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

Call the base of the rectangle $b$ and the height $h$. If the distance from A to P is $x$, can you write an equation linking $b$, $h$ and $x$ in order to make $AP=PQ$? How many possible values can x take in your equation?