Similarity and Scale Factor If I have two triangles ABC and ADE and I know that each of the triangles is equilateral, each of the angles is 60 degrees. Is there a way to determine the scale factor of the two triangles, for example:
The ratio:
Area of ADE $:$Area of ABC
Thanks in advance.


If side $DC$$ = $ $x$, then $AC$ must be $2x$ because both triangles are equilateral. By Pythagorean theorem, $AD$ is $3^\frac{1}{2}x$. Then $AD:AC=3^\frac{1}{2}:2=\frac{3^\frac{1}{2}{}}{2}:1$. So the scale factor $k$, is $\frac{3^\frac{1}{2}}{2}$. Using the formula:
$\frac{\text{Area of }AED}{\text{Area of }ABC}=k^2$
$\frac{\text{Area of }AED}{\text{Area of }ABC}=(\frac{3^\frac{1}{2}}{2})^2$
$\frac{\text{Area of }AED}{\text{Area of }ABC}=\frac{3}{4}$
Therefore $k^2= \frac{3}{4}$ and so the ratio between triangle $ADE$ and $ABC$ is $1:\frac{3}{4}$
