Here is some information about regular pentagons.

ABCDE is a regular pentagon.

We first prove that triangles AEX and ADC are similar and that the ratio AD/AE is equal to the golden ratio
1+Ö5
2

.

As ABCDE is a regular pentagon, triangles AEX and ADC have angles 36^o, 72^o, 72^o and so they are similar isosceles triangles. Label the side length of the pentagon s so AE=AX=CD=CX=s and the chord length c so AD=EC = c and EX=c-s. From the similar triangles

c-s

so writing AD/AE=c/s = x gives

x - 1 =

which is the quadratic equation x²-x-1=0 and hence, as it must be positive, the ratio x=AD/AE is the golden ratio

1+Ö5

Now we can find the exact values of cos72^o and cos36^o.

Drawing a perpendicular from E to AD gives a right angled triangle from which

cos36^o = c
2s
= Ö5+1
4
.

Drawing a perpendicular from A to DC gives a right angled triangle from which

cos72^o = s
2c
= 1
2f
= Ö5-1
4
.

In explaining the construction of a regular pentagon coordinates are useful.

If the pentagon is inscribed in a unit circle with A the point (0,1) you can find the exact coordinates of B and C using the exact values of cos72^o and cos36^o which we have just found.

If the pentagon is inscribed in a unit circle with A the point (0,1) then B=(sin72^o, cos72^o) and C=(sin36^o, -cos 36^o).

Hint: When you explain the construction focus on the ¢y¢ coordinates of B, C, D and E.