\documentclass[a4paper,11pt]{article} \usepackage{francois_meria} \usepackage[dvips]{graphicx} \usepackage[dvips]{epsfig} \setlength{\parindent}{0mm} \lhead{\textsf{Collège Château Forbin} - \textit{Mathématiques} - \textsf{6\ieme}} \chead{} \rhead{\textit{Année} 2005/2006} \pagestyle{fancy} \renewcommand{\headrulewidth}{0.5pt} % \begin{document} \begin{center} \begin{tabularx}{\textwidth}{|X|} \hline \vskip 0.3cm \begin{center} {\Large\textbf{Construction géométrique et symétrie - 1}}\\ \end{center}\\ \hline \end{tabularx} \end{center} % \vskip 1cm \begin{enumerate}[(a)] \item \begin{multicols}{2} \begin{itemize} \item [$\circ$] Tracer au centre de la feuille un carré $ABCD$ de $6$~cm de côté ainsi que ses deux diagonales d'intersection $O$. \item [$\circ$] Construire les trois bissectrices du triangle $ABD$. elles se coupent en $I$. \item [$\circ$] Tracer le cercle de centre $I$ et de rayon $IO$. On obtient la figure 1. \end{itemize} \begin{center} \psset{unit=0.6cm} \pspicture(-0.5,-0.5)(6.5,6.5) \pstGeonode[PointSymbol=none,PosAngle={225,-45,45,135}](0,0){D}(6,0){C}(6,6){B}(0,6){A} \pstLineAB{A}{B} \pstLineAB{C}{B} \pstLineAB{A}{D} \pstLineAB{D}{C} \pstLineAB{A}{C} \pstLineAB{D}{B} \pstInterLL[PointSymbol=none,PosAngle=-90]{D}{B}{A}{C}{O} \pstBissectBAC[PointName=none,PointSymbol=none,linestyle=dotted]{A}{B}{D}{M} \pstBissectBAC[PointName=none,PointSymbol=none,linestyle=dotted,nodesepB=2.5]{B}{D}{A}{N} \pstInterLL[PointSymbol=+]{A}{C}{B}{M}{I} \pstCircleOA{I}{O} \put(2,-0.7){Figure 1} \endpspicture \end{center} \end{multicols} \item Compléter cette figure par symétrie par rapport à la droite $(BD)$, faire de même avec la nouvelle figure par rapport à la droite $(BC)$ et enfin par rapport à la droite $(DC)$. \item Colorier la figure avec deux couleurs différentes que l'on alternera afin d'obtenir la figure 2. \end{enumerate} \vskip 1cm \begin{center} \psset{unit=1cm} \pspicture(0,-6)(12,6) \pstGeonode[PointSymbol=none,PointName=none](0,0){D}(6,0){C}(6,6){B}(0,6){A} \pstLineAB{A}{B} \pstLineAB{C}{B} \pstLineAB{A}{D} \pstLineAB{D}{C} \pstLineAB{A}{C} \pstLineAB{D}{B} \pstInterLL[PointSymbol=none,PosAngle=-90,PointName=none]{D}{B}{A}{C}{O} \pstBissectBAC[PointName=none,PointSymbol=none,linestyle=none]{A}{B}{D}{M} \pstBissectBAC[PointName=none,PointSymbol=none,linestyle=none]{B}{D}{A}{N} \pstInterLL[PointSymbol=none,PointName=none]{A}{C}{B}{M}{I} \pstCircleOA{I}{O} \pstOrtSym[PointSymbol=none,PointName=none]{D}{B}{I}{J} \pstCircleOA{J}{O} \pstOrtSym[PointName=none,PointSymbol=none]{B}{C}{D}{E} \pstOrtSym[PointName=none,PointSymbol=none]{B}{C}{A}{F} \pstOrtSym[PointName=none,PointSymbol=none]{B}{C}{I}{K} \pstOrtSym[PointName=none,PointSymbol=none]{B}{C}{J}{L} \pstOrtSym[PointName=none,PointSymbol=none]{B}{C}{O}{O'} \pstCircleOA{K}{O'} \pstCircleOA{L}{O'} \pstLineAB{B}{E} \pstLineAB{B}{F} \pstLineAB{E}{C} \pstLineAB{E}{F} \pstLineAB{C}{F} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{A}{Z} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{B}{Y} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{F}{X} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{J}{T} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{I}{Q} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{O}{P} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{O'}{S} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{L}{H} \pstOrtSym[PointName=none,PointSymbol=none]{D}{C}{K}{H'} \pstCircleOA{T}{P} \pstCircleOA{H}{S} \pstCircleOA{H'}{S} \pstCircleOA{Q}{P} \pstLineAB{D}{Z} \pstLineAB{C}{Y} \pstLineAB{E}{X} \pstLineAB{E}{F} \pstLineAB{C}{F} \pstLineAB{C}{X} \pstLineAB{Z}{X} \pstLineAB{C}{Z} \pstLineAB{D}{Y} \pstLineAB{Y}{E} \put(5.4,-6.7){Figure 2} \endpspicture \end{center} \newpage \begin{center} \begin{tabularx}{\textwidth}{|X|} \hline \vskip 0.3cm \begin{center} {\Large\textbf{Construction géométrique et symétrie - 2}}\\ \end{center}\\ \hline \end{tabularx} \end{center} \vskip 1cm \begin{multicols}{2} À partir de la figure 1 ci-contre, on veut obtenir la figure 2 puis la figure 3, uniquement à l'aide de la symétrie axiale. \begin{center} \psset{unit=1cm} \pspicture(-2,-0.5)(5,2.2) \rput{25}{ \pstGeonode[PointSymbol=none,PosAngle={-45,-45,-45,-45,-45,-45,90}](0,0){O}(1,0){I}(2,0){J}(3,0){K}(4,0){L}(5,0){M}(0,2){A} \pstSegmentMark[SegmentSymbol=pstslashh]{O}{I} \pstSegmentMark[SegmentSymbol=pstslashh]{I}{J} \pstSegmentMark[SegmentSymbol=pstslashh]{J}{K} \pstSegmentMark[SegmentSymbol=pstslashh]{K}{L} \pstSegmentMark[SegmentSymbol=pstslashh]{L}{M} \pstLineAB{A}{O} \pstLineAB{A}{I} \pstLineAB{A}{J} \pstLineAB{A}{K} \pstLineAB{A}{L} \pstLineAB{A}{M} } \put(2,0){Figure 1} \endpspicture \end{center} \end{multicols} \begin{enumerate}[(a)] \item Reproduire la figure 1 en prenant $OA=2$~cm et $OI=1$~cm. \item Quels sont les axes de symétrie de la figure 2 ? Compléter la figure 1 afin d'obtenir la figure 2. \item Décrire avec précision les axes de symétrie de la figure 3. Compléter la figure 2 pour obtenir la figure 3. \item Colorier la figure 3 à l'aide de deux couleurs en alternant les couleurs. \item Combien d'axes de symétrie possède la figure 3 ? Et la figure coloriée ? \end{enumerate} \begin{center} \psset{unit=1cm} \pspicture(-5,-3)(5,2.5) \rput{25}{ \pstGeonode[PointSymbol=none,PosAngle={-40,-40,-40,-40,-40,-40,90}](0,0){O}(1,0){I}(2,0){J}(3,0){K}(4,0){L}(5,0){M}(0,2){A} \pstLineAB{A}{O} \pstLineAB{A}{I} \pstLineAB{A}{J} \pstLineAB{A}{K} \pstLineAB{A}{L} \pstLineAB{A}{M} \pstLineAB{O}{M} \pstOrtSym[PointSymbol=none,PosAngle=-45]{O}{M}{A}{B} \pstLineAB{B}{O} \pstLineAB{B}{I} \pstLineAB{B}{J} \pstLineAB{B}{K} \pstLineAB{B}{L} \pstLineAB{B}{M} \pstLineAB{B}{M} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{I}{I_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{J}{J_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{K}{K_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{L}{L_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{M}{M_1} \pstLineAB{O}{M_1} \pstLineAB{A}{I_1} \pstLineAB{A}{J_1} \pstLineAB{A}{K_1} \pstLineAB{A}{L_1} \pstLineAB{A}{M_1} \pstLineAB{B}{I_1} \pstLineAB{B}{J_1} \pstLineAB{B}{K_1} \pstLineAB{B}{L_1} \pstLineAB{B}{M_1} } \put(2,-2){Figure 2} \endpspicture \end{center} \begin{center} \psset{unit=1cm} \pspicture(-5,-5)(6,2.5) \rput{25}{ \pstGeonode[PointSymbol=none,PointName=none](0,0){O}(1,0){I}(2,0){J}(3,0){K}(4,0){L}(5,0){M}(0,1.4){A} \pstLineAB{A}{O} \pstLineAB{A}{I} \pstLineAB{A}{J} \pstLineAB{A}{K} \pstLineAB{A}{L} \pstLineAB{A}{M} \pstLineAB{O}{M} \pstOrtSym[PointSymbol=none,PointName=none]{O}{M}{A}{B} \pstLineAB{B}{O} \pstLineAB{B}{I} \pstLineAB{B}{J} \pstLineAB{B}{K} \pstLineAB{B}{L} \pstLineAB{B}{M} \pstLineAB{B}{M} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{I}{I_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{J}{J_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{K}{K_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{L}{L_1} \pstOrtSym[PointSymbol=none,PointName=none]{A}{B}{M}{M_1} \pstLineAB{O}{M_1} \pstLineAB{A}{I_1} \pstLineAB{A}{J_1} \pstLineAB{A}{K_1} \pstLineAB{A}{L_1} \pstLineAB{A}{M_1} \pstLineAB{B}{I_1} \pstLineAB{B}{J_1} \pstLineAB{B}{K_1} \pstLineAB{B}{L_1} \pstLineAB{B}{M_1} \pstSymO[PointSymbol=none,PointName=none]{B}{A}{C} \pstSymO[PointSymbol=none,PointName=none]{B}{I_1}{I_2} \pstSymO[PointSymbol=none,PointName=none]{B}{J_1}{J_2} \pstSymO[PointSymbol=none,PointName=none]{B}{K_1}{K_2} \pstSymO[PointSymbol=none,PointName=none]{B}{L_1}{L_2} \pstSymO[PointSymbol=none,PointName=none]{B}{M_1}{M_2} \pstLineAB{B}{C} \pstLineAB{C}{I_2} \pstLineAB{C}{J_2} \pstLineAB{C}{K_2} \pstLineAB{C}{L_2} \pstLineAB{C}{M_2} \pstLineAB{B}{I_2} \pstLineAB{B}{J_2} \pstLineAB{B}{K_2} \pstLineAB{B}{L_2} \pstLineAB{B}{M_2} \pstOrtSym[PointSymbol=none,PointName=none]{B}{C}{I_2}{I_3} \pstOrtSym[PointSymbol=none,PointName=none]{B}{C}{J_2}{J_3} \pstOrtSym[PointSymbol=none,PointName=none]{B}{C}{K_2}{K_3} \pstOrtSym[PointSymbol=none,PointName=none]{B}{C}{L_2}{L_3} \pstOrtSym[PointSymbol=none,PointName=none]{B}{C}{M_2}{M_3} \pstLineAB{B}{I_3} \pstLineAB{B}{J_3} \pstLineAB{B}{K_3} \pstLineAB{B}{L_3} \pstLineAB{B}{M_3} \pstLineAB{C}{I_3} \pstLineAB{C}{J_3} \pstLineAB{C}{K_3} \pstLineAB{C}{L_3} \pstLineAB{C}{M_3} \pstLineAB{M_2}{M_3} } \put(3,-4){Figure 3} \endpspicture \end{center} \newpage \begin{center} \begin{tabularx}{\textwidth}{|X|} \hline \vskip 0.3cm \begin{center} {\Large\textbf{Construction géométrique et symétrie - 3}}\\ \end{center}\\ \hline \end{tabularx} \end{center} \vskip 1cm \begin{enumerate}[1.] \begin{multicols}{2} \item \begin{enumerate}[(a)] \item Tracer un carré $ABCD$ de $15$~cm de côté et ses quatre axes de symétrie. Appeler $O$ leur point d'intersection. \item Placer le point $I$, milieu du segment $[AB]$ et le point $J$, milieu du segment $[BC]$. \item Construire les bissectrices des angles $\widehat{OAD}$, $\widehat{OAB}$, $\widehat{IOB}$ et $\widehat{JOB}$. \item Compléter la construction pour obtenir la figure 1. \end{enumerate} \begin{center} \psset{unit=0.4cm} \pspicture(15,15) \pstGeonode[PointSymbol=none,PosAngle={235,-45,45,135,0,90}](0,0){D}(15,0){C}(15,15){B}(0,15){A}(15,7.5){J}(7.5,15){I} \pspolygon(A)(B)(C)(D) \pstInterLL[PointSymbol=none,PosAngle=-118]{A}{C}{B}{D}{O} \psline(A)(C) \psline(B)(D) \pstGeonode[PointSymbol=none,PointName=none](7.5,0){I_1}(0,7.5){J_1} \psline(J)(J_1) \psline(I)(I_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{D}{A}{O}{M_1}% \pstInterLL[PointName=none,PointSymbol=none]{A}{M_1}{J}{O}{T_1}% \psline(A)(T_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{O}{A}{B}{M_2}% \pstInterLL[PointName=none,PointSymbol=none]{A}{M_2}{I}{O}{T_2}% \psline(A)(T_2) \psline(T_1)(T_2) \pstMarkAngle[MarkAngleRadius=1.2]{T_1}{A}{O}{} \pstMarkAngle[MarkAngleRadius=1.4]{T_1}{A}{O}{} \pstMarkAngle[MarkAngleRadius=1.4]{O}{A}{T_2}{} \pstMarkAngle[MarkAngleRadius=1.6]{O}{A}{T_2}{} \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{B}{O}{I}{N_1}% \pstInterLL[PointName=none,PointSymbol=none]{O}{N_1}{I}{B}{P_1}% \psline(O)(P_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{J}{O}{B}{N_2}% \pstInterLL[PointName=none,PointSymbol=none]{O}{N_2}{J}{B}{P_2}% \psline(O)(P_2) \psline(P_1)(P_2) \pstMarkAngle[MarkAngleRadius=1.2]{B}{O}{P_1}{} \pstMarkAngle[MarkAngleRadius=1.4]{B}{O}{P_1}{} \pstMarkAngle[MarkAngleRadius=1.4]{P_2}{O}{B}{} \pstMarkAngle[MarkAngleRadius=1.6]{P_2}{O}{B}{} \put(6.2,-1){Figure 1} \endpspicture \end{center} \end{multicols} \item \begin{enumerate}[(a)] \item Compléter la figure 1 par symétrie par rapport aux deux diagonales du carré $ABCD$. \item Colorier la figure 2 à l'aide de deux couleurs que l'on alternera. \end{enumerate} \end{enumerate} \begin{center} \psset{unit=1cm} \pspicture(15,15) \pstGeonode[PointSymbol=none,PointName=none,PosAngle={235,-45,45,135,0,90}](0,0){D}(15,0){C}(15,15){B}(0,15){A}(15,7.5){J}(7.5,15){I} \pspolygon(A)(B)(C)(D) \pstInterLL[PointSymbol=none,PointName=none,PosAngle=50]{A}{C}{B}{D}{O} \psline(A)(C) \psline(B)(D) \pstGeonode[PointSymbol=none,PointName=none](7.5,0){I_1}(0,7.5){J_1} \psline(J)(J_1) \psline(I)(I_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{D}{A}{O}{M_1}% \pstInterLL[PointName=none,PointSymbol=none]{A}{M_1}{J}{O}{T_1}% \psline(A)(T_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{O}{A}{B}{M_2}% \pstInterLL[PointName=none,PointSymbol=none]{A}{M_2}{I}{O}{T_2}% \psline(A)(T_2) \psline(T_1)(T_2) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{B}{O}{I}{N_1}% \pstInterLL[PointName=none,PointSymbol=none]{O}{N_1}{I}{B}{P_1}% \psline(O)(P_1) \pstBissectBAC[linestyle=none,PointSymbol=none,PointName=none]{J}{O}{B}{N_2}% \pstInterLL[PointName=none,PointSymbol=none]{O}{N_2}{J}{B}{P_2}% \psline(O)(P_2) \psline(P_1)(P_2) \pstOrtSym[PointName=none,PointSymbol=none]{B}{D}{T_1}{Q_1} \pstOrtSym[PointName=none,PointSymbol=none]{B}{D}{T_2}{Q_2} \pspolygon(Q_1)(Q_2)(C) \pstOrtSym[PointName=none,PointSymbol=none]{A}{C}{P_1}{R_1} \pstOrtSym[PointName=none,PointSymbol=none]{A}{C}{P_2}{R_2} \pspolygon(R_1)(R_2)(O) \pstInterLL[PointName=none,PointSymbol=none]{T_1}{T_2}{O}{A}{S} \pspolygon[fillstyle=solid,fillcolor=lightgray](A)(T_2)(S) \pspolygon[fillstyle=solid,fillcolor=lightgray](O)(T_1)(S) \pspolygon[fillstyle=solid,fillcolor=lightgray](A)(J_1)(T_1) \pstInterLL[PointName=none,PointSymbol=none]{P_1}{P_2}{O}{B}{S_1} \pspolygon[fillstyle=solid,fillcolor=lightgray](O)(P_1)(I) \pspolygon[fillstyle=solid,fillcolor=lightgray](O)(S_1)(P_2) \pspolygon[fillstyle=solid,fillcolor=lightgray](P_1)(S_1)(B) \pstInterLL[PointName=none,PointSymbol=none]{R_1}{R_2}{O}{D}{U} \pspolygon[fillstyle=solid,fillcolor=lightgray](O)(R_1)(U) \pspolygon[fillstyle=solid,fillcolor=lightgray](D)(U)(R_2) \pspolygon[fillstyle=solid,fillcolor=lightgray](R_2)(O)(I_1) \pstInterLL[PointName=none,PointSymbol=none]{Q_1}{Q_2}{O}{C}{U_1} \pspolygon[fillstyle=solid,fillcolor=lightgray](Q_1)(U_1)(C) \pspolygon[fillstyle=solid,fillcolor=lightgray](O)(U_1)(Q_2) \pspolygon[fillstyle=solid,fillcolor=lightgray](Q_2)(J)(C) \put(6.7,-0.7){Figure 2} \endpspicture \end{center} \end{document}