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

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?

• In the picture, below the triangle there is a $1$. – Rodrigo Pizarro Feb 21 at 18:07
• But since the question asks about ratio of areas, the measurement that @RodrigoPizarro has pointed out doesn’t matter. – Lubin Feb 21 at 18:39
• Anyways, that isn't the point of my comment. The OP said that "I can't use that since there are no numbers", which in this case is not true. – Rodrigo Pizarro Feb 22 at 19:04

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 $$x=\frac{1+\sqrt5}{2}.$$