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Question Number 185768 by Mingma last updated on 27/Jan/23

Answered by JDamian last updated on 27/Jan/23

$$l=\frac{2}{3}L$$$$\mathrm{outer}\mathrm{triangle}\mathrm{area}$$$$A={kL}^2$$$$\mathrm{inner}\mathrm{triangle}\mathrm{area}$$$$a={kl}^2=k{\left(\frac{2}{3}L\right)}^2=k\frac{4}{9}{L}^2$$$$$$$$\frac{A}{a}=\frac{{kL}^2}{k\frac{4}{9}{L}^2}=\frac{9}{4}\Rightarrow A=\frac{9}{4}a$$$$$$$$\mathrm{trapezoid}\mathrm{area}$$$$s=\frac{A-a}{3}=\frac{\frac{9}{4}a-a}{3}=\frac{5}{12}a$$$$$$$$\frac{s}{a}=\frac{5}{12}$$

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