\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage{lmodern,amsmath,amssymb,tikz,xcolor,fancyhdr,qrcode,hyperref}
\usepackage[margin=1.5cm,headheight=15pt,footskip=20pt]{geometry}
\usetikzlibrary{calc}
\definecolor{utspe}{HTML}{345BC7}
\pagestyle{fancy}\fancyhf{}\fancyhead[L]{\small UTSPE -- Terminale -- Chapitre 3}\fancyfoot[R]{\thepage/4}
\fancyfoot[L]{\scriptsize $[AB]$ : segment ; $(AB)$ : droite ; $AB$ : longueur.}
\setlength{\parindent}{0pt}
\newcommand{\justifications}{\textbf{Justifications :}\par\medskip\foreach \n in {1,...,6}{\noindent\textcolor{gray!40}{\rule{\linewidth}{.3pt}}\par\vspace{.34cm}}}
\hypersetup{pdfborder={0 0 0}}
\begin{document}
\shorthandoff{:;!?}
{\Large\bfseries Construire des sections de solides}\par\medskip
Nom : \dotfill\quad Classe : \dotfill\quad Date : \dotfill\par\medskip
{\small Tracer à la règle. Conserver les traits auxiliaires. Utiliser les pointillés pour les parties cachées. Les points donnés sont déjà placés ; aucune équation cartésienne n’est demandée.}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 1. Cube - section oblique}\par\smallskip
$\overrightarrow{AI}=\cfrac14\overrightarrow{AE}$, $\overrightarrow{BJ}=\cfrac12\overrightarrow{BF}$ et $\overrightarrow{CK}=\cfrac34\overrightarrow{CG}$. Construire la section par $(IJK)$ et justifier le quatrième sommet.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-1}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 1.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (6.5000,1.3000);
\coordinate (D) at (2.0000,1.3000);
\coordinate (E) at (0.0000,4.5000);
\coordinate (F) at (4.5000,4.5000);
\coordinate (G) at (6.5000,5.8000);
\coordinate (H) at (2.0000,5.8000);
\draw[dashed] (A) -- (D);
\draw[dashed] (D) -- (C);
\draw[solid] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (E) -- (F);
\draw[solid] (F) -- (G);
\draw[solid] (G) -- (H);
\draw[solid] (H) -- (E);
\draw[solid] (B) -- (F);
\draw[solid] (E) -- (A);
\draw[solid] (C) -- (G);
\draw[dashed] (D) -- (H);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\node[below right] at (E) {$ E $};
\node[below right] at (F) {$ F $};
\node[below right] at (G) {$ G $};
\node[below right] at (H) {$ H $};
\coordinate (I) at ($(A)!0.25000000!(E)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(B)!0.50000000!(F)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(C)!0.75000000!(G)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 2. Cube - six côtés}\par\smallskip
$\overrightarrow{AI}=\cfrac12\overrightarrow{AB}$, $\overrightarrow{BJ}=\cfrac12\overrightarrow{BC}$ et $\overrightarrow{CK}=\cfrac12\overrightarrow{CG}$. Construire la section par $(IJK)$. Utiliser un point auxiliaire extérieur si nécessaire.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-2}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 2.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (6.5000,1.3000);
\coordinate (D) at (2.0000,1.3000);
\coordinate (E) at (0.0000,4.5000);
\coordinate (F) at (4.5000,4.5000);
\coordinate (G) at (6.5000,5.8000);
\coordinate (H) at (2.0000,5.8000);
\draw[dashed] (A) -- (D);
\draw[dashed] (D) -- (C);
\draw[solid] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (E) -- (F);
\draw[solid] (F) -- (G);
\draw[solid] (G) -- (H);
\draw[solid] (H) -- (E);
\draw[solid] (B) -- (F);
\draw[solid] (E) -- (A);
\draw[solid] (C) -- (G);
\draw[dashed] (D) -- (H);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\node[below right] at (E) {$ E $};
\node[below right] at (F) {$ F $};
\node[below right] at (G) {$ G $};
\node[below right] at (H) {$ H $};
\coordinate (I) at ($(A)!0.50000000!(B)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(B)!0.50000000!(C)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(C)!0.50000000!(G)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\newpage
{\Large\bfseries Construire des sections de solides}\par\medskip
Nom : \dotfill\quad Classe : \dotfill\quad Date : \dotfill\par\medskip
{\small Tracer à la règle. Conserver les traits auxiliaires. Utiliser les pointillés pour les parties cachées. Les points donnés sont déjà placés ; aucune équation cartésienne n’est demandée.}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 3. Cube - section pentagonale}\par\smallskip
$\overrightarrow{BI}=\cfrac12\overrightarrow{BC}$, $\overrightarrow{CJ}=\cfrac12\overrightarrow{CD}$ et $\overrightarrow{AL}=\cfrac34\overrightarrow{AE}$. Construire la section par $(IJL)$ et expliquer chaque nouveau sommet.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-3}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 3.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (6.5000,1.3000);
\coordinate (D) at (2.0000,1.3000);
\coordinate (E) at (0.0000,4.5000);
\coordinate (F) at (4.5000,4.5000);
\coordinate (G) at (6.5000,5.8000);
\coordinate (H) at (2.0000,5.8000);
\draw[dashed] (A) -- (D);
\draw[dashed] (D) -- (C);
\draw[solid] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (E) -- (F);
\draw[solid] (F) -- (G);
\draw[solid] (G) -- (H);
\draw[solid] (H) -- (E);
\draw[solid] (B) -- (F);
\draw[solid] (E) -- (A);
\draw[solid] (C) -- (G);
\draw[dashed] (D) -- (H);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\node[below right] at (E) {$ E $};
\node[below right] at (F) {$ F $};
\node[below right] at (G) {$ G $};
\node[below right] at (H) {$ H $};
\coordinate (I) at ($(B)!0.50000000!(C)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(C)!0.50000000!(D)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (L) at ($(A)!0.75000000!(E)$);\fill[blue!65!black] (L) circle (1.8pt) node[above left] {$ L $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 4. Cube - autre plan de coupe}\par\smallskip
$\overrightarrow{AI}=\cfrac3{10}\overrightarrow{AE}$, $\overrightarrow{BJ}=\cfrac7{10}\overrightarrow{BF}$ et $\overrightarrow{CK}=\cfrac35\overrightarrow{CG}$. Construire la section par $(IJK)$ et préciser les parallélismes utilisés.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-4}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 4.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (6.5000,1.3000);
\coordinate (D) at (2.0000,1.3000);
\coordinate (E) at (0.0000,4.5000);
\coordinate (F) at (4.5000,4.5000);
\coordinate (G) at (6.5000,5.8000);
\coordinate (H) at (2.0000,5.8000);
\draw[dashed] (A) -- (D);
\draw[dashed] (D) -- (C);
\draw[solid] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (E) -- (F);
\draw[solid] (F) -- (G);
\draw[solid] (G) -- (H);
\draw[solid] (H) -- (E);
\draw[solid] (B) -- (F);
\draw[solid] (E) -- (A);
\draw[solid] (C) -- (G);
\draw[dashed] (D) -- (H);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\node[below right] at (E) {$ E $};
\node[below right] at (F) {$ F $};
\node[below right] at (G) {$ G $};
\node[below right] at (H) {$ H $};
\coordinate (I) at ($(A)!0.30000000!(E)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(B)!0.70000000!(F)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(C)!0.60000000!(G)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\newpage
{\Large\bfseries Construire des sections de solides}\par\medskip
Nom : \dotfill\quad Classe : \dotfill\quad Date : \dotfill\par\medskip
{\small Tracer à la règle. Conserver les traits auxiliaires. Utiliser les pointillés pour les parties cachées. Les points donnés sont déjà placés ; aucune équation cartésienne n’est demandée.}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 5. Tétraèdre - un trapèze}\par\smallskip
$\overrightarrow{AI}=\cfrac13\overrightarrow{AB}$, $\overrightarrow{AJ}=\cfrac13\overrightarrow{AC}$ et $\overrightarrow{BK}=\cfrac12\overrightarrow{BD}$. Tracer $[IK]$, puis construire la section par $(IJK)$. Justifier le parallélisme avec le théorème du toit.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-5}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 5.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (2.9000,1.3000);
\coordinate (D) at (2.2750,4.9550);
\draw[dashed] (A) -- (C);
\draw[dashed] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (B) -- (D);
\draw[solid] (D) -- (A);
\draw[dashed] (C) -- (D);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\coordinate (I) at ($(A)!0.33333333!(B)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(A)!0.33333333!(C)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(B)!0.50000000!(D)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 6. Tétraèdre - quatre côtés}\par\smallskip
$\overrightarrow{AI}=\cfrac12\overrightarrow{AB}$, $\overrightarrow{AJ}=\cfrac12\overrightarrow{AC}$ et $\overrightarrow{BK}=\cfrac12\overrightarrow{BD}$. Construire la section par $(IJK)$, déterminer sa nature et justifier le quatrième sommet.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-6}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 6.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (2.9000,1.3000);
\coordinate (D) at (2.2750,4.9550);
\draw[dashed] (A) -- (C);
\draw[dashed] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (B) -- (D);
\draw[solid] (D) -- (A);
\draw[dashed] (C) -- (D);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\coordinate (I) at ($(A)!0.50000000!(B)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(A)!0.50000000!(C)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(B)!0.50000000!(D)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\newpage
{\Large\bfseries Construire des sections de solides}\par\medskip
Nom : \dotfill\quad Classe : \dotfill\quad Date : \dotfill\par\medskip
{\small Tracer à la règle. Conserver les traits auxiliaires. Utiliser les pointillés pour les parties cachées. Les points donnés sont déjà placés ; aucune équation cartésienne n’est demandée.}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 7. Tétraèdre - traces sans parallélisme}\par\smallskip
$\overrightarrow{AI}=\cfrac13\overrightarrow{AB}$, $\overrightarrow{AJ}=\cfrac23\overrightarrow{AC}$ et $\overrightarrow{BK}=\cfrac12\overrightarrow{BD}$. Justifier que $(IJ)$ n’est pas parallèle à $(BC)$. Construire la section par $(IJK)$ en utilisant les droites supports des arêtes. Ne supposer aucun parallélisme de la trace dans $BCD$.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-7}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 7.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (2.9000,1.3000);
\coordinate (D) at (2.2750,4.9550);
\draw[dashed] (A) -- (C);
\draw[dashed] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (B) -- (D);
\draw[solid] (D) -- (A);
\draw[dashed] (C) -- (D);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\coordinate (I) at ($(A)!0.33333333!(B)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(A)!0.66666667!(C)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(B)!0.50000000!(D)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\textcolor{utspe}{\rule{\linewidth}{.4pt}}\par\smallskip
{\bfseries\color{utspe} 8. Tétraèdre - points à des proportions différentes}\par\smallskip
$\overrightarrow{AI}=\cfrac25\overrightarrow{AB}$, $\overrightarrow{AJ}=\cfrac34\overrightarrow{AC}$ et $\overrightarrow{BK}=\cfrac23\overrightarrow{BD}$. Construire la section par $(IJK)$. Indication : construire $T=(IJ)\cap(BC)$ sur les droites prolongées, puis utiliser $(KT)$ pour trouver le quatrième sommet sur $[CD]$. Justifier chaque intersection.\par\medskip
\noindent\qrcode[height=1.6cm]{https://utspe.com/sections-espace/index.html#feuille-exercice-8}\quad\begin{minipage}{.75\linewidth}\small Correction et animation de l’exercice 8.\par Scanne ce code après avoir cherché la construction.\end{minipage}\par\smallskip
\begin{minipage}[t]{.53\linewidth}\vspace{0pt}
\begin{tikzpicture}[line width=.65pt, every node/.style={font=\small}]
\path[use as bounding box] (-.5,-.5) rectangle (7,5.9);
\coordinate (A) at (0.0000,0.0000);
\coordinate (B) at (4.5000,0.0000);
\coordinate (C) at (2.9000,1.3000);
\coordinate (D) at (2.2750,4.9550);
\draw[dashed] (A) -- (C);
\draw[dashed] (C) -- (B);
\draw[solid] (B) -- (A);
\draw[solid] (B) -- (D);
\draw[solid] (D) -- (A);
\draw[dashed] (C) -- (D);
\node[below right] at (A) {$ A $};
\node[below right] at (B) {$ B $};
\node[below right] at (C) {$ C $};
\node[below right] at (D) {$ D $};
\coordinate (I) at ($(A)!0.40000000!(B)$);\fill[blue!65!black] (I) circle (1.8pt) node[above left] {$ I $};
\coordinate (J) at ($(A)!0.75000000!(C)$);\fill[blue!65!black] (J) circle (1.8pt) node[above left] {$ J $};
\coordinate (K) at ($(B)!0.66666667!(D)$);\fill[blue!65!black] (K) circle (1.8pt) node[above left] {$ K $};
\end{tikzpicture}
\end{minipage}\hfill\begin{minipage}[t]{.43\linewidth}\vspace{.4cm}\justifications\end{minipage}\par\medskip
\end{document}