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b/2nd/10_Geometrie_reperee/3B_abs_distance.tex new file mode 100644 index 0000000..f20fd7f --- /dev/null +++ b/2nd/10_Geometrie_reperee/3B_abs_distance.tex @@ -0,0 +1,109 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Géométrie repérée - Cours} +\date{2022-01-13} + +\pagestyle{empty} + +\begin{document} + +\section*{Distance entre deux points d'une droite} + +\begin{center} + \includegraphics[scale=0.8]{./fig/eleve_distance} +\end{center} + +\begin{propriete}[Valeur absolue] + La \textbf{valeur absolue d'une nombre $a$}, noté $|a|$ est égale à + \begin{itemize} + \item $a$ si $a \geq 0$ + \item $-a$ si $a < 0$ + \end{itemize} +\end{propriete} + +\paragraph{Exemples:} +\[ + |3| = 3 \qquad \qquad |0| = 0 \qquad \qquad |-4| = - (-4) = 4 +\] + +\begin{propriete}[Distance entre deux points sur une droite] + $a$ et $b$ deux nombres. Alors la distance entre $a$ et $b$ est égale à $| b - a |$. +\end{propriete} + +\paragraph{Exemples:}~ + +\begin{multicols}{2} + \begin{itemize} + \item La distance entre $-3$ et $4$ est + % \[ + % | 4 - (-3) | = | 4 + 3 | = | 7 | = 7 + % \] + \item La distance entre $-3$ et $-7$ est + % \[ + % | -7 - (-3) | = | -7 + 3 | = | -4 | = 4 + % \] + \end{itemize} +\end{multicols} +\vspace{1cm} + + +% \begin{propriete}[Lien avec la racine carré] +% Soit $x$ un nombre réel, Alors +% \[ +% \sqrt{x^2} = |x| +% \] +% \end{propriete} + +\section*{Distance entre deux points du plan} + + +\begin{propriete}[Distance entre deux points] + \begin{minipage}{0.5\linewidth} + Soit $M (x_M; y_M)$ et $N (x_N; y_N)$ deux points quelconques. Alors la distance entre $M$ et $N$ se calcule + \[ + NM = \sqrt{(x_M - x_N)^2 + (y_M - y_N)^2} + \] + \end{minipage} + \hfill + \begin{minipage}{0.4\linewidth} + \begin{tikzpicture}[scale=1.2] + \draw[->, very thick] (-1, 0) -- (4, 0); + \draw[->, very thick] (0, -1) -- (0, 4); + \draw (0, 0) node [below left] {0}; + + \draw (1.3, 1.4) node {+} node [below left] {$M$}; + \draw (1.3, 0) node {+} node [below] {$x_M$}; + \draw (0, 1.4) node {+} node [left] {$y_M$}; + + \draw[dashed] (1.3, 1.4) --(1.3, 0); + \draw[dashed] (1.3, 1.4) --(0, 1.4); + + \draw (3.3, 3.4) node {+} node [above right] {$N$}; + \draw (3.3, 0) node {+} node [below] {$x_N$}; + \draw (0, 3.4) node {+} node [left] {$y_N$}; + + \draw[dashed] (3.3, 3.4) --(3.3, 0); + \draw[dashed] (3.3, 3.4) --(0, 3.4); + + \draw (1.3, 1.4) -- (3.3, 3.4); + + \draw (1.3, 1.4) -- node [midway, below] {$|x_M - x_N|$} + (3.3, 1.4); + \draw (3.3, 1.4) -- node [midway, below, sloped] {$|y_M - y_N|$} + (3.3, 3.4); + \end{tikzpicture} + \end{minipage} +\end{propriete} + +\paragraph{Exemple:} Distance entre $A (3; 4)$ et $B(-2; 0)$ + +% \[ +% AB = \sqrt{(3 - (-2))^2 + (4 - 0)^2} = \sqrt{ 25 + 16 } = \sqrt{41} \approx 6.4 +% \] + +\vfill + + +\end{document} diff --git a/2nd/10_Geometrie_reperee/fig/eleve_distance.png b/2nd/10_Geometrie_reperee/fig/eleve_distance.png new file mode 100644 index 0000000000000000000000000000000000000000..199de89fe0d0b2441c9f37b327e0b8ae4c918ab5 GIT binary patch literal 55694 zcmXtg2RzpA_db%n_ns-qCK-=CLy}}Ady}0V*?T23BZP$TLC9X&SxI&_$=(_L&-?p( z{a>Fxy-JVgx!?DF?sKklUDpw*si91WM~8=ohDLZ_ML`P<4MPb2`~n9Hes35s$v{Ix z53`k**Ss$;&*JRrWM%7MiH63R_%cyKr9+b<>|bFNF&B{s$wor1xIPIbsbMvq;s+AD zP#s2f$G<^03GYWm=l?Qr-mTPo;q3BbyzL1ICNoQ%hDH>x)?)qrT*Vk{-jk)4wSV;| zhB6;g)2&{_pl3L~Emklbu+Z)Lq>D)+OPrUMm~J_6UyH2y=*7cXZBK#hmH89vqhhgW zbo4~86xzZEjk2g3Tl&nC$c|FZlFqfx{F~1bP7^nyx}7rddGS#~4uy{oI!(rHS)>c!Z3=90V}>;vKU#yhYo`uYcpDrHmYJHe0@^?1 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