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\def\examtitle{Devoir 01 - S01}
\def\schoolname{\textbf{Lycée :} Lycée : Taghzirt}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
\textbf{Partie 1}: (8 points)\\
Dans le plan rapporté à un repère orthonormé \((O; \vec{i}; \vec{j})\), soient \(ABC\) un triangle, \(H\) un point du plan tel que \(\overrightarrow{AH} = 2\overrightarrow{AC}\) et \(G = bar\{(A; 2); (B; -2); (C; -4)\}\).
\begin{enumerate}
    \item Construire le point \(H\). \dotfill (1pt)
    \begin{enumerate}
        \item Montrer que \(H = bar\{(A; 2); (C; -4)\}\). \dotfill (1pt)
        \item Montrer que \(G\) est le milieu du segment \([HB]\). \dotfill (1pt)
    \end{enumerate}
    \item Soit \(F = bar\{(A; 2); (B; -1); (C; -2)\}\). \dotfill (1pt)
    \begin{enumerate}
        \item Écrire les vecteurs \(\overrightarrow{AG}\) et \(\overrightarrow{AF}\) en fonction de \(\overrightarrow{AB}\) et \(\overrightarrow{AC}\). \dotfill (1pt)
        \item Déduire que les points \(A, F\) et \(G\) sont alignés. \dotfill (1pt)
    \end{enumerate}
    \item Déterminer l’ensemble des points \(M\) du plan tels que :
    \begin{enumerate}
        \item \(\|2\overrightarrow{MA} - 2\overrightarrow{MB} - 4\overrightarrow{MC}\| = 4\). \dotfill (1pt)
        \item \(\|2\overrightarrow{MA} - 4\overrightarrow{MC}\| = \|5\overrightarrow{MA} - \overrightarrow{MB} - 2\overrightarrow{MC}\|\). \dotfill (1pt)
    \end{enumerate}
\end{enumerate}
\textbf{Partie 2}: (4 points)\\
Sachant que \(A(2; 5), B(-1; 5)\) et \(C(-1; 2)\).
\begin{enumerate}
    \setcounter{enumi}{3}
    \item Montrer que \(F(-7; -1)\) et que \(H(-4; -1)\). \dotfill (1pt)
    \item Calculer \(BA, BC\) et \(\overrightarrow{BA} \cdot \overrightarrow{BC}\). \dotfill (1pt)
    \item Calculer \(\cos(\overrightarrow{AB}; \overrightarrow{AC})\) et \(\sin(\overrightarrow{AB}; \overrightarrow{AC})\) puis déduire une mesure de l’angle
      \((\widehat{\overrightarrow{AB}; \overrightarrow{AC}})\). \dotfill (1pt)
    \item Quelle est la nature du triangle \(ABC\) ? Calculer sa surface. \dotfill (1pt)
\end{enumerate}
\textbf{Partie 3}: (6 points)
\begin{enumerate}
    \setcounter{enumi}{7}
    \item Déterminer une équation cartésienne de la droite \((\Delta)\) médiatrice du segment \([AC]\). \dotfill (1pt)
    \item Déterminer une équation cartésienne de la droite \((AH)\). \dotfill (1pt)
    \item Déterminer une équation cartésienne de la droite \((D)\) passant par \(F\) \\et parallèle à la droite \((AH)\). \dotfill (1pt)
    \item Prouver que la droite \((\Delta)\) est perpendiculaire à la droite \((D)\) au point \(B\). \dotfill (1pt)
    \item Déterminer une équation cartésienne de la droite \((\Delta')\) passant par \(C\) \\et de vecteur normal \(\vec{n}(1; 1)\). \dotfill (1pt)
    \item Montrer que \((\Delta')\) et \((\Delta)\) sont parallèles, puis déduire que \((\Delta')\) et \((D)\) sont perpendiculaires. \dotfill (1pt)
\end{enumerate}
\textbf{Partie 4}: (3 points)\\
On considère le cercle \((\zeta)\) de centre \(\Omega(1; 2)\) et de rayon \(R = 2\).
\begin{enumerate}
    \setcounter{enumi}{13}
    \item Montrer que \(C \in (\zeta)\). \dotfill (1pt)
    \item Donner une équation cartésienne du cercle \((\zeta)\). \dotfill (1pt)
    \item Vérifier que le point \(A\) est à l’extérieur du cercle \((\zeta)\). \dotfill (1pt)
    \item Résoudre géométriquement le système :
    ~$
    (S) \left\{
    \begin{array}{l}
        x^2 + y^2 - 2x - 4y + 1 \leq 0 \\
        -x + y - 3 < 0 \\
        y \geq 3
    \end{array}
    \right.
    $~\dotfill (1pt)
\end{enumerate}
\centering
Pr: Adam Abdelrhafour OUSSAMA
}








\end{document}