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The bisected equilateral triangle (30-60-90) is an important shape for students to become familiar with, and the early questions in this problem draw out the compound forms built from this triangle.
The final question, to which the other questions serve as an introduction, explores surd forms and the circumstances when one length can, or cannot, be a rational multiple of another.
The second of the key questions above is challenging, able students should be encouraged to reason this thoroughly.
The following problems make a natural follow on from this activity: The Square Hole and Impossible Triangles?
There are a number of activities which can provide valuable auxiliary experiences for students working on this problem :