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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?

Contact

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?

Wedge on Wedge

Age 14 to 16 Challenge Level:

This problem, based on a structure that includes a compound angle, could be a standard post-16 trigonometry question, but offered here, with side lengths selected to make ratio calculations easy, we hope to encourage students to explore the construction on which the compound angle formulae rest.

The numbers were chosen to make calculation as easy as possible. It seemed that once students reached for a calculator to help them multiply or divide, they would think that the SIN, COS or TAN buttons were the answer here and this is not a terribly useful directon to take.

But there may be value in asking students how they might solve the problem with numbers that they make up for themselves (or for each other). Can they describe ageneral strategy? It is also useful at this point to ask whether all the side length data is necessary.

A short distance beyond Stage 4 students will have a range of formulae to apply to problems like this, our aim at this stage is to help them spend some time exploring the constructions that make those formulae possible or valid.