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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.

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

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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:


Two right-angled triangles are connected together as part of a structure.

two connected right-angled triangles


The green triangle has side lengths of 65, 52, and 39 cm and the blue triangle has side lengths of 52, 48, and 20 cm.

An object is dropped from the top of the green triangle, where does it pass the base of the blue triangle?

In what proportion is the blue base split and how far has the object fallen?

This problem does not require a calculator or any special formula.