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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:

This problem was originally created within a theme of trigonometry. Trigonometry can sometimes become routinised and leave students without a good feel for trig values as ratios. This problem is deliberately couched in ratio terms rather than Sine or Cosine expressions. Connecting up again with Sine and Cosine could be very valuable, especially if the extension at the end of these notes has been explored.

Increasing the scope of this enquiry, it may be useful to ask students whether the problem can be solved when any three of the eight ratios are given, or must it be this particular three ?

As a very enriching extension, the eight ratios can be grouped to produce a surprising result :

Make a group of four ratio values starting with 0.43, 0.88, and 0.62, then continue clock-wise to include the unknown ratio value which compares the left portion of the horizontal line with the hypotenuse of the green triangle. Form the product of these four ratios and compare that value with the product of the other four ratios.