%========================================= %Macros personnelles %christophe.poulain@melusine.eu.org %création : 25 Septembre 1999 %dernière modification : 28 Avril 2005 %========================================= \RequirePackage{xlop,fancybox,color,amssymb,ifthen} \input{xlopsqrt} \newtheorem{ppte}{Propriété} \newtheorem{theo}{Théorème} \newtheorem{defi}{\color{green}Définition} \newtheorem{lemme}{Lemme} \newtheorem{coro}{Corollaire} \newtheorem{prop}{Proposition} \newtheorem{reg}{Règle} \newtheorem{conj}{Conjecture} \newtheorem{remar}{Remarque} \newtheorem{exem}{Exemple} \newcommand{\rema}{\underline{Remarque} } \newcommand{\exe}{\underline{Exemple} } \newcommand{\pre}{\underline{Preuve}} \newcommand{\cas}{\underline{Cas particulier}} \newcommand{\cass}{\underline{Cas particuliers}} \newcommand{\Not}{\underline{Notation} } \newcommand{\Si}{\underline{Si} } \newcommand{\si}{\underline{si} } \newcommand{\alors}{\underline{alors} } \newcommand{\cons}{\underline{Conséquence}} \newcommand{\Comme}{\underline{Comme} } \newcommand{\comme}{\underline{comme} } \def\qed{\hfill\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width4pt\vfill\hrule}\vrule}} \def\cqfd{\hfill\unskip\kern 6pt\penalty 500\qed\par} \catcode`\@=11 \def\Eqalign#1{\null\,\vcenter{\openup\jot\m@th\ialign{ \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil &&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$ \hfil\crcr #1\crcr}}\,} \catcode`\@=12 \newcommand{\vecteur}[1] {\overrightarrow{\strut #1}} \font\tenbb=msbm10 \font\sevenbb=msbm7 \font\fivebb=msbm5 \newfam\bbfam \textfont\bbfam=\tenbb \scriptfont\bbfam=\sevenbb \scriptscriptfont\bbfam=\fivebb \def\bb{\fam\bbfam\tenbb} \let\oldbb=\bb \def\bb #1{{\oldbb #1}} \def\tvi{\vrule height 12pt depth 5pt width 0pt} \def\tvj{\vrule height 12pt depth 5pt width 1pt} \def\hfq{\hfill\,\,} \def\cc#1{\hfq #1\hfq} \def\tv{\tvi\vrule} \def\tw{\tvj\vrule} \def\traithorizontal{\noalign{\hrule}} \def\traithorizontale{\noalign{\hrule height 1pt}} \newcommand{\encadre}[1] {\begin{center} \fbox{\begin{minipage}{\linewidth} {#1} \end{minipage}} \end{center} } \def\pgcd{\mathop{\rm PGCD}\nolimits} \def\ppcm{\mathop{\rm PPCM}\nolimits} \def\cut{{}\hfill\cr \hfill{}} \newcommand{\biindice}[3]% { \renewcommand{\arraystretch}{0.5} \begin{array}[t]{c} #1\\ {\scriptstyle #2}\\ {\scriptstyle #3} \end{array} \renewcommand{\arraystretch}{1} } \newlength{\ltxt} \newcommand{\compo}[4]{ \setlength{\ltxt}{\linewidth} \setbox#1=\hbox{\includegraphics[scale=#3]{#2.#1}} \addtolength{\ltxt}{-\wd#1} \addtolength{\ltxt}{-10pt} \begin{minipage}{\wd#1} \includegraphics[scale=#3]{#2.#1} \end{minipage} \hfill \begin{minipage}{\ltxt} #4 \end{minipage} } \newlength{\lntxt} \newcommand{\Compo}[4]{ \setlength{\lntxt}{\linewidth} \setbox#1=\hbox{\includegraphics[scale=#3]{#2}} \addtolength{\lntxt}{-\wd#1} \addtolength{\lntxt}{-10pt} \begin{minipage}{\wd#1} \includegraphics[scale=#3]{#2} \end{minipage} \hfill \begin{minipage}{\lntxt} #4 \end{minipage} } \newlength{\lnttxt} \newcommand{\dispo}[3]{ \setlength{\lnttxt}{\linewidth} \setbox#1=\hbox{#2} \addtolength{\lnttxt}{-\wd#1} \addtolength{\lnttxt}{-20pt} \begin{minipage}{\wd#1} #2 \end{minipage} \hfill \begin{minipage}{\lnttxt} #3 \end{minipage} } \newcounter{num}[section] \newcommand{\exo}{\addtocounter{num}{1} \par \par\underline{\bf Exercice~\thenum} } \newcommand{\titrage}[2]{ {\Large #1}\hfill#2 \par\rule[+6pt]{\linewidth}{0.5mm} \par } \newcommand{\titragedossier}[1]{ {\small #1}\hfill{\small www.melusine.eu.org/syracuse/poulecl/} \par\rule[+6pt]{\linewidth}{0.5mm} \par } \newcommand{\partie}[2]{ \begin{center} \begin{minipage}{#1pt} \begin{center} \boxput*(0,0){\colorbox{white}{#2}} {\rule{\linewidth}{0.5mm}} \end{center} \end{minipage} \end{center} \par } \newenvironment{myenumerate}{ \renewcommand{\theenumi}{\arabic{enumi}} \def\labelenumi{{\bf \theenumi /}} \begin{enumerate}}{\end{enumerate}} \newenvironment{Myenumerate}{ \renewcommand{\theenumi}{\arabic{enumi}} \def\labelenumi{$\rhd${\bf \theenumi /}} \begin{enumerate}}{\end{enumerate}} \newdimen\shadeshift\shadeshift=1pt \def\shadedtext#1{{\setbox0=\hbox{#1}\leavevmode \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}% \kern0.1em\lower0.1em\copy0 \special{color pop}}\vss}\box0}}% \long\def\shadedparagraph#1\par{{\setbox0=\vbox{\hsize=\hsize#1}% \noindent\leavevmode \vtop to 0pt{\rlap{\special{color push rgb 0.75 0.75 0.75}% \kern0.1em\lower0.1em\copy0 \special{color pop}}\vss}\box0\par}}% \newboolean{exact} \setboolean{exact}{true} \newcommand{\pythahypo}[5]{ \opset{decimalsepsymbol={,}} \opcopy{#4}{A1} \opcopy{#5}{A2} Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire : $$\Eqalign{ #1#3^2&=#1#2^2+#2#3^2\cr #1#3^2&=\opprint{A1}^2+\opprint{A2}^2\cr #1#3^2&=\opmul*{A1}{A1}{a1}\opprint{a1}+\opmul*{A2}{A2}{a2}\opprint{a2}\cr #1#3^2&=\opadd*{a1}{a2}{a3}\opprint{a3}\cr #1#3&=\sqrt{\opprint{a3}}\cr \ifthenelse{\boolean{exact}}{#1#3&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#3&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4} \opprint{a4}}\cr }$$ } \newcommand{\pythadroit}[5]{ \opset{decimalsepsymbol={,}} \opcopy{#4}{A1} \opcopy{#5}{A2} Dans le triangle $#1#2#3$ rectangle en $#2$, le théorème de Pythagore permet d'écrire : $$\Eqalign{ #1#3^2&=#1#2^2+#2#3^2\cr \opprint{A1}^2&=#1#2^2+\opprint{A2}^2\cr \opmul*{A1}{A1}{a1}\opprint{a1}&=#1#2^2+\opmul*{A2}{A2}{a2}\opprint{a2}\cr #1#2^2&=\opmul*{A1}{A1}{a1}\opprint{a1}-\opmul*{A2}{A2}{a2}\opprint{a2}\cr #1#2^2&=\opsub*{a1}{a2}{a3}\opprint{a3}\cr #1#2&=\sqrt{\opprint{a3}}\cr \ifthenelse{\boolean{exact}}{#1#2&=\opsqrt[maxdivstep=3]{a3}{a4}\opunzero{a4}\opprint{a4}}{#1#2&\approx\opsqrt[maxdivstep=3]{a3}{a4}\opround{a4}{2}{a4}\opunzero{a4} \opprint{a4}}\cr }$$ } \newcommand{\Thales}[5]{ Dans le triangle $#1#2#3$, $#4$ est un point de la droite $(#1#2)$, $#5$ est un point de la droite $(#1#3)$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles. Le théorème de Thalès permet d'écrire : $$\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}$$ } \newcommand{\Thalesf}[5]{ Dans le triangle $#1#2#3$, $#4$ est un point du segment $[#1#2]$, $#5$ est un point du segment $[#1#3]$ ; les droites $(#4#5)$ et $(#2#3)$ sont parallèles. L'égalité des 3 rapports permet d'écrire : $$\frac{#1#4}{#1#2}=\frac{#1#5}{#1#3}=\frac{#4#5}{#2#3}$$ } \newcommand{\ResolThales}[6]{ \opset{decimalsepsymbol={,}} \opcopy{#3}{a3} \opcopy{#4}{a4} \opcopy{#5}{a5} On utilise $$\Eqalign{ \frac{#1#2}{\opprint{a3}}&=\frac{\opprint{a4}}{\opprint{a5}}\cr #1#2&=\frac{\opprint{a3}\times\opprint{a4}}{\opprint{a5}}\cr #1#2&=\frac{\opmul*{a3}{a4}{a6}\opprint{a6}}{\opprint{a5}}\cr #1#2&=\opdiv*{a6}{a5}{a7}{a8}\opprint{a7}\cr }$$ La longueur $#1#2$ mesure \opprint{a7}\, #6 } %%QCM \newcounter{qqcm} %définir un booléen qui permet de choisir la correction ou non \newboolean{solution} %définir une commande \V qui permet de changer le carré en carré coché suivant la valeur du booléen. \newcommand{\V}[1]{\ifthenelse{\boolean{solution}}{$\boxtimes$\kern2mm #1}{$\Box$\kern2mm #1}} \newcommand{\F}[1]{$\Box$\kern2mm #1} \newcommand{\vr}{\ifthenelse{\boolean{solution}}{$\boxtimes$}{$\Box$}} \newcommand{\fa}{$\Box$} %%QCM Version 2 \newenvironment{Qcm}[1][2]{\par\setboolean{solution}{false} \setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5} \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular} \renewcommand{\arraystretch}{1}} \newenvironment{Qcmcor}[1][2]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5} \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/#1}|*{#1}{l|}}\hline}{\hline\end{tabular} \renewcommand{\arraystretch}{1}} \newcommand{\QCM}[3]{\setboolean{solution}{false} \setcounter{qqcm}{0} \renewcommand{\arraystretch}{1.5} \newcounter{taille} \setcounter{taille}{#1} \addtocounter{taille}{1} \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}} \hline \multicolumn{\thetaille}{|c|}{{\sc #2}}\\ #3 \hline \end{tabularx} } \newcommand{\QCMcor}[3]{\setboolean{solution}{true} \setcounter{qqcm}{0} \renewcommand{\arraystretch}{1.5} \newcounter{taille} \setcounter{taille}{#1} \addtocounter{taille}{1} \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}} \hline \multicolumn{\thetaille}{|c|}{{\sc #2}}\\ #3 \hline \end{tabularx} } \newcommand{\QCMvar}[4]{\setboolean{solution}{false} \setcounter{qqcm}{0} \renewcommand{\arraystretch}{#2} \newcounter{taille} \setcounter{taille}{#1} \addtocounter{taille}{1} \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}} \hline \multicolumn{\thetaille}{|c|}{{\sc #3}}\\ #4 \hline \end{tabularx} } \newcommand{\QCMvarcor}[4]{\setboolean{solution}{true} \setcounter{qqcm}{0} \renewcommand{\arraystretch}{#2} \newcounter{taille} \setcounter{taille}{#1} \addtocounter{taille}{1} \begin{tabularx}{\linewidth}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}X|*{#1}{l|}} \hline \multicolumn{\thetaille}{|c|}{{\sc #3}}\\ #4 \hline \end{tabularx} } \newenvironment{VF}[1]{\par\setboolean{solution}{false} \setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5} \begin{center} \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline \multicolumn{3}{|c|}{#1}\\ \hline \multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\ \hline }{\hline\end{tabular} \end{center}\renewcommand{\arraystretch}{1}} \newenvironment{VFcor}[1]{\par\setboolean{solution}{true}\setcounter{qqcm}{0}\renewcommand{\arraystretch}{1.5} \begin{tabular}{|>{\small\stepcounter{qqcm}{\bf \theqqcm/}\,}b{\linewidth/2}|*{2}{c|}}\hline \multicolumn{3}{|c|}{#1}\\ \hline \multicolumn{1}{|c|}{\bf Question}&\multicolumn{1}{c|}{\bf Vrai}&\multicolumn{1}{c|}{\bf Faux}\\ \hline }{\hline\end{tabular} \renewcommand{\arraystretch}{1}}