We were given the following true or false question in our Algebra exam
Q) The regular $5$-gon is not constructible by using a straight edge and compass.
The answer is false but we also had to provide an explanation. Followings was the explanation I gave
A regular $5$-gon is constructible since $5=2^{2}+1$ and for any prime $p$ if $p=2^{r}+1$ for some $ r\in Z$ then a regular $p$-gon can be constructed.
The above explanation I believe is the statement of a theorem by Gauss. But my professor didn't find the answer satisfactory and wrote the words "Why ?"around the theorem. I am assuming because the theorem was only stated and not proved in our class. How else can we show that a regular $5$-gon is constructible by using compass and straight edge ?
If you assume the side of the pentagon has length 1, then you can show that the length of a diagonal is the golden ratio $(1+\sqrt{5})/2$ by just drawing all the diagonals and comparing similar triangles.
Given a unit, construct a segment of length $2$ and then perpendicular to one endpoint, construct a segment of length $1$. Draw the hypotenuse and you have segment of length $\sqrt{5}$. Add one to that, then bisect and you have the golden ratio. Use it and two segments of length $1$ to make a triangle. Bisect the sides of length one and find the intersection of the bisectors. This is the center of the pentagon. Make a circle with this center and radius any vertex of the triangle. The three vertices of the triangle are also vertices of the pentagon. Just copy those lengths about the circle. Done.