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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.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?


A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Eight Ratios

Age 14 to 16 Challenge Level:

Because the data for this problem only mentions ratio, not length, you can imagine the diagram being scaled up or down without any of the ratio values changing. So choose one length somewhere as something convenient and then try to calculate other lengths from that.