Similar Lengths, Similar Areas

If the ratio of their areas is $9:25,$ the ratio of their corresponding side lengths is $3:5.$ Therefore, if we let $\lvert\overline{AB}\rvert$ denote the length of $\overline{AB},$ then $\begin{aligned} \lvert\overline{AB}\rvert:\lvert\overline{DE}\rvert &= 3:5\\ \lvert\overline{AB}\rvert: 60 &=3:5\\ \Rightarrow \lvert\overline{AB}\rvert &= \frac{60 \times 3}{5}\\ &= 36. \end{aligned}$

The areas of similar plane figures have the same ratio as the squares of any two corresponding sides. We have

$\dfrac{A_{ABC}}{A_{DEF}}=\dfrac{(AB)^2}{(DE)^2}$

$\dfrac{9}{25}=\dfrac{(AB)^2}{60^2}$

$\boxed{AB=36}$

$\frac{A_1}{A_2} =$ $\frac{(s_1)^2}{(s_2)^2}$

$\frac{9}{25} = \frac{(AB)^2}{60^2}$

$(9)(60^2) = 25(AB)^2$

$AB = 36$