Pentagon Construction

Why do this problem?

This problem provides an excellent opportunity for students to practise using dynamic geometry tools such as GeoGebra as well as introducing an interesting construction they may not have seen before. The proof that the construction yields a regular pentagon is quite challenging and will require students to persevere through a tricky blend of coordinate geometry and trigonometry.

Possible approach

This problem works best if students have access to their own devices with GeoGebra. For students who are complete novices, the article Using GeoGebra would provide useful pre-session reading. 

Once they have followed each step to complete their pentagon, they may find it useful to print it out in order to annotate it with what they know and what they can deduce. The goal is to calculate the relevant angles based on lengths that can be deduced using coordinate geometry.

Key questions

If it is a regular pentagon, which angles need to be $72^{\circ}$ or $144^{\circ}$?
How do the identities $\cos 72^\circ = \frac{\sqrt{5}-1}4$ and 
$\cos 144^\circ = \frac{-1-\sqrt{5}}4$ help?

Possible extension

Students could explore the derivation of the identities for $\cos 72^{\circ}$ and $\cos 144^{\circ}$.

Possible support

Students could focus on trying to prove that the side lengths of the pentagon are equal.