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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?

  • $\begingroup$ In the picture, below the triangle there is a $1$. $\endgroup$ – Rodrigo Pizarro Feb 21 at 18:07
  • $\begingroup$ But since the question asks about ratio of areas, the measurement that @RodrigoPizarro has pointed out doesn’t matter. $\endgroup$ – Lubin Feb 21 at 18:39
  • $\begingroup$ 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. $\endgroup$ – 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 $$x=\frac{1+\sqrt5}{2}.$$


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