How can I find the ratio of a medium triangle to a smaller triangle?

Question

I know area is $\frac{1}{2} bh$ but I can't use that since there are no numbers. The triangles are isosceles and the medium triangle and the small triangle share a side. How can I find the ratio of the areas?

Answer 1

Let $$\frac{S_{\Delta DGO}}{S_{\Delta DLO}}=\frac{GO}{OL}=x$$ and $OL=a$.

Thus, $$DL=DO=GO=ax,$$ $$DG=GL=ax+a=a(x+1)$$ and since $$\Delta DOL\sim\Delta GDL,$$ we obtain: $$\frac{DL}{GL}=\frac{OL}{DL}$$ or $$\frac{ax}{a(x+1)}=\frac{a}{ax}$$ or $$x^2=x+1,$$ which gives $$x=\frac{1+\sqrt5}{2}.$$