Question

if the ratio of the areas of 2 similar polygons is 32:50, what is the ration of the corresponding side lengths?

Answer 1

$\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------$

$\bf \cfrac{small}{large}\qquad \cfrac{s}{s}=\cfrac{\sqrt{s^2}}{\sqrt{s^2}}\implies \cfrac{s}{s}=\cfrac{\sqrt{32}}{\sqrt{50}}\implies \cfrac{s}{s}=\cfrac{\sqrt{16\cdot 2}}{\sqrt{25\cdot 2}} \\\\\\ \cfrac{s}{s}=\cfrac{\sqrt{4^2\cdot 2}}{\sqrt{5^2\cdot 2}}\implies \cfrac{s}{s}=\cfrac{4\sqrt{2}}{5\sqrt{2}}\implies \cfrac{s}{s}=\cfrac{4}{5}$