Question

The construction of the regular pentagon is equivalent to the construction of the point in the unit circle with x coordinate x1 = the square root of 5 - 1 divided by 4. Use this to construct a regular pentagon that has all its vertices inside the unit circle. Write a construction protocol and justify all your construction steps. How do I construct the square root of 5-1?

Answer 1

You must mean
√5 - 1

draw a square ABCD with sides 2 , where A and B are the endpoints of the base
Extend the length of the base to the right

Draw a vertical MN where M is the midpoint of AB and N is the midpoint of CD
You should see a right-angled triangle with legs AB = 1 and MN = 2
by Pythagoras AN = √5

Use your compass, centre at A and radius AN , draw an arc to cut the entension of AB at P
so AP = √5
and MP = √5 - 1