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a/2nd/04_Demontrastion_Geometrique/4E_problemes.tex +++ b/2nd/04_Demontrastion_Geometrique/4E_problemes.tex @@ -24,4 +24,7 @@ \printcollection{banque} \vfill +\pagebreak +\printsolutions{exercise} + \end{document} diff --git a/2nd/04_Demontrastion_Geometrique/exercises.tex b/2nd/04_Demontrastion_Geometrique/exercises.tex index 24aeab5..b02895a 100644 --- a/2nd/04_Demontrastion_Geometrique/exercises.tex +++ b/2nd/04_Demontrastion_Geometrique/exercises.tex @@ -62,6 +62,22 @@ \end{enumerate} \end{exercise} +\begin{solution} + \begin{enumerate} + \item + \item On note $H$ le projeté orthogonal de $B$ sur $(AC)$ et $P$ celui de $D$ sur $(AC)$. + + \textbf{On sait que} $H$ est le projeté orthogonal de $B$ sur $(AC)$. \textbf{Or} le projeté orthogonal est le point d'intersection de la perpendiculaire passant par le point et de la droite. \textbf{Donc} $(BH)$ est perpendiculaire à $(AC)$. + + \textbf{On sait que} $P$ est le projeté orthogonal de $D$ sur $(AC)$. \textbf{Or} le projeté orthogonal est le point d'intersection de la perpendiculaire passant par le point et de la droite. \textbf{Donc} $(DP)$ est perpendiculaire à $(AC)$. + + \textbf{On sait que} $B$ et $D$ ont le même projeté orthogonal sur la droite $(AC)$. \textbf{Donc} $H$ et $P$ sont un même point. + + \textbf{On sait que} $(BH)$ est perpendiculaire à $(AC)$, $(DP)$ est perpendiculaire à $(AC)$ et que $H$ et $P$ sont un même point. \texbf{Donc} $(BD)$ est perpendiculaire à $(AC)$. + \item \textbf{On sait que } $ABCD$ est un parallélogramme et que $(BD)$ est perpendiculaire à $(AC)$. \textbf{Or} un parallélogramme qui a ses diagonales qui se coupent en angle droit est un losange. \textbf{Donc} $ABCD$ est un losange. + \end{enumerate} +\end{solution} + \begin{exercise}[subtitle={Longueurs et aire}, step={4}, origin={Magnard 2nd 41p124}, topics={ Demontrastion Geometrique }, tags={ Géométrie, Démonstration }] On considère un rectangle $ABCD$ avec $AB=6$ et $BC=3$. On projette orthogonalement le point $B$ sur $(AC)$ en un point $H$. \begin{enumerate} @@ -71,6 +87,26 @@ \end{enumerate} \end{exercise} +\begin{solution} + \begin{enumerate} + \item Aire du triangle $ABC$: $\dfrac{AB\times BC}{2} = \dfrac{6\times 3}{2} = 9$ + \item \textbf{On sait que} $ABC$ est un triangle rectangle. \textbf{Or} d'après le théorème de Pythagore, on a \textbf{donc} + \begin{enarray*} + + AC^2 = AB^2 + BC^2\\ + AC^2 = 6^2 + 3^2 = 36 + 9 = 45\\ + AC \approx 6.7 + \end{enarray*} + \item On sait que l'aire du triangle $ABC$ est égale à 9. + \begin{eqnarray*} + 9 = \frac{AC\times BH}{2} = \frac{6,7\tilmes BH}{2}\\ + BH = \frac{9 \times 2}{6,7} \approx 2,7 + \end{eqnarray*} + Donc $BH = 2,7$ + \end{enumerate} + +\end{solution} + \begin{exercise}[subtitle={Longueurs}, step={4}, origin={Magnard 2nd 42p124}, topics={ Demontrastion Geometrique }, tags={ Géométrie, Démonstration }] On considère deux droites $d$ et $d'$ sécantes en un point $O$ et un point $A$ n'appartenant ni à $d$ ni à $d'$.