85 lines
2.7 KiB
TeX
85 lines
2.7 KiB
TeX
|
\documentclass[a4paper,12pt]{article}
|
||
|
|
\usepackage{myXsim}
|
||
|
|
|
||
|
|
\author{Benjamin Bertrand}
|
||
|
|
\title{Géométrie repérée - Cours}
|
||
|
|
\date{2022-01-13}
|
||
|
|
|
||
|
|
\pagestyle{empty}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
|
||
|
|
\maketitle
|
||
|
|
|
||
|
|
\bigskip
|
||
|
|
|
||
|
|
\section*{Milieu d'un segment}
|
||
|
|
|
||
|
|
\begin{propriete}[Coordonnée du milieu d'un segment]
|
||
|
|
\begin{minipage}{0.5\linewidth}
|
||
|
|
Soit $M$ et $N$ deux points quelconques et $K$ le milieu du segment $[MN]$. Alors
|
||
|
|
\begin{itemize}
|
||
|
|
\item L'abscisse de $K$ est la moyenne des abscisses de $M$ et $N$
|
||
|
|
\[
|
||
|
|
x_K = \frac{x_M + x_N}{2}
|
||
|
|
\]
|
||
|
|
\item L'ordonnée de $K$ est la moyenne des ordonnées de $M$ et $N$
|
||
|
|
\[
|
||
|
|
y_K = \frac{y_M + y_N}{2}
|
||
|
|
\]
|
||
|
|
\end{itemize}
|
||
|
|
\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, 0) node {+} node [below left] {1};
|
||
|
|
\draw (0, 1) node {+} node [below left] {1};
|
||
|
|
|
||
|
|
\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 [below 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 (2.3, 2.4) node {+} node [below right] {$K$};
|
||
|
|
\draw (2.3, 0) node {+} node [below] {$x_K$};
|
||
|
|
\draw (0, 2.4) node {+} node [left] {$y_K$};
|
||
|
|
|
||
|
|
\draw[dashed] (2.3, 2.4) --(2.3, 0);
|
||
|
|
\draw[dashed] (2.3, 2.4) --(0, 2.4);
|
||
|
|
|
||
|
|
\draw (1.3, 1.4) -- node [midway, sloped] {//}
|
||
|
|
(2.3, 2.4) -- node [midway, sloped] {//}
|
||
|
|
(3.3, 3.4);
|
||
|
|
\draw (1.3, 0) -- node [midway, sloped] {$\bullet$}
|
||
|
|
(2.3, 0) -- node [midway, sloped] {$\bullet$}
|
||
|
|
(3.3, 0);
|
||
|
|
\draw (0, 1.4) -- node [midway, sloped] {$\diamond$}
|
||
|
|
(0, 2.4) -- node [midway, sloped] {$\diamond$}
|
||
|
|
(0, 3.4);
|
||
|
|
\end{tikzpicture}
|
||
|
|
\end{minipage}
|
||
|
|
\end{propriete}
|
||
|
|
|
||
|
|
\paragraph{Exemple}: Coordonnée de $I$ milieu du segment $[AB]$ avec $A(23; 45)$ et $B (-3; 12)$
|
||
|
|
|
||
|
|
\[
|
||
|
|
x_I = \frac{x_A + x_B}{2} = \frac{23 + (-3)}{2} = 10
|
||
|
|
\qquad \qquad
|
||
|
|
y_I = \frac{y_A + y_B}{2} = \frac{45 + 12}{2} = 28.5
|
||
|
|
\]
|
||
|
|
Les coordonnées de $I$ sont $(10; 28.5)$.
|
||
|
|
|
||
|
|
\end{document}
|