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The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

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Golden Mathematics

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.


Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2
The angles are all multiples of $36^o$ and the figure has many symmetries. Form a quadratic equation by equating ratios of corresponding sides in the similar triangles.

In explaining the construction look for $\sqrt 5$ appearing from triangle $YOS$ and the golden ratio appearing as the radius of circle $C_5$ (see Note). Concentrate on cosines which, in this case, are easier than the sines to find exactly.