110 lines
2.9 KiB
TeX
110 lines
2.9 KiB
TeX
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\documentclass[a4paper,12pt]{article}
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\usepackage{myXsim}
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\author{Benjamin Bertrand}
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\title{Géométrie repérée - Cours}
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\date{2022-01-13}
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\pagestyle{empty}
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\begin{document}
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\section*{Distance entre deux points d'une droite}
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\begin{center}
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\includegraphics[scale=0.8]{./fig/eleve_distance}
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\end{center}
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\begin{propriete}[Valeur absolue]
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La \textbf{valeur absolue d'une nombre $a$}, noté $|a|$ est égale à
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\begin{itemize}
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\item $a$ si $a \geq 0$
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\item $-a$ si $a < 0$
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\end{itemize}
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\end{propriete}
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\paragraph{Exemples:}
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\[
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|3| = 3 \qquad \qquad |0| = 0 \qquad \qquad |-4| = - (-4) = 4
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\]
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\begin{propriete}[Distance entre deux points sur une droite]
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$a$ et $b$ deux nombres. Alors la distance entre $a$ et $b$ est égale à $| b - a |$.
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\end{propriete}
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\paragraph{Exemples:}~
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\begin{multicols}{2}
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\begin{itemize}
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\item La distance entre $-3$ et $4$ est
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% \[
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% | 4 - (-3) | = | 4 + 3 | = | 7 | = 7
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% \]
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\item La distance entre $-3$ et $-7$ est
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% \[
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% | -7 - (-3) | = | -7 + 3 | = | -4 | = 4
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% \]
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\end{itemize}
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\end{multicols}
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\vspace{1cm}
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% \begin{propriete}[Lien avec la racine carré]
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% Soit $x$ un nombre réel, Alors
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% \[
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% \sqrt{x^2} = |x|
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% \]
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% \end{propriete}
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\section*{Distance entre deux points du plan}
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\begin{propriete}[Distance entre deux points]
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\begin{minipage}{0.5\linewidth}
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Soit $M (x_M; y_M)$ et $N (x_N; y_N)$ deux points quelconques. Alors la distance entre $M$ et $N$ se calcule
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\[
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NM = \sqrt{(x_M - x_N)^2 + (y_M - y_N)^2}
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\]
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\end{minipage}
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\hfill
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\begin{minipage}{0.4\linewidth}
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\begin{tikzpicture}[scale=1.2]
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\draw[->, very thick] (-1, 0) -- (4, 0);
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\draw[->, very thick] (0, -1) -- (0, 4);
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\draw (0, 0) node [below left] {0};
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\draw (1.3, 1.4) node {+} node [below left] {$M$};
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\draw (1.3, 0) node {+} node [below] {$x_M$};
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\draw (0, 1.4) node {+} node [left] {$y_M$};
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\draw[dashed] (1.3, 1.4) --(1.3, 0);
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\draw[dashed] (1.3, 1.4) --(0, 1.4);
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\draw (3.3, 3.4) node {+} node [above right] {$N$};
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\draw (3.3, 0) node {+} node [below] {$x_N$};
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\draw (0, 3.4) node {+} node [left] {$y_N$};
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\draw[dashed] (3.3, 3.4) --(3.3, 0);
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\draw[dashed] (3.3, 3.4) --(0, 3.4);
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\draw (1.3, 1.4) -- (3.3, 3.4);
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\draw (1.3, 1.4) -- node [midway, below] {$|x_M - x_N|$}
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(3.3, 1.4);
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\draw (3.3, 1.4) -- node [midway, below, sloped] {$|y_M - y_N|$}
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(3.3, 3.4);
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\end{tikzpicture}
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\end{minipage}
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\end{propriete}
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\paragraph{Exemple:} Distance entre $A (3; 4)$ et $B(-2; 0)$
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% \[
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% AB = \sqrt{(3 - (-2))^2 + (4 - 0)^2} = \sqrt{ 25 + 16 } = \sqrt{41} \approx 6.4
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% \]
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\vfill
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\end{document}
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