Problema (nível 3): O Sebastião traçou, começando em $A$, uma linha poligonal alternadamente perpendicular a cada um dos lados de um ângulo, como indicado na figura. Os dois primeiros segmentos mediam $2$ cm e $6$ cm. Quanto media o vigésimo segmento? (image1)


Conteúdo referenciado:
[image1]
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Resolução: A figura indica os primeiros seis segmentos desenhados pelo Sebastião. (image2) À medida que vai traçando segmentos alternadamente perpendiculares aos lados do ângulo, o Sebastião vai criando triângulos retângulos todos semelhantes entre si, com razão de semelhança igual a $$ \frac{\overline{DE}}{\overline{DC}}=\frac{\overline{DC}}{\overline{BC}}=3. $$ Assim, o terceiro segmento media $ \displaystyle \overline{DE}=\frac{\overline{DC}}{\overline{BC}}\times \overline{DC}=3\times 6=3^2\times 2$, o quarto media $\displaystyle \overline{EF}=\frac{\overline{DC}}{\overline{BC}}\times \overline{DE}=3^3\times 2$, e assim sucessivamente. Portanto o vigésimo segmento media $3^{20-1}\times 2=3^{19}\times 2$ cm.


Conteúdo referenciado:
[image2]
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