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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:



We contruct a new triangle as shown. Angle $x$ is equal to $90^\circ - (90^\circ - y) = y$, so the highlighted triangles are similar. Therefore $$p = 48 \times \frac{39}{52} = 36$$ $$q =20\times \frac{39}{52} = 15$$


The base is split $48 - q : q = 33 : 15 = 11 : 5$ and the ball falls a distance of $p + 20 = 36 + 20 = 56$.