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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:1

2identity 3rotation subgroup 4reflection subgroup
List the symmetries in each of the four diagrams, that is the transformations which map the picture into itself.

What transformations do you get if you combine the transformations from diagrams 3 and 4 in all possible ways?

A set of transformations, with the operation of combining transformations, becomes a group when the set includes all possible combinations of the transformations in the set.

Do the sets of transformations you have listed form groups when their transformations are combined?

Can you find any other groups of transformations which map a regular pentagon to itself?

How many distinctly different groups are there which map a regular pentagon into itself?

Note: You don't need to do more in order to submit a solution but why not pose some similar questions for yourself? The article Paint Rollers for Frieze Patterns explores in 3D the same idea of groups of symmetries.