Question

The shortest side of a polygon of area 196 in² is 4" long find the area of similar polygon who shortest side is 8" long

Answer 1

`\bf ~\hspace{5em} \textit{ratio relations of two similar shapes} \\[2em] \begin{array}{ccccllll} &\stackrel{ratio~of~the}{Sides}&\stackrel{ratio~of~the}{Areas}&\stackrel{ratio~of~the}{Volumes}\\ \cline{2-4}&\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array}\\\\[-0.35em] ~\dotfill`

`\bf \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{√(s^2)}{√(s^2)}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{sides}{\cfrac{4^2}{8^2}}=\stackrel{Areas}{\cfrac{196}{a}}\implies \cfrac{16}{64}=\cfrac{196}{a}\implies \cfrac{1}{4}=\cfrac{196}{a}\implies a=784`