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\begin{document}%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 14385049285134--
%%--Etude de fonction et géométrie plane--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
$\vspace*{-0.35cm}$

\begin{minipage}{9cm}
Sur la figure ci-contre, le triangle $ABC$ est rectangle et isocèle en $A$.
On donne $BC=9$ et on appelle $I$ le milieu de $[BC]$.
Le point $M$ est un point de $[BI]$ et on construit le rectangle $MNPQ$ tel que $N$ soit un point
de $[AB]$, $P$ soit un point de $[AC]$ et $Q$ soit un point de $[BC]$.
\begin{enumerate}
\item \begin{enumerate}
\item Donner la mesure de $\widehat{NBM}$.
\item En déduire que $MN=BM=QC$.
\end{enumerate}
\end{enumerate}
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\begin{enumerate}
\setcounter{enumi}{1}

\item On pose $BM=x$.

\begin{enumerate}
\item Pourquoi le r\'{e}el $x$ appartient-il \`{a} $[0;4,5]$?

\item Exprimer $MQ$ et $MN$ en fonction de $x$.

\item D\'{e}terminer $x$ afin que $MNPQ$ soit un rectangle.

\item Montrer que l'aire du rectangle $MNPQ$, not\'{e}e $f(x)$, est donn\'{e}e
par: $f(x)=9x-2x^{2}$
\end{enumerate}

\item Calculer $f\left(  \frac{9}{4}\right)  $ et $f(1+\sqrt{2})$.

\item D\'{e}terminer l'image de $2$ par $f$.

\item Comment placer $M$ pour que l'aire de $MNPQ$ soit \'{e}gale \`{a} $4$.

\item Sur le graphique ci-contre, on a trac\'{e} la repr\'{e}sentation
graphique de $f$ not\'{e}e $\mathcal{C}_{f}$.

\begin{enumerate}
\item Par lecture graphique, donner le tableau de variations de $f$?

\item D\'{e}velopper $\frac{81}{8}-2\left(  x-\frac{9}{4}\right)  ^{2}$.

\item En d\'{e}duire la valeur de $x$ pour laquelle l'aire du rectangle $MNPQ$
est maximale ainsi que la valeur de ce maximum.

\item D\'{e}montrer que l'aire $g(x)$ du triangle $ANP$ est donn\'{e}e en
fonction de $x$ par: $g(x)=x^{2}-9x+\frac{81}{4}$

\item Tabuler $g$ au pas de $0.5$ puis repr\'{e}senter graphiquement $g$ dans
le m\^{e}me rep\`{e}re que $f$.

\item D\'{e}duire graphiquement les valeurs de $x$ pour lesquelles les aires
du triangle $ANP$ et du rectangle $MNPQ$ sont \'{e}gales.
\end{enumerate}
\end{enumerate}

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\end{exercise}


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