Use the symmetry of the regular pentagon to find similar triangles implying $\frac{x}{1}=\frac{1}{x-1}$

Question

That is $x^2-x-1=0$

By definition, regular pentagon has equal sides, therefore equal diagonals.

I'm not sure what the question is asking me to do. Can someone guide me? give me a hint.

Answer 1

HINT:

join the endpoints of a side with the opposite vertex, compare the resulting isosceles triangle with another similar isosceles triangle, which is already present in your diagram.

Answer 2

There are two similar triangles in your figure. One has sides x,x,1 and the other has sides 1,1,(1-x). Therefore, if follows that

$$\frac{x}{1}=\frac{1}{x-1}$$