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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.

One and Three

Age 14 to 16 Challenge Level:

This problem stimulates students into creating some kind of diagram as an aid to visualisation. Hopefully the phrase "ratio of the speeds" will encourage students to think through how the simulation is not significantly altered if the speeds of both boats were, for example, doubled or halved. This is a useful problem for connecting speed, distance and time, and may also offer students a useful context in which to use algebra to express relationships containing known and unknown quantities.