\documentclass[12pt]{article}\usepackage[Aymar]{ammaths}\en\begin{document}



\enteteseconde{6}{Classic Solids}{3D Geometry 1}


\begin{center}
\fbox{\parbox{14cm}{\sf\small A la fin de ce chapitre, vous devez être capable de :
\begin{itemize}
\item manipuler, construire et représenter en perspective des solides ;
\item effectuer des calculs, de longueur, d'aire et de volumes.
\end{itemize}}}
\end{center}


\exocpart{Representation of 3D solids}

\exoc{The solid $SABCD$ is a squared-based pyramid with vertex $S$, $EFGH$ are $IJKL$ two tetrahedra. Complete their representations.\\
\begin{center}
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        %pyramide SABCD
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        \psline[linestyle=dashed](-0.5,1)(1.7,1.2)
        \psdots(-0.5,1)(0.5,3)(1.5,0)(1.7,1.2)
        \uput[l](-0.5,1){$A$}
        \uput[dr](1.7,1.2){$B$}
        \uput[r](0.5,3){$S$}
        \uput[r](1.5,0){$D$}
        %tétraèdre EFGH
        \uput[l](4,0){$E$}
        \uput[d](5.5,1){$F$}
        \uput[r](7.5,0.5){$G$}
        \uput[ul](5,2.5){$H$}
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        \psline(4,0)(5.5,1)\psline(7.5,0.5)(5.5,1)\psline(5.5,1)(5,2.5)
        %tétraèdre IJKL
        \uput[l](9.5,0){$I$}
        \uput[d](11,1){$J$}
        \uput[r](13,0.5){$K$}
        \uput[ul](10.5,2.5){$L$}
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        \psline[linestyle=dashed](9.5,0)(11,1)
        \psline[linestyle=dashed](13,0.5)(11,1)\psline[linestyle=dashed](11,1)(10.5,2.5)
        \endpspicture
\end{center}}

\exoc{Give the  number of faces for each of the the following solids.\\
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\psline(5,4)(5.46,4.7)\psline(0,4)(0,1)(5,1)\psline(1,3)(4,3)(4,2)
\end{pspicture*}}
\hspace{1cm}
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\hspace{1cm}
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\vspace{0.5cm}

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\hspace{1cm}
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\pagebreak

\exoc{Draw in cavalier representation each of the following solids :
\begin{enumerate}
	\item A cube with edge $2$ cm, vanishing angle $45^{o}$, report $0,7$.
	\item A cuboid with length $5$ cm, width and depth $2$ cm, vanishing angle $45^{o}$, report $0,7$.
	\item A hexagonal-based prismoid (free measures).
	\item A pentagonal-based pyramid.
\end{enumerate}}



\exocpart{Length, area and volume}
\begin{multicols}{2}
\exoc{Below is a box made of a cuboid and a semi-cylinder.
\begin{enumerate}
 \item Compute the volume of the box (rounded to the unit).
 %\item Dessiner en vraie grandeur la base de cette boîte.  
 \item Draw a net of the box.
 \item Compute the area of the box.
\end{enumerate}}

\columnbreak  
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\aput{:U}{22\ mm} 
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% End of figure 
\end{multicols}
 

\exoc{Same question as in the previous exercise for the following solids.\\
\begin{multicols}{2}
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\end{center}
\end{multicols}}

\begin{minipage}{8cm}
\exoc{Let $ABCDA'B'C'D'$ be a cuboid with dimensions $AB=7$, $AD=3$ and
$AA'=5$, the unit being the centimeter.}
\end{minipage}
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        \uput[r](8,5){$C'$}
        \uput[ul](2,5){$D'$}
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\end{center}
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\begin{enumerate}
\item Computing a few lengths.
    \begin{enumerate}
    \item Compute $D'B$, using twice a famous theorem.
    \item Let $I$ be the midpoint of $AB$ and $K'$ the midpoint of $A'D'$.\\
    Compute the length $IK'$ with the same method as in question 1.
    \end{enumerate}
\item Areas and volumes.
    \begin{enumerate}
    \item Compute the volume of $ABCDA'B'C'D'$.
    \item Compute the area of the triangle $IBC$.
    \item Deduce the volume of the prism $IBCI'B'C'$, where $I'$ is the midpoint of $A'B'$.
    \item Deduce from the previous questions the volume of the prism $AICDA'I'C'D'$.
    \item Check the previous result by computing the volume of the prism $AICDA'I'C'D'$ directly.
    \end{enumerate}
\end{enumerate}


% \pagebreak
% \exoc{$ABCDEFGH$ est un cube de côté 4 cm.
% On note $O$ le point d'intersection des diagonales $[AG]$ et $[BH]$.
% On admet que $O$ est le milieu des diagonales.
% \begin{enumerate}
% 	\item \begin{enumerate}
% 		  \item Calculer la longueur de la diagonale $[AG]$.
% 		  \item En déduire la valeur de $GO^2+FO^2$.
% 		  \end{enumerate}
% 	\item Le triangle $GOF$ est-il rectangle ?
% 	\item Les diagonales d'un cube sont-elles perpendiculaires ?
% \end{enumerate}}



\exoc{Let $ABCDEFGH$ be a cube with side $4$m.
Let $I$ and $K$ be the midpoints of the line segments $BF$ and $AB$, and $J$ the point on $EF$ such that
$EJ=\frac{1}{4}EF$.
% \begin{center}
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% \uput[l](0,0){\small $E$}
% \uput[dr](6,0){\small $F$}
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% \uput[ul](2.5,2){\small $H$}
% \uput[l](0,6){\small $A$}
% \uput[dr](6,6){\small $B$}
% \uput[r](8.5,8){\small $C$}
% \uput[ul](2.5,8){\small $D$}
% \endpspicture
% \end{center}

\begin{enumerate}
\item What can you say about the lines $IK$ and $AF$? Explain your answer.
\item Four ants are moving on the surface of the cube, from $A$ to $G$, along different paths :
			\vspace*{-0.8cm}
			{\setlength{\columnseprule}{0pt}
			\begin{multicols}{2}
			\begin{enumerate}
			\item $AJ+JG$;
			\item $AI+IF+FG$;
			\item $AF+FG$;
			\item $AK+KI+IG$.
			\end{enumerate}
			\end{multicols}}
			\vspace*{-0.3cm}
			Compute the length of the path followed by each ant, first as an exact value, then an
approximate value to 2DP.
\item \begin{enumerate}
			\item Draw the net of the cube to the scale $1:200$.
			\item Use the net to find the length of the shortest path from $A$ to $G$.
			\end{enumerate}
\item Knowing that a competition ant moves at the incredible speed of \mbox{$5\times
10^{-2}$\,km$\cdot$h$^{-1}$}, find out how long it will take for such an ant to go from $A$ to $G$ along the shortest
path. The answer will be given in minutes and seconds.
\end{enumerate}}

\begin{multicols}{2}
\exoc{\emph{The cuboctahedron}\\
Let $ABCDEFGH$ be a cube with side 4 cm.
The points $I$, $J$ and $K$ are respectively the midpoints of the edges $[FE]$, $[FG]$ and $[FB]$.
\begin{enumerate}
\item Count the numbers of vertices $V$, the number of edges $E$ and the number of faces $F$ for the cube. Then compute the value of $V - E + F$.
\end{enumerate}

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\end{multicols}



%\item Déterminer le volume exact du solide obtenu en enlevant le solide $FIJK$ du cube. Pour ce nouveau solide, donner
%le nombre de faces, d'arêtes et de sommets, et calculer $S- A + F$.
\begin{enumerate}\setcounter{enumi}{1}
\item \begin{enumerate}
		\item What is the name of the solid $FIJK$ ?
		\item Compute the volume of the solid $FIJK$.
		\item Compute, for this solid, the value of $V - E + F$.
		\end{enumerate}
\item We remove the eight ``corners''~of the cube $ABCDEFGH$, which gives the solid shown below. It is called a \emph{cuboctahedron}.
	\begin{center}
		\scalebox{0.25}{\includegraphics{cuboctahedron.eps}}
		\end{center}
	\begin{enumerate}
	\item Compute the value of $V - E + F$ for the cuboctahedron.
	\item Compute the volume of this cuboctahedron and its area.\\ Then, try to draw its area.
	\end{enumerate}
\end{enumerate}
}




% exos à finir
% \exoc{\emph{Patron d'un tronc de cône.}\\
% Voici un schéma en perspective cavalière d'un tronc de cône compris entre deux plans parallèles dont le cercle de base
% a pour centre $O$ et le petit cercle de base a pour centre $O'$. Les dimensions sont indiqués sur la figure. Le but
% est de construire un patron de ce cône.
% 
% 
% 
% \begin{enumerate}
% \item En utilisant le théorème de Pythagore, déterminer au centième de millimètre près la longueur de la directrice
% $AA'$.
% \item Que peut-on dire des longueurs de toutes les directrices %
% \item Déterminer les périmètres des cercles de rayon $4$ et $2$.
% \item Ce tronc est issu d'un cône de sommet $S$ dont la partie haute a été ôtée. Justifier que l'on a $SO=6$ cm.
% \item Construire un patron du cône de sommet $S$ et dont la base est le cercle de centre $O$ et de rayon $4$ cm. On
% pourra pour cela utiliser la proportionnalité entre l'angle et la longeur de l'arc.
% \item Déduire des réalisations précedentes, un patron du tronc du cône.
% \end{enumerate}}





% \exoc{\textbf{About solids and polyhedra.}
% 
% \probpart{-- Solids}
% 
% In geometry, a \emph{solid} is a three-dimensional object. Solids
% may be separated into two main families, the polyhedra and
% non-polyhedra.
% 
% A \emph{polyhedron} is a three-dimensional shape that is made up of
% a finite number of polygonal \emph{faces} which are parts of planes.
% Its faces meet in \emph{edges} which are straight-line segments, and
% the edges meet in points called \emph{vertices}. A polyhedron is a
% three-dimensional analog of a polygon. Cubes, prisms and pyramids
% are examples of polyhedra. Cones, cylinders and spheres are examples
% of solids that are not polyhedra.
% 
% \probpart{-- Polyhedra}
% 
% Polyhedra are mostly named after their number of faces :
% tetrahedron, pentahedron, hexahedron, octahedron, decahedron, etc.
% Such terms may represent different types of polyhedra which have not
% much in common except having the same number of faces.
% 
% A polyhedron is \emph{convex} if any straight line between two
% points in the polyhedron lies entirely inside it. It's
% \emph{concave} otherwise. Almost all the polyhedra we will study are
% convex so any polyhedron not specified as concave will be considered convex.
% There are many different classifications of the polyhedra, some of
% them overlapping. Here are a few important families.
% 
% \probpart{-- Prismatoids}
% 
% A \emph{prismatoid} is a polyhedron where all vertices lie in two
% parallel planes. Under some circumstances it is called a prismoid
% (if both planes have the same number of elements.) Families of
% prismatoids include :
% \begin{itemize}
% \itemb pyramids, where one plane contains only a single point ;
% \itemb wedges, where one plane contains only two points ;
% \itemb prisms, where the polygons in each plane are congruent and
% joined by rectangles or parallelograms ;
% \itemb anti-prisms, where the polygons in each plane are congruent and joined by an
% alternating strip of triangles ;
% \itemb cupolas, where the polygon in one plane contains twice as many
% points as the other and is joined to it by alternating triangles and
% rectangles ;
% \itemb quadrilateral-faced hexahedral prismatoids : parallelepiped (six parallelogram
% faces), rhombohedron (six rhombi faces), hexahedral traperohedron
% (six congruent rhombi faces), cuboid (six rectangular faces),
% quadrilateral frustum (an apex-truncated square pyramid), cube (six
% square faces).
% \end{itemize}
% 
% \probpart{-- Platonic, Archimedean and Johnson solids}
% 
% A polyhedron is a \emph{Platonic solid} if all its faces are
% congruent regular polygons and if the same number of faces meet at
% each of the vertices. There are only 5 Platonic solids, a fact well
% known by the ancient Greeks. A proof of this result can be found in
% \emph{Euclid's Elements}.
% 
% A polyhedron is an \emph{Archimedean solid} if it's composed of two
% or more types of regular polygons and if the same number of faces
% meet at each of the vertices, and if it's not a prism or an anti-prism. There are 13 Archimedean solids (or 15
% if you count separately the mirror images of two of them). They were
% first listed by Archimedes in a now-lost work and rediscovered
% during the Renaissance, notably by Johannes Kepler who completed the
% study in 1619.
% 
% A polyhedron is a \emph{Johnson solid} if all its faces are regular
% polygons and if it's not a platonic solid, an archimedean solid, a
% prism or an anti-prism. There are 92 Johnson solids, that were
% listed by Norman Johnson in 1966.
% 
% \probpart{-- Questions}
% 
% \begin{enumerate}
% \item Draw a quick sketch of a concave polyhedron.
% \item For each of the following solids, find out what family (or
% families) it is part of.
%     \begin{enumerate}
%     \item A square-based regular pyramid.
%     \item A cube.
%     \item A triangle-based regular prism.
%     \item A regular tetrahedron.
%     \item A cylinder.
%     \item A football, or soccer ball, or truncated icosahedron.
%     \end{enumerate}
% \item What is the difference between a prism and an anti-prism ?
% \item What is the French for ``cuboid'' ?
% \item What is the other name of the cube ?
% \item How many Platonic solids are there ?\\ In what famous work is
% the proof of this result available ?
% \item Give two simple Platonic solids.
% \item Who discovered and listed the Archimedean solids ?
% \item What is the difference between an Archimedean solid and a
% Johnson solid ?
% \item Try to find one simple Johnson solid.
% \item Give the volume formulae for the following solids.
%     \begin{enumerate}
%     \item A cube of side $s$.
%     \item A rectangular prism with dimensions $l$, $L$ and $h$.
%     \item A cylinder of radius $r$ and height $h$.
%     \item A prism with base of area $A$ and height $h$.
%     \item A pyramid with a base of area $A$ and height $h$.
%     \item A cone with base of radius $r$ and height $h$.
%     \item A sphere of radius $r$.
%     \end{enumerate}
% \end{enumerate}}


%\emph{Notes :} 1 dm$^3$=1L.



\partie{Homework \#4 -- Barycentres of two points}

Let $[AB]$ be a segment with length 6 cm. All the points in this exercise will be
placed on the same figure.
\begin{enumerate}
\item Let $G$ be the point defined by the equality
$2\vect{GA}+3\vect{GB}=\vect{0}$
    \begin{enumerate}
    \item Prove that $5\vect{GA}+3\vect{AB}=\vect{0}$ and deduce a simple relation between $\vect{AG}$~and~$\vect{AB}$.
    \item Is there more than one position for the point $G$ ? Show it on the figure.
    \end{enumerate}
\item Let $H$ the point such that $B$ is the midpoint of $[AH]$.
    \begin{enumerate}
    \item Build the point $H$ on the figure.
    \item What can you say about the vectors $\vect{AH}$ and $\vect{HB}$ ?
    \item Find two numbers $\alpha$ and $\beta$ such that $\alpha\vect{HA}+\beta\vect{HB}=\vect{0}$.
    \end{enumerate}
\end{enumerate}

The point $G$ defined in question $1$ by the equality
$2\vect{GA}+3\vect{GB}=\vect{0}$ is called the \emph{barycentre of
the points $A$ and $B$ with respective coefficients $2$ and $3$}.
More generally, any point $K$ defined by an equality
$\alpha\vect{KA}+\beta\vect{KB}=\vect{0}$ where $\alpha$ and $\beta$
are two real numbers such that $\alpha+\beta\neq 0$ is called the
\emph{barycentre of the points $A$ and $B$ with respective
coefficients $\alpha$ and~$\beta$}.

\begin{enumerate}\setcounter{enumi}{2}
\item What can you say about the point $H$ ?
\item Let $P$ be the barycentre of $A$ and $B$ with respective coefficients $2$ and $1$.
    \begin{enumerate}
    \item What equality defines the point $P$ ?
    \item Deduce from this equality a relation between the vectors $\vect{AP}$ and $\vect{AB}$.
    \item Place the point $P$.
    \end{enumerate}
\item Let $Q$ be the barycentre of $A$ and $B$ with respective coefficients $-2$ and $1$.
    \begin{enumerate}
    \item What equality defines the point $Q$ ?
    \item Deduce from this equality a relation between the vectors $\vect{AQ}$ and $\vect{AB}$.
    \item Place the point $Q$.
    \end{enumerate}
\item Let $R$ be the barycentre of $A$ and $B$ with respective coefficients $-2$ and $2$.
    \begin{enumerate}
    \item What equality defines the point $R$ ?
    \item Deduce from this equality a relation between the vectors $\vect{AR}$ and $\vect{AB}$.
    \item Could you place the point $R$ ? What is the problem ?
    \end{enumerate}
\item Let $S$ be the point such that $\vect{AS}=\dfrac{1}{4}\vect{AB}$.
     \begin{enumerate}
      \item Place the point $S$.
      \item Prove that $4\vect{AS}=\vect{AS}+\vect{SB}$.
      \item Deduce two numbers $m$ and $n$ such that $m\vect{SA}+n\vect{SB}=\vect{0}$.
      \item Give a definition of $S$ using barycentre.
     \end{enumerate}
%\item Find on the Internet some uses of barycentres.
\end{enumerate}






\end{document}

