Problema (nível 3): Na figura, $[ABCD]$ é um quadrado de lado 16 $cm$. A circunferência é tangente a $[BC]$ e passa pelos vértices $A$ e $D$. (image1) Quanto mede o raio da circunferência?


Conteúdo referenciado:
[image1]
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Resolução: Sejam $r$ e $O$ o raio e o centro da circunferência, respectivamente. Designemos por $E$ o ponto de tangência da circunferência com o quadrado e por $F$ o ponto de intersecção de $[AD]$ com o diâmetro traçado por $E$. Como $[AD]$ é perpendicular a $[FO]$ temos $\overline{AF}=\overline{FD}=8$ $cm$. (image2) Então, atendendo ao Teorema de Pitágoras, $$ r^2=\overline{OA}^2=\overline{OF}^2+\overline{FA}^2= (\overline{EF}-\overline{OE})^2+\overline{FA}^2= (16-r)^2+8^2=256-32r+r^2+64 $$ donde $32r=320$, isto é, $r=10$. $\,$$\,$ Sociedade Portuguesa de Matemática $\;$ XVII ONM - Categoria A (8${\bf ^{\underline o}}$ e 9${\bf ^{\underline o}}$ anos) 2$^{\bf ^{\underline a}}$ Eliminatória $\;\;$ 27/01/99 (table1) $\,$$\,$


Conteúdo referenciado:
[image2]
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[table1]
\begin{tabular}{|p{7,7cm}|} \hline SUGEST\~{O}ES para a resolu\c{c}\~{a}o dos problemas\\ \hline \end{tabular}