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$B (...; ...)$ + \item $C (...; ...)$ + \end{itemize} + + + Points à placer + + \begin{itemize} + \item $D (-2; 3)$ + \item $E (-1; -1)$ + \item $F (2; -3)$ + \end{itemize} +\end{minipage} + +\afaire{Trouver les coordonnées des points et placer les points} + +Cette géométrie introduite par René Descarte au XVII$^e$ siècle, permet d'introduire le calcul dans la géométrie. Dans la suite nous allons voir comment \textbf{calculer} le milieu d'un segment, une distance entre deux points ou encore si un point est sur une droite. + + +\end{document} diff --git a/2nd/11_Geometrie_reperee/2B_milieu.pdf b/2nd/11_Geometrie_reperee/2B_milieu.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0df35b36d38d8dfc4a8ca47ed1780b2b3aa5dcce GIT binary patch literal 27737 zcmce-1CVUZ)+XGxZQJ%~+ox^Ywr$(CZQHhOpVRK1^L{hm-22D8|BZ=>xVIwqt{u6v za@Dif%B*^xwUSg`Sd^BLjunb@YPo-Dcx`TI1d4@#fxyAzY;Z5^x$=>IzZR}V%4dO<;3Hv-K+A7vn5WcZ(lHUYhizLSIbU+s+l z^Ui;r|FkO@JK8!s7#cegu>NUM_%oEbm6NfgzwZ3G`s@3%{~7r|hlAtahJ*7z zii7bl5B_@`!uVGZ{O3Hv$n@_+@}G?@#(!UMG5zK4|C~>l{tAJAPjoRc|7Vf>wM_n- zkg)w{k^B|R{~jdFe@(&vj;?>T#4|H-u>WI`|Hl%4tvPP9C63g6N&NymHRBBk0@y%` zc_(8C?8xAy5ZlQZhh?Yx_IxTrXT;P_qRa|Bm9L*#5rG;NSB#WbS<1b12?OnDT{pm|^@$@^WMzeKxj! zEbr&f%hL#v!>ZqVeN_2ZPk8QIUy#zgKHabF2NRYhK{z@)HfJ3_b1-Yc28&h)bHA~C zws=%49nPBoB`Jf5mloXU7b`aTlnYP;Kn;T`MJhq|1cG3vG&4o02ZRuVzx+~%E+L8R 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\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} diff --git a/2nd/11_Geometrie_reperee/3B_abs_distance.pdf 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+\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/11_Geometrie_reperee/3B_distance.tex b/2nd/11_Geometrie_reperee/3B_distance.tex new file mode 100644 index 0000000..ae80983 --- /dev/null +++ b/2nd/11_Geometrie_reperee/3B_distance.tex @@ -0,0 +1,64 @@ +\documentclass[a4paper,12pt]{article} +\usepackage{myXsim} + +\author{Benjamin Bertrand} +\title{Géométrie repérée - Cours} +\date{2022-01-13} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\bigskip + +\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 $B(-2; 0)$ + +\[ + AB = \sqrt{(3 - (-2))^2 + (4 - 0)^2} = \sqrt{ 25 + 16 } = \sqrt{41} \approx 6.4 +\] + + +\end{document} diff --git a/2nd/11_Geometrie_reperee/4B_ensembles_points.pdf b/2nd/11_Geometrie_reperee/4B_ensembles_points.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f1840622d2f64eb98d9a0f90b85789865c3ac88e GIT binary patch literal 22990 zcmce-1CS_9*5}=}ZQHhOci*;cyZg3v+qP}nwr$&X@AJI-?KiWr6K`y6#7sq0Rz(&v zPn>_9$ogd-QUwt)T1GlnDALKLzQv){*}-8b76JwWJ3~t-9v%XE6%Tt80(y0MLrW7Q zX99X<7enX&su8nuv>~AXyZo;^7zyZwgzVf2wElXOfq;?Wf9^U2^s)xdjuwB{Gycyj z|5g4~uVmt6=i+E&;zYpuSC!IVud=XqHgWvxXl>wZB4T1>XZ%;YGA6cW&gKM+tOR^~ z1pjM;a&mSwF|dJhUtfTqR-l;P1p)y70=$JXu{HkxuKacOcmLV{dh>tI4vv4D9i0DB zI~f1=;J?QqjDL&ZKj#rfrhlI$|5-j4cf&N6`OeNZ-zcT-tfB&Y2+Z& zfIz@-Jzb00OLiO`ytlK3g%^7HovbXAar=`jfam)oNW|M5<8db&8mB%d@aV42x||mw z3i?4t2tXl-hfLoCU~_N2{SiFZ@i1ONxcN)SfF6_n#Raz^q?xxACS*QEFz&qo&d>XM 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Nous reviendrons dessus plus en détails plus tard dans l'année. + +\bigskip +\hspace{-1cm} +\begin{minipage}{0.5\linewidth} + \begin{itemize} + \item On a noté $(a)$ \textbf{l'ensemble des points d'ordonnée égal à 2}. + \begin{itemize} + \item $U(2; 4)$ n'est pas un point de l'ensemble $(a)$ car son ordonnée est 4 et non 2. On note $U \not\in (a)$ + \item $A(-2; 2)$ est un point de l'ensemble $(a)$ car son ordonnée est 2. On note $A \in (a)$ + \item Un point quelconque $M$ de coordonnées $(x; y)$ est un point de $(a)$ si et seulement si $y=2$ + \end{itemize} + On dit que $(a)$ a pour \textbf{équation} $y = 2$ + + \item On a noté $(b)$ \textbf{l'ensemble des points d'ordonnée égal à l'abscisse}. + \begin{itemize} + \item $U(2; 4)$ n'est pas un point de l'ensemble $(b)$ car son ordonnée est 4 et son abscisse est 2. On note $U \not\in (b)$ + \item $B(2; 2)$ est un point de l'ensemble $(b)$ car son ordonnée est 2 et son abscisse est 2. On note $B \in (b)$ + \item Un point quelconque $M$ de coordonnées $(x; y)$ est un point de $(b)$ si et seulement si $y=x$ + \end{itemize} + On dit que $(b)$ a pour \textbf{équation} $y = -x$ + \end{itemize} +\end{minipage} +\hfill +\begin{minipage}{0.45\linewidth} + \begin{tikzpicture}[scale=0.8] + \repere{-5}{5}{-5}{5} + + + \end{tikzpicture} +\end{minipage} + +\afaire{Placer les points $A(-2; 2)$, $B(2; 2)$, $C(-4; 3)$ et $U(2; 4)$ dans le repère. Puis tracer les ensembles $(a)$ et $(b)$} + +\end{document} diff --git a/2nd/11_Geometrie_reperee/exercises.tex b/2nd/11_Geometrie_reperee/exercises.tex new file mode 100644 index 0000000..d3095df --- /dev/null +++ b/2nd/11_Geometrie_reperee/exercises.tex @@ -0,0 +1,611 @@ +\begin{exercise}[subtitle={Milieu d'un segment}, step={1}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnées, milieu}, mode={\faIcon{search}}] + On définit les points suivants + \begin{multicols}{5} + \begin{enumerate}[label={$\Alph* $}] + \item $(2; 4)$ + \item $(-1; 4)$ + \item $(2; -1)$ + \item $(0; 3)$ + \item $(-2; -3)$ + \end{enumerate} + \end{multicols} + \begin{enumerate} + \item Tracer un repère orthonormé et y placer les points. + \item Déterminer les coordonnées des points suivants + \begin{multicols}{4} + \begin{enumerate} + \item $W$ milieu de $[AB]$ + \item $X$ milieu de $[AC]$ + \item $Y$ milieu de $[AD]$ + \item $Z$ milieu de $[BE]$ + \end{enumerate} + \end{multicols} + \item Proposer une méthode pour déterminer les coordonnées du milieu d'un segment sans avoir à faire un dessin. + \item Appliquer cette méthode pour déterminer les coordonnées du milieu du segment $[MN]$ où $M(456; 289)$ et $N (251; - 20)$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{tikzpicture} + \draw (-3, -4) grid (3, 5); + \draw[->, very thick] (-3, 0) -- (3, 0); + \draw[->, very thick] (0, -4) -- (0, 5); + \draw (0, 0) node [below left] {0}; + \draw (1, 0) node [below left] {1}; + \draw (0, 1) node [below left] {1}; + + \draw (2, 4) node {x} node [below left] {$A$}; + \draw (-1, 4) node {x} node [below left] {$B$}; + \draw (2, -1) node {x} node [below left] {$C$}; + \draw (0, 3) node {x} node [below left] {$D$}; + \draw (-2, -3) node {x} node [below left] {$E$}; + \end{tikzpicture} +\end{solution} + +\begin{exercise}[subtitle={Bilan sur les coordonnées le milieu d'un segment}, step={1}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnées, milieu}, mode={\faIcon{users}}] + En groupe, expliquer votre méthode pour déterminer les coordonnées du milieu d'un segment en connaissant les coordonnées de ses extrémités. Vous illustrerez votre méthode en traitant un exemple que vous vérifierez avec un dessin. +\end{exercise} + +\begin{exercise}[subtitle={Exercice technique}, step={1}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnées, milieu}, mode={\faIcon{tools}}] + On définit les points suivants + \begin{multicols}{5} + \begin{enumerate}[label={$\Alph*$}] + \item $(2; 6)$ + \item $(-4; 0)$ + \item $(0; 3)$ + \item $(-2; -2)$ + \item $(23; 95)$ + \end{enumerate} + \end{multicols} + Calculer les coordonnées du milieu des segments suivants + \begin{multicols}{6} + \begin{enumerate} + \item $[AB]$ + \item $[CD]$ + \item $[AD]$ + + \item $[CE]$ + \item $[EA]$ + \item $[EB]$ + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{center} + \begin{tikzpicture} + \draw (-5, -3) grid (3, 6); + \draw[->, very thick] (-5, 0) -- (3, 0); + \draw[->, very thick] (0, -3) -- (0, 6); + \draw (0, 0) node [below left] {0}; + \draw (1, 0) node [below left] {1}; + \draw (0, 1) node [below left] {1}; + + \draw (2, 6) node {x} node [below left] {$A$}; + \draw (-4, 0) node {x} node [below left] {$B$}; + \draw (0, 3) node {x} node [below left] {$C$}; + \draw (-2, -2) node {x} node [below left] {$D$}; + \end{tikzpicture} + \end{center} + \begin{enumerate} + \item Coordonnées du milieu du segment $[AB]$ + \[ + x = \frac{x_A + x_B}{2} = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1 \qquad + y = \frac{y_A + y_B}{2} = \frac{6 + 0}{2} = \frac{6}{2} = 3 + \] + Les coordonnées du milieu sont $\left(-1; 3\right)$ + + \item Coordonnées du milieu du segment $[CD]$ + \[ + x = \frac{x_C + x_D}{2} = \frac{0 + (-2)}{2} = \frac{-2}{2} = -1 \qquad + y = \frac{y_C + y_D}{2} = \frac{3 + (-2)}{2} = \frac{1}{2} + \] + Les coordonnées du milieu sont $\left(-1; \dfrac{1}{2}\right)$ + + \item Coordonnées du milieu du segment $[AD]$ + \[ + x = \frac{x_A + x_D}{2} = \frac{2 + (-2)}{2} = 0 \qquad + y = \frac{y_A + y_D}{2} = \frac{6 + (-2)}{2} = \frac{4}{2} = 2 + \] + Les coordonnées du milieu sont $\left(0; 2\right)$ + + \item Coordonnées du milieu du segment $[CE]$ + \[ + x = \frac{x_C + x_E}{2} = \frac{0 + 23}{2} = 11.5 \qquad + y = \frac{y_C + y_E}{2} = \frac{3 + 95}{2} = 49 + \] + Les coordonnées du milieu sont $\left(11.5; 49\right)$ + + + \item Coordonnées du milieu du segment $[EA]$ + \[ + x = \frac{x_A + x_E}{2} = \frac{2 + 23}{2} = 25 \qquad + y = \frac{y_A + y_E}{2} = \frac{6 + 95}{2} = 50.5 + \] + Les coordonnées du milieu sont $\left(25; 50.5\right)$ + + \item Coordonnées du milieu du segment $[EB]$ + \[ + x = \frac{x_B + x_E}{2} = \frac{-4 + 23}{2} = 9.5 \qquad + y = \frac{y_B + y_E}{2} = \frac{0 + 95}{2} = 47.5 + \] + Les coordonnées du milieu sont $\left(9.5; 47.5 \right)$ + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Exercice technique}, step={1}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, milieu}, mode={\faIcon{tools}}] + On considère les points $E(1; -1)$, $F(5; 3)$, $C(3; 1)$ et $H(1; 3)$. + \begin{enumerate} + \item Construire un repère puis y placer les points. + \item Démontrer que $C$ est le milieu du segment $[EF]$. + \item Quelles sont les coordonnées du point $G$ tel que $C$ soit le milieu de $[HG]$? + \item Quelle est la nature du quadrilatère $EGFH$? + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item ~ + \begin{center} + \begin{tikzpicture} + \draw (0, -2) grid (6, 4); + \draw[->, very thick] (0, 0) -- (6, 0); + \draw[->, very thick] (0, -2) -- (0, 4); + \draw (0, 0) node [below left] {0}; + \draw (1, 0) node [below left] {1}; + \draw (0, 1) node [below left] {1}; + + \draw (1, -1) node {x} node [below left] {$E$}; + \draw (5, 3) node {x} node [below left] {$F$}; + \draw (3, 1) node {x} node [below left] {$C$}; + \draw (1, 3) node {x} node [below left] {$H$}; + \end{tikzpicture} + \end{center} + \item \textbf{On sait que} $E(1; -1)$, $F(5; 3)$ et que $C(3; 1)$ + + \textbf{Or} le milieu du segment $[EF]$ se calcule de la manière suivante + \[ + x = \frac{x_E + x_F}{2} = \frac{1 + 5}{2} = 3 \qquad + y = \frac{y_E + y_F}{2} = \frac{-1 + 3}{2} = 1 + \] + \textbf{Donc} $C$ est bien le milieu du segment $[EF]$. + \item On note $(x_G; y_G)$ les coordonnées du point $G$. + + \textbf{On sait que} $C$ est le milieu de $HG$ + + \textbf{Or} d'après la formule du milieu + \begin{align*} + x_C = \frac{x_H + x_G}{2} &\qquad y_C = \frac{y_H + y_G}{2} \\ + 3 = \frac{1 + x_G}{2} & \qquad 1 = \frac{3 + y_G}{2} \\ + 6 = 1 + x_G & \qquad 2 = 3 + y_G \\ + 5 = x_G & \qquad -1 = y_G \\ + \end{align*} + \textbf{Donc} $G(5; -1)$ + \item On sait que $C$ est le milieu des diagonales de $EGFH$ + + Or un quadrilatère qui a ses diagonales qui se coupent en leur milieu est un parallélogramme. + + Donc $EGFH$ est un parallélogramme. + + + \textit{Remarque:} On voit que c'est aussi un carré mais il faudrait encore du travail pour démontrer que s'en est un. + \end{enumerate} +\end{solution} + + +% ---- étape 2 +\begin{exercise}[subtitle={Distance sur une droite}, step={2}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, distance}, mode={\faIcon{search}}] + On considère une droite munie d'un repère et deux points $A$ et $B$ de cette droite. + + Comme la droite est munie d'un repère, on peut considérer les abscisses $x_A$ et $x_B$ de ces deux points. + + \begin{enumerate} + \item Dans cette question, on suppose que $x_A = 2$ et $x_B = 9$. + \begin{center} + \begin{tikzpicture} + \foreach \x in {0, 1, ..., 10} {% + \draw (\x, 0.1) -- (\x, -0.1) node [below] {\x}; + } + \draw[->] (-0.5, 0) -- (10.5, 0); + \draw (2, 0) node {x} node [above] {$A$}; + \draw (9, 0) node {x} node [above] {$B$}; + \end{tikzpicture} + \end{center} + Proposer une formule utilisant $x_A$ et $x_B$ pour calculer la distance $AB$. + \item Même question pour $x_A = 58$ et $x_B = 9$. + \item Même question pour $x_A = 3$ et $x_B = -2$. + \item On suppose que $x_A$ et $x_B$ peuvent prendre n'importe quelle valeur. Déterminer une façon de calculer la distance $AB$ en utilisant $x_A$ et $x_B$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Bilan sur distance sur une droite}, mode={\faIcon{users}}, step={2}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, distance}] + Faire le bilan des méthodes trouvées dans l'exercice précédent puis rédiger en groupe une méthode commune pour calculer la distance entre deux points placés sur l'axe des abscisses. +\end{exercise} + +\begin{exercise}[subtitle={Distance entre deux points}, step={2}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnées, distance}, mode={\faIcon{search}}] + On définit les points suivants + \begin{multicols}{5} + \begin{enumerate}[label={$\Alph*$}] + \item $(1; 1)$ + \item $(-1; 1)$ + \item $(2; 4)$ + \item $(-1; 3)$ + \item $(2; -1)$ + \end{enumerate} + \end{multicols} + \begin{enumerate} + \item Tracer un repère orthonormé et y placer les points. + \item Calculer les distances suivantes + \begin{multicols}{3} + \begin{enumerate} + \item $AB$ + \item $BD$ + \item $DE$ + \end{enumerate} + \end{multicols} + \item On souhaite calculer la longueur $AC$ + \begin{enumerate} + \item On note $P$ le projeté orthogonal de $C$ sur la droite $(AB)$. Placer ce point. + \item Quelle est la nature du triangle $APC$? + \item Calculer les longueurs $AP$ et $CP$. + \item En déduire la longueur $AC$. + \end{enumerate} + \item En utilisant le même procédé, calculer les distances + \begin{multicols}{3} + \begin{enumerate} + \item $BC$ + \item $EA$ + \item $DA$ + \end{enumerate} + \end{multicols} + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{tikzpicture} + \draw (-2, -1) grid (5, 5); + \draw[->, very thick] (-2, 0) -- (5, 0); + \draw[->, very thick] (0, -1) -- (0, 5); + \draw (0, 0) node [below left] {0}; + \draw (1, 0) node [below left] {1}; + \draw (0, 1) node [below left] {1}; + + \draw (4, 1) node {x} node [below left] {$A$}; + \draw (-1, 1) node {x} node [below left] {$B$}; + \draw (2, 4) node {x} node [below left] {$C$}; + \draw (-1, 3) node {x} node [below left] {$D$}; + \draw (2, -1) node {x} node [below left] {$E$}; + \end{tikzpicture} +\end{solution} + +\begin{exercise}[subtitle={Bilan sur distance entre deux points}, step={2}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, distance}, mode={\faIcon{users}}] + Proposer une formule pour calculer le distance entre deux points du plan. Vous illustrerez la formule avec un dessin et vous l'appliquerez à un exemple de votre choix. +\end{exercise} + +\begin{exercise}[subtitle={Exercice technique}, step={2}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, distance}, mode={\faIcon{tools}}] + Soit les points $M(3; -2)$, $N(-2; -3)$ et $P(-4; 3)$. + \begin{enumerate} + \item Placer ces points dans un repère. + \item Calculer les distance $MN$, $MP$ et $NP$. + \item Le triangle $MNP$ est-il rectangle? + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item ~ + \begin{tikzpicture} + \repere{-5}{4}{-4}{4} + + \draw (3, -2) node {x} node [below left] {$M$}; + \draw (-2, -3) node {x} node [below left] {$N$}; + \draw (-4, 3) node {x} node [below left] {$P$}; + \end{tikzpicture} + + \item + Distance $MN$ + \[ + MN = \sqrt{\left(3 - (-2)\right)^2 + \left( -2 - (-3)\right)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} + \] + Distance $MP$ + \[ + MP = \sqrt{\left(3 - (-4)\right)^2 + \left( -2 - 3\right)^2} = \sqrt{7^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74} + \] + Distance $NP$ + \[ + NP = \sqrt{\left(-2 - (-4)\right)^2 + \left( -3 - 3\right)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} + \] + \item On sait que $NM = \sqrt{26}$, $MP = \sqrt{74}$ et $NP = \sqrt{40}$ + + Or + \[ + NM^2 + NP^2 = \sqrt{26}^2 + \sqrt{40}^2 = 26 + 40 = 76 \qquad \qquad MP^2 = \sqrt{74}^2 = 74 + \] + Donc $NM^2 + NP^2 \neq MP^2$ + + Donc d'après le théorème de Pythagore le triangle $MNP$ n'est pas un triangle rectangle. + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Quadrilatère}, step={2}, origin={Sesamath 60p125}, topics={Géométrie Repérée}, tags={Coordonnées, distance}, mode={\faIcon{tools}}] + On considère les points $A(1; 2)$, $B(-6; 3)$, $C(6;7)$ et $D(-1; 8)$. + + Déterminer la nature du quadrilatère $BACD$. +\end{exercise} + +\begin{solution} + \begin{tikzpicture} + \repere{-7}{7}{0}{9} + \draw (1, 2) node {x} node [below left] {$A$}; + \draw (-6, 3) node {x} node [below left] {$B$}; + \draw (6, 7) node {x} node [below left] {$C$}; + \draw (-1, 8) node {x} node [below left] {$D$}; + \end{tikzpicture} + + On a l'impression que le quadrilatère $BACD$ est un losange. Pour le démontrer on va calculer la longueur de ses côtés. + + \begin{itemize} + \item + Distance $AB$ + \[ + AB = \sqrt{\left(1 - (-6)\right)^2 + \left( 2 - 3\right)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} + \] + \item + Distance $BD$ + \[ + BD = \sqrt{\left(-6 - (-1)\right)^2 + \left( 3 - 8\right)} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} + \] + \item + Distance $DC$ + \[ + DC = \sqrt{\left(6 - (-1)\right)^2 + \left( 7 - 8\right)} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} + \] + \item + Distance $CA$ + \[ + CA = \sqrt{\left(6 - 1\right)^2 + \left( 7 - 2\right)} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} + \] + \end{itemize} + Les quatre côtés du quadrilatère ont la même longueur, c'est donc un losange. +\end{solution} + +% ---- étape 3 + +\begin{exercise}[subtitle={BEAU rectangle}, step={3}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, milieu, distance}, mode={\faIcon{tools}}] + Soit $B(3; 2)$, $E(-1; -2)$, $A(-3; 0)$ et $U(1; 4)$ quatre points du plan. + \begin{enumerate} + \item Calculer les coordonnées du milieu de $[BA]$ + \item Calculer les coordonnées du milieu de $[EU]$ + \item Déterminer la nature du triangle $BEA$. + \item En déduire que $BEAU$ est un rectangle. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{center} + \begin{tikzpicture} + \repere{-4}{4}{-3}{5} + \draw (3, 2) node {x} node [below left] {$B$}; + \draw (-1, -2) node {x} node [below left] {$E$}; + \draw (-3, 0) node {x} node [below left] {$A$}; + \draw (1, 4) node {x} node [below left] {$U$}; + \end{tikzpicture} + \end{center} + \begin{enumerate} + \item Coordonnées du milieu de $[BA]$ + \[ + x = \frac{x_A + x_B}{2} = \frac{-3 + 3}{2} = 0 \qquad + y = \frac{y_A + y_B}{2} = \frac{0 + (-2)}{2} = 1 + \] + \item Coordonnées du milieu de $[EU]$ + \[ + x = \frac{x_E + x_U}{2} = \frac{-1 + 1}{2} = 0 \qquad + y = \frac{y_E + y_U}{2} = \frac{-2 + 4}{2} = 1 + \] + \item On a l'impression que le triangle $BEA$ est un triangle rectangle. + + \begin{itemize} + \item Longueur $BE$ + \[ + BE = \sqrt{\left(3 - (-1)\right)^2 + \left( 2 - (-2)\right)} = \sqrt{4^2+ 4^2} = \sqrt{16 + 16} = \sqrt{32} + \] + \item Longueur $EA$ + \[ + EA = \sqrt{\left(-1 - (-3)\right)^2 + \left( -2 - 0\right)} = \sqrt{2^2+ (-2)^2} = \sqrt{4 + 4} = \sqrt{8} + \] + \item Longueur $AB$ + \[ + AB = \sqrt{\left(3 - (-3)\right)^2 + \left( 2 - 0\right)} = \sqrt{6^2+ 2^2} = \sqrt{36+4} = \sqrt{40} + \] + \end{itemize} + Donc + \[ + BE^2 + EA^2 = \sqrt{32}^2 + \sqrt{8}^2 = 32 + 8 = 40 \qquad AB^2 = \sqrt{40}^2 = 40 + \] + Donc on sait que $BE^2 + EA^2 = AB^2$ + + Donc d'après le théorème de Pythagore, le triangle $BEA$ est rectangle en $E$. + + \item On sait que $BEAU$ est un quadrilatère et que les diagonales se coupent en leur milieu (questions 1 et 2) + + Or un quadrilatère qui a ses diagonales qui se coupent en leur milieu est un parallélogramme. + + Donc $BEAU$ est un parallélogramme. + + De plus, on sait que l'angle $\widehat{BEA}$ est un angle droit (question 3) + + Or un parallélogramme qui a un angle droit est un rectangle. + + Donc $BEAU$ est un rectangle. + \end{enumerate} +\end{solution} + +\begin{exercise}[subtitle={Presque}, step={3}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnées, milieu, distance}, mode={\faIcon{tools}}] + On a tracer la figure ci-dessous avec géogébra. + + \begin{center} + \includegraphics[scale=0.5]{./fig/deux_triangles} + \end{center} + \begin{multicols}{2} + \begin{enumerate} + \item Démontrer que $AC = \sqrt{\np{50 000}}$ + \item Le triangle $ABC$ est-il rectangle? + \item Le triangle $ACD$ est-il rectangle? + \item Peut on affirmer que $ABCD$ est un carré? + \end{enumerate} + \end{multicols} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Longueur $AC$ + \[ + AC = \sqrt{(-50 - 50)^2 + (100 - (-100))^2} = \sqrt{(-100)^2 + 200^2} = \sqrt{50 000} + \] + \item Il faut calculer les longueurs $AB$ et $BC$ puis appliquer le théorème de Pythagore. C'est la même rédaction que la question 3 de l'exercice 11. Le triangle est rectangle. + \item Il faut calculer les longueurs $AD$ et $DC$ puis appliquer le théorème de Pythagore. C'est la même rédaction que la question 3 de l'exercice 11. Le triangle n'est pas rectangle. + \item On sait que $ABCD$ est un quadrilatère et le triangle $ACD$ n'est pas rectangle. Or un carré est un quadrilatère qui a 4angles droits et 4côté de même longueur. Donc $ABCD$ n'est pas un carré. + \end{enumerate} +\end{solution} + +% ---- étape 4 + +\begin{exercise}[subtitle={Ensemble de points}, step={4}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnée, équation, ensemble}, mode={\faIcon{search}}] + \begin{enumerate} + \item Tracer un repère orthonormé. + \item Représenter sur le repère les ensembles suivants + \begin{enumerate} + \item L'ensemble $(a)$ constitué des points dont l'abscisse vaut 2. + \item L'ensemble des points dont l'ordonnée faut 3, on l'appelle $(b)$ + \item $(c) = \left\{ \mbox{points dont l'abscisse vaut -2} \right\}$ + \item $(d) = \left\{ \mbox{points dont l'ordonnée vaut 0} \right\}$ + \item L'ensemble $(e)$ des points dont l'ordonnée est égal à l'abscisse. + \end{enumerate} + \item À quels ensembles appartiennent les points suivants: + \[ + U(2, 4) \qquad \qquad + V(0, 4) \qquad \qquad + W(-2, -2) \qquad \qquad + X(2, 2) + \] + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Décrire un ensemble}, step={4}, origin={dMeedC}, topics={Géométrie repérée}, tags={Coordonnée, équation, ensemble}, mode={\faIcon{users}}] + Soit un $M$ un point du plan quelconque. On note $(x, y)$ ses coordonnées. + \begin{enumerate} + \item À quelle condition sur $x$ et $y$, le point $M$ est un point de la droite $(a)$? + \item Même question pour les ensembles $(b)$, $(c)$, $(d)$ et $(e)$. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Ensemble $y = 2x$}, step={4}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnée, équation, ensemble}, mode={\faIcon{tools}}] + On note $(a)$ l'ensemble des points tel que $y = 2x$. Cette ensemble est une droite. + \begin{enumerate} + \item Lesquels des points suivants sont dans cet ensemble. + \[ + U(2, 4) \qquad \qquad + V(1, -1) \qquad \qquad + W(-1, -2) \qquad \qquad + X(0, 0) + \] + \item Placer les points qui sont dans cet ensemble dans un repère puis tracer la droite $(a)$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Les points de l'ensemble $(a)$ vérifie $y=2x$ donc leur ordonnée doit être deux fois plus grand que leur abscisse. + \begin{itemize} + \item Pour le point $U$ + \[ + 2x = 2\times 2 = 4 = y + \] + Donc le point $U$ appartient à $(a)$ + \item Pour le point $V$ + \[ + 2x = 2\times 1 = 2 \neq -1 = y + \] + Donc le point $V$ n'appartient pas à $(a)$ + \item Pour le point $W$ + \[ + 2x = 2\times -2 = -4 = y + \] + Donc le point $W$ appartient à $(a)$ + \item Pour le point $X$ + \[ + 2x = 2\times 0 = 0 = y + \] + Donc le point $X$ appartient à $(a)$ + \end{itemize} + \item ~ + + \begin{tikzpicture} + \repere{-5}{5}{-5}{5} + + \draw (2, 4) node {x} node [below left] {$U$}; + \draw (1, -1) node {x} node [below left] {$V$}; + \draw (-1, -2) node {x} node [below left] {$W$}; + \draw (0, 0) node {x} node [above left] {$X$}; + + \draw(-2, -4) node [above left] {$(a)$}; + \draw[domain=-2.5:2.5] plot(\x, {2*\x}); + \end{tikzpicture} + \end{enumerate} +\end{solution} + + +\begin{exercise}[subtitle={Ensemble $y = -x$}, step={4}, origin={Création}, topics={Géométrie repérée}, tags={Coordonnée, équation, ensemble}, mode={\faIcon{tools}}] + On note $(b)$ l'ensemble des points tel que $y = -x$. Cette ensemble est une droite. + \begin{enumerate} + \item Lesquels des points suivants sont dans cet ensemble. + \[ + U(2, 4) \qquad \qquad + V(1, -1) \qquad \qquad + W(-1, -2) \qquad \qquad + X(0, 0) + \] + \item Placer les points qui sont dans cet ensemble dans un repère puis tracer la droite $(b)$. + \end{enumerate} +\end{exercise} + +\begin{solution} + \begin{enumerate} + \item Les points de l'ensemble $(b)$ vérifie $y=-x$ donc leur ordonnée doit être opposé à leur abscisse. + \begin{itemize} + \item Pour le point $U$ + \[ + -x = 2 = -2 \neq 4 = y + \] + Donc le point $U$ n'appartient pas à $(b)$ + \item Pour le point $V$ + \[ + - x = -1 = y + \] + Donc le point $V$ appartient à $(b)$ + \item Pour le point $W$ + \[ + -x = -(-1) = 1 \neq -2 = y + \] + Donc le point $W$ n'appartient pas à $(b)$ + \item Pour le point $X$ + \[ + -x = - 0 = 0 = y + \] + Donc le point $X$ appartient à $(b)$ + \end{itemize} + \item ~ + + \begin{tikzpicture} + \repere{-5}{5}{-5}{5} + + \draw (2, 4) node {x} node [below left] {$U$}; + \draw (1, -1) node {x} node [below left] {$V$}; + \draw (-1, -2) node {x} node [below left] {$W$}; + \draw (0, 0) node {x} node [above right] {$X$}; + + \draw(-4, 4) node [above right] {$(b)$}; + \draw[domain=-5:5] plot(\x, {-\x}); + \end{tikzpicture} + \end{enumerate} +\end{solution} diff --git a/2nd/11_Geometrie_reperee/fig/deux_triangles.png b/2nd/11_Geometrie_reperee/fig/deux_triangles.png new file mode 100644 index 0000000000000000000000000000000000000000..553b969439fbfd81ba3bc0511a5d28c3b608d368 GIT binary patch literal 33361 zcmZs@1yq!6)HXUGAOcECh%^F{iimVeN_VFSlF}fJNS6poOM`@Tr$~25hlq5y^x61+ z-+$JLan^!`GxOZf9eeL9_5>-&N!-S`hk-yKZc9muDIpNZod^UH2|6{CQtp_sHQ}DUxQo|Dh5Ug6^{5g)va4Jv{e23m%HN}x2bO7 zNTM&FNeJ56+4W(cA_V(>lvh^XK=-ADCrk4>58~giSiy>_sxgJe-q(Nbu>JK;B(>0Z z1uvx4?#-J)M*6=+uyvg#0%k=5JIe@}R3)7I8@q)S^LLHNXlT&up1#YB*r3VrClTzH`|(w7h$ zlb`eEcy*IJXrq#iju}!kF#O7>=^95SR@SYh-h|;S*_O+<>!U20tN!yIyEnyXY*Wxt zgmE&W$atF$R)6)8k;zwf;)UcYXCouB 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+------------------- + +- Calculer la distance entre deux points. Calculer les coordonnées du milieu d’un segment +- Tracer une droite connaissant son équation cartésienne ou réduite. +- Déterminer une équation de droite à partir de deux points + + +Progression +=========== + +Avant de commencer les élèves écriront sur le cahier de bord un paragraphe sur les repères et les coordonnées des points. + +.. image:: ./1B_coordonees.pdf + :height: 200px + :alt: introduction à la géométrie repérée + + +Les étapes 1, 2 et 4 peuvent être fait en parallèles. +L'étape 3 suit les étapes 1 et 2. +L'étape 5 suit l'étape 4. + +.. image:: ./plan_de_travail.pdf + :height: 200px + :alt: Tout en 1! + + +.. image:: ./solutions.pdf + :height: 200px + :alt: Solution des exercices techniques + + +Étape 1: Coordonnée du milieu +----------------------------- + +.. image:: ./2E_milieu.pdf + :height: 200px + :alt: Exercice autour des coordonnées du milieu d'un segment. + + +On fait tracer un repère orthonormée pour y placer des points. Les élèves cherchent le milieu de segments d'abord sur des exemples simples (verticaux ou horizontaux) puis en diagonale. Ils doivent déterminer un calcul qui permet de trouver les coordonnées exactes du milieu. Pour cela, on pourra utiliser des coordonnées impossibles à placer dans un repère. + +Bilan: formule de calcul des coordonnées du milieu. + +.. image:: ./2B_milieu.pdf + :height: 200px + :alt: Cours sur les coordonnées du milieu d'un segment. + + +Exercices techniques. + +Étape 2: Distance entre 2 points +-------------------------------- + +On fait tracer un repère orthonormée pour y placer des points. Les élèves cherchent la longueur de segments d'abord sur des exemples simples (verticaux ou horizontaux) puis en diagonale. Cela pourrait être l'occasion de reparler de projeté orthogonal pour avoir l'angle droite et appliquer Pythagore. + +Bilan: formule de calcul de la distance entre deux points + +.. image:: ./3B_val_abs.pdf + :height: 200px + :alt: Définition de la valeur absolue + + +.. image:: ./3B_distance.pdf + :height: 200px + :alt: Distance entre deux points + +Exercices techniques (voir 54 p 125 sesamath) + +Étape 3: Mélange des deux étapes précédentes +-------------------------------------------- + +Problèmes de géométrie utilisant les coordonnées. + +Étape 4: Ensemble de points +--------------------------- + +.. image:: ./4B_ensembles_points.pdf + :height: 200px + :alt: Bilan sur les ensembles de points diff --git a/2nd/11_Geometrie_reperee/plan_de_travail.pdf b/2nd/11_Geometrie_reperee/plan_de_travail.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ede18e09e7af9dd362d2017c7653acea30e447f3 GIT binary patch literal 75628 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+\title{Géométrie repérée - Solutions} +\tribe{2nd} +\date{Janvier 2023} + +\DeclareExerciseCollection{banque} +\xsimsetup{ + exercise/print=false, + solution/print=true, +} + +\pagestyle{empty} + + +\begin{document} + +\maketitle + +\input{exercises.tex} +%\printcollection{banque} +%\printsolutions{exercises} + +\end{document}