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\enteteseconde{8}{Vectors and coordinates}{Plane geometry 3}




\begin{center}
\fbox{\parbox{14cm}{\sf\small A la fin de ce chapitre, vous devez être capable de :
\begin{itemize}
\itemb connaître les coordonnées $(x_B-x_A,y_B-y_A)$ d'un vecteur $\vect{AB}$ ;
\itemb calculer les coordonnées de la somme de deux vecteurs dans un repère ;
\itemb utiliser la notation $\lambda\vect{u}$ ;
\itemb établir la colinéarité de deux vecteurs.
\end{itemize}}}
\end{center}


\exocpart{Vector coordinates}

\exoc{In a Cartesian coordinate system, consider the points :
$$A(1,-3) \quad ; \quad B(4,-1) \quad ; \quad C(2,1) \quad \textrm{and} \quad D(-1,-1)$$
\begin{enumerate}
 \item Show that quadrilateral $ABCD$ is a parallelogram :
  \begin{enumerate}
   \item using lengths ;
   \item using the coordinates of the midpoints of the diagonals ;
   \item using the coordinates of the vectors $\vect{AB}$ and $\vect{DC}$.
  \end{enumerate}
\item Same question with the points $A(1,2)$, $B(1,-1)$, $C(-1,-2)$ and $D(-1,1)$.
\end{enumerate}
}

\exoc{\tice\ Consider the algorithm below :
 \begin{enumerate}
 \item Apply the algorithm in the following situations : 
	\begin{enumerate}
	\item $A(-4,3)$ ; $B(1,5)$ ; $C(3,2)$ ; $D(-2,0)$.
	\item $A(-3,3)$ ; $B(0,5)$ ; $C(3,2)$ ; $D(-2,1)$.
	\end{enumerate}
\end{enumerate}

\begin{multicols}{2}
 \begin{enumerate}\setcounter{enumi}{1}
 \item What is the aim of this algorithm ?
 \item \begin{enumerate}
	\item Apply this algorithm to the points $A(-3,3)$ ; $B(2,3)$ ; $C(2,1)$ ; $D(-3,1)$.
	\item What problem are we facing in this situation ?
	\item Add a conditional instruction to avoid that problem.
       \end{enumerate}
 \item \begin{enumerate}
	\item Apply this algorithm to the points $A(-2;2)$ ; $B(1;3)$ ; $C(7;5)$ ; $D(4;4)$.
	\item Place the points in a Cartesian coordinate system and check the output of the algorithm. Comment the result.
       \end{enumerate}
\item This algorithm is not always working. Find some input points for which there is a problem.
\end{enumerate}

\begin{minipage}[t]{8cm}
\IncMargin{1em}
\begin{algorithm}[H]
\Begin{\KwIn{$(x_A,y_A)$, $(x_B,y_B)$, $(x_C,y_C)$, $(x_D,y_D)$ coordinates of four points\;}
$x_B-x_A\rightarrow a$ \;
$y_B-y_A\rightarrow b$ \;
$x_C-x_D\rightarrow a'$ \;
$y_C-y_D\rightarrow b'$ \;
$\frac{a}{a'}\rightarrow r_1$ \;
$\frac{b}{b'}\rightarrow r_2$ \;
\eIf{$r_1=r_2$}{\KwOut{``$ABCD$ is a parallelogram'' \;}}
{\KwOut{``$ABCD$ is not a parallelogram'' \;}}

}
\end{algorithm}
\DecMargin{1em}
\end{minipage}
\end{multicols}}

\exoc{In a Cartesian coordinate system $\OIJ$ with unit 1 cm, we consider the points :
\begin{center} $E(-3,0)$ ; $B(2,0)$ ; $T(0,4)$ and $U(5,4)$. \end{center}
\begin{enumerate}
\item Find the coordinates of vectors $\vect{ET},~\vect{EB},~\vect{UE}$ and $\vect{BU}$.
\item  \begin{enumerate}
	\item Compute the lengths $ET$ and $EB$. 
	\item What type of quadrilateral is $TUBE$ ? Explain.
	\item Find the coordinates of $F$, centre of $TUBE$, and plot it.
	\end{enumerate}
\item Let ($\Co$) be the circle centered at $E$ passing through $B$, and $A$ the second intersection point of that circle with the $x$-axis.
\begin{enumerate}
	\item What type of triangle is $ATB$ ? Explain.
	\item Prove that the lines $(AT)$ and $(EF)$ are parallel.
	\item Compare the lengths $EF$ and $AT$.
\end{enumerate}
\item What is the image of triangle $ATE$ under the translation that shifts $A$ to $E$ ?
\end{enumerate}}



\exoc{Let $A(2,3)$, $B(1,-1)$, $C(-3,-3)$ be three points in a coordinate system $\OIJ$ and 
$D$ a fourth point such that $ABCD$ is a parallelogram. The coordinates of $D$ are unknown and will be denoted 
$(a,b)$.
\begin{enumerate}
\item Compute the coordinates of vector $\vect{AB}$.
\item Write the coordinates of $\vect{DC}$ using the unknowns $a$ and $b$.
\item Find out the values of $a$ and $b$.
\item Compute the lengths $AB$, $BC$, $CD$ and $DA$ and check that the opposite sides are equal.
\end{enumerate} }



\exoc{Let $\vect{u}\coor{-2}{5}$ and $\vect{v}\coor{3}{\frac{1}{4}}$ be two vectors defined by their coordinates in a Cartesian coordinate system $\OIJ$.
Compute the coordinates of the following vectors : $$\vect{u}+\vect{v} \quad,\quad 2\vect{u}\quad,\quad
-\frac{3}{7}\vect{v}\quad,\quad-3\vect{u}+\vect{v}\quad,\quad 2\vect{v}+0,8\vect{u}\quad,\quad
1,5\vect{u}-\frac{3}{5}\vect{v}.$$}


\exoc{In a coordinate system $\OIJ$, consider the points $A$, $B$ and $C$ with respective coordinates $(-2,-1)$, $(0,1)$ and $(-4,0)$, and another point $D$ with coordinates $(x,y)$ such that $\vect{CD}=2\vect{AB}$.
\begin{enumerate}
\item Plot the point $D$.
\item Find the coordinates of vectors $\vect{AB}$ and $\vect{CD}$.
\item Compute the coordinates of point $D$ and check your result with the graph.
\end{enumerate} }


\exoc{Let $A(-2,-1)$, $B(3,2)$ and $C(1,5)$ be three points in a coordinate system $\OIJ$.
\begin{enumerate}
	\item Compute the coordinates of vectors $\vect{AB}$ and $\vect{AC}$.
	\item Compute the coordinates of the point $I$, midpoint of segment $AB$.
	\item Compute the length $AC$.
	\item Compute the coordinates of vectors $\vect{AB}+\vect{AC}$, $3\vect{AB}$ and $2\vect{AB}+3\vect{AC}$ .
	\item Let $K$ be the point with coordinates $(x,y)$, such that $\vect{CK}=3\vect{AB}$.
		\begin{enumerate}
		\item Using $x$ and $y$, give the coordinates of the vector $\vect{CK}$.
		\item Find out the values of $x$ and $y$.
		\end{enumerate}
	\item Compute the coordinates of the point $L$ defined by $\vect{AL}+2\vect{BL}+\vect{CL}=\vect{0}$.
\end{enumerate}}

\exocpart{Collinearity and coordinates}

\exoc{Let $A(-2,1)$, $B(2,3)$, $C(0,-2)$, $D(2,-1)$ be four points in a Cartesian coordinate system.
\begin{enumerate}
 \item Plot the points $A, B, C$, $D$.
\item \begin{enumerate}
      \item What can you say about the vectors $\vect{AB}$ and $\vect{CD}$ ?
      \item Find the coordinates of vectors $\vect{AB}$ and $\vect{CD}$, denoted respectively $(x,y)$ and $(x',y')$. Are these two couples of coordinates proportional ?
      \item Compute $xy'-yx'$.
      \end{enumerate}
\item \begin{enumerate}
      \item Do the vectors $\vect{AC}$ and $\vect{BD}$ look collinear ?
      \item Find the coordinates of vectors $\vect{AC}$ and $\vect{BD}$, denoted respectively $(s,t)$ and $(s',t')$.  Are these two couples of coordinates proportional ?
      \item Compute $st'-ts'$.
      \end{enumerate}
\item Use the examples from the previous questions to copy and fill the property below.\par
\begin{center}
\framebox{\parbox{13cm}{Two vectors $\vect{u}$ and $\vect{v}$ with respective coordinates $(x,y)$ and $(x',y')$ are collinear if and only if their coordinates \ldots, that is if \ldots.}}
\end{center}
\end{enumerate}
}



\exoc{Using the property discovered in the previous exercise, decide whether the two vectors in each pair below are collinear or not.
\begin{multicols}{2} 
\begin{enumerate}
\item $\vect{u}(2,5)$ and $\vect{v}(-4,-10)$ ;
\item $\vect{u}(-3,9)$ and $\vect{v}(2,6)$ ;
\item $\vect{u}(5,1)$ and $\vect{v}(-10,2)$ ;
\item $\vect{u}(-4,12)$ and $\vect{v}(6,-18)$ ;
\item $\vect{u}(0,7)$ and $\vect{v}(0,-8)$ ;
\item $\vect{u}(0,7)$ and $\vect{v}(-8,0)$.
\end{enumerate}
\end{multicols}
}


\exoc{Let $A(-7,4)$, $B(-4,10)$, $C(10,13)$ and $D(6,5)$ be four points in a coordinate system~$\OIJ$.
\begin{enumerate}
\item \begin{enumerate}
	\item Compute the coordinates of the vectors $\vect{AB}$ and $\vect{CD}$.
	\item Deduce that the quadrilateral $ABCD$ is a trapezium.
	\end{enumerate}
\item Let $I$ be the point such that $\vect{IA}=3\vect{AD}$, $J$ and $K$ the midpoints of the segments $AB$ and $CD$.
	\begin{enumerate}
	\item Compute the coordinates of $I$.
	\item Compute the coordinates of $J$ and $K$.
	\item Prove that $I$, $J$ and $K$ are collinear.
	\end{enumerate}
\end{enumerate}}

\pagebreak

\begin{multicols}{2}
\exoc{\tice\ The aim of the algorithm on the right-hand side is to decide if three points are collinear or not.
 \begin{enumerate}
 \item Copy and fill the conditional instruction.
 \item Apply the algorithm to the points $A(-2;-1)$ ; $B(1;1)$ ; $C(7;5)$.
 \item Apply the algorithm to the points $A(-2;5)$ ; $B(2;2)$ ; $C(5;0)$.
 \item What instruction should you add at the beginning of the algorithm to make sure that the three points are distinct ?
\end{enumerate}}
\vfill\ 

\begin{minipage}[t]{8cm}
\IncMargin{1em}
\begin{algorithm}[H]
\Begin{\KwIn{$(x_A,y_A)$, $(x_B,y_B)$, $(x_C,y_C)$ coordinates of three distinct points\;}
$x_B-x_A\rightarrow a$ \;
$y_B-y_A\rightarrow b$ \;
$x_C-x_A\rightarrow a'$ \;
$y_C-y_A\rightarrow b'$ \;
$\frac{a}{a'}\rightarrow r_1$ \;
$\frac{b}{b'}\rightarrow r_2$ \;
\eIf{\ldots}{\KwOut{\ldots \;}}
{\KwOut{\ldots \;}}

}
\end{algorithm}
\DecMargin{1em}
\end{minipage}
\end{multicols}

\exoc{In the picture below, $ABCD$ is a square with side one unit, $AIB$ and $BJC$ are two equilateral triangles. The aim of this exercise is to prove, by two different methods, that the points $D$, $I$, $J$ are
collinear.
%\smallskip

\begin{minipage}{6cm}
\begin{center}
\psset{xunit=1.0cm,yunit=1.0cm}
\begin{pspicture*}(0,0.02)(7,4.5)
%\psgrid[subgriddiv=0,gridlabels=0,gridcolor=lightgray](0,0)(0,0.02)(7,4.98)
\psset{xunit=0.85cm,yunit=0.85cm,algebraic=true,dotstyle=x,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt
2,arrowinset=0.25}
\pspolygon(1,1)(4,1)(4,4)(1,4)
\pspolygon(1,1)(4,1)(2.5,3.6)
\pspolygon(4,4)(4,1)(6.6,2.5)
\psline(1,1)(4,1)
\psline(4,1)(4,4)
\psline(4,4)(1,4)
\psline(1,4)(1,1)
\psline(1,1)(4,1)
\psline(4,1)(2.5,3.6)
\psline(2.5,3.6)(1,1)
\psline(4,4)(4,1)
\psline(4,1)(6.6,2.5)
\psline(6.6,2.5)(4,4)
\psdots(1,1)\uput[dl](1,1){$A$}
\psdots(4,1)\uput[dr](4,1){$B$}
\psdots(4,4)\uput[ur](4,4){$C$}
\psdots(1,4)\uput[ul](1,4){$D$}
\psdots(2.5,3.6)\uput[l](2.5,3.6){$I$}
\psdots(6.6,2.5)\uput[r](6.6,2.5){$J$}
\end{pspicture*}
\end{center}
\end{minipage}
\begin{minipage}{9cm}
\begin{enumerate}
\item \textbf{Method 1 : With geometric angles}
      \begin{enumerate}
      \item Compute the measures of the angles $\widehat{DIA}$, $\widehat{AIB}$ and $\widehat{BIJ}$.
      \item Prove that $\widehat{DIJ}=180^o$. Conclude.
      \end{enumerate}
\item \textbf{Method 2 : In a coordinate system} \\
      Use a convenient coordinate system to prove that the points $D$, $I$ and $J$ are collinear.\\
      \emph{You may use the fact that in an equilateral triangle with side $a$, the length of an altitude is $a\frac{\sqrt{3}}{2}$.}
\end{enumerate}
\end{minipage}
}

\exoc{Let $ABC$ be a triangle with $AB=$ 8 cm, $AC=$ 6 cm and $BC=$ 5 cm. We consider the points $M$ and $N$ such that :
$\vect{CN}=-\frac14\vect{AC}$ and $\vect{AM}=\frac34\vect{AB}$.
\begin{enumerate}
\item Place the points $M$ and $N$.
\item \begin{enumerate}
    \item Write $\vect{BC}$ as a linear combination of $\vect{AB}$ and $\vect{AC}$.
    \item Write $\vect{MN}$ as a linear combination of $\vect{AB}$ and $\vect{AC}$.
    \item Deduce that the line $MN$ is parallel to the line $BC$.
    \end{enumerate}
\item \begin{enumerate}
    \item Give the coordinates of points $A$, $B$, $C$, $M$, $N$ in the coordinate system~$(A,B,C)$.
    \item Compute the coordinates of $\vect{MN}$ and $\vect{BC}$. 
    \item Use the vectors to prove that the line $MN$ is parallel to the line $BC$.
    \end{enumerate}
\end{enumerate}}



\exoc{Let $ABC$ be a non-flat triangle, $M$ and $N$ two points such that :
$$\vect{AM}=\cfrac35\vect{AB} \qquad \textrm{and} \qquad \vect{AN}=\cfrac25\vect{AB}+\cfrac15\vect{AC}$$
\begin{enumerate}
\item \begin{enumerate}
      \item Why is $(A,B,C)$ a coordinate system of the plane ?
      \item Give the coordinates of the points $A$, $B$, $C$, $M$ and $N$.
      \end{enumerate}
\item Prove that the lines $MN$ and $BC$ are parallel one to another :
      \begin{enumerate}
      \item first, using the coordinates of $\vect{MN}$ and $\vect{BC}$ ;
      \item second, decomposing $\vect{MN}$ and $\vect{BC}$ as linear combinations of $\vect{AB}$ and $\vect{AC}$.
      \end{enumerate}
\item \begin{enumerate}
      \item What are the coordinates of the point $P$ such that $MNPB$ is a parallelogram ?
      \item Prove that $P$ is on the line $BC$.
      \end{enumerate}
\end{enumerate}}



\exoc{Let $ABC$ be a scalene, non-flat, triangle. Let $M$ be the midpoint of segment $[AB]$ and $N$ the midpoint of segment $[AC]$. We call $K$ and $L$ the respective midpoints of the segments $[CM]$ and $[BN]$.
Let $P$ and $Q$ be the points defined by the relations :
$$\vect{NP}=\cfrac13\vect{NM} \qquad \textrm{and} \qquad \vect{MQ}=\cfrac13\vect{MN}.$$
\begin{enumerate}
\item \begin{enumerate}
      \item Draw a picture.
      \item Explain why $(A;B,C)$ is a Cartesian coordinate system in the plane.
      \item Give the coordinates of $A$, $B$, $C$, $M$, $N$, $K$, $L$, $P$ and $Q$ in this coordinate system. 
      \end{enumerate}
\item Prove that the lines $(PQ)$ and $(KL)$ are parallel.
\item Prove that the points $A$, $P$, $K$ on one hand and $A$, $Q$, $L$ on the other hand, are collinear.
\end{enumerate}}

\exoc{In a Cartesian coordinate system $\OIJ$, with unit 1 cm, consider the points $A(-1,0)$, $B(1,-6)$ and $H(3,-2)$.
\begin{enumerate}
 \item Draw a picture and fill it all along the exercise.
  \item \begin{enumerate}
	\item Compute the coordinates of the point $K$ such that $ABHK$ is a parallelogram.
	\item Compute the coordinates of the point $L$ symmetric of $H$ around $A$.
	\item Compute the coordinates of the point $C$ such that $\vect{HC}=-2\vect{HB}$.
	\item Are the points $C$, $K$ and $L$ collinear ?
	\end{enumerate}
 \item Let $I$ be the midpoint of $[AB]$ and $J$ the midpoint of $[AC]$. Let $G$ be the point such that $\vect{GA}+\vect{GB}+\vect{GC}=\vect{0}$ 
      \begin{enumerate}
      \item Compute the coordinates of the points $I$, $J$ and $G$.
      \item Prove that $C$, $G$ and $I$ are collinear, and then that $B$, $G$ and $J$ are also collinear.
      \item What is the point $G$ in the triangle $ABC$ ?
      \end{enumerate}
\item Let $M$ be the point defined by the relation $3\vect{BM}=3\vect{BA}+2\vect{BC}$.
      \begin{enumerate}
      \item Compute the coordinates of point $M$.
      \item Prove that the lines $(AM)$ and $(BC)$ are parallel.
      \end{enumerate}
\item \begin{enumerate}
      \item Compute $AH^2$, $HB^2$ and $AB^2$. What can you deduce about the lines $(AH)$ and $(BC)$ ?
      \item Is the triangle $ABC$ right-angled ? Compute the area of triangle $ABC$.
      \end{enumerate}
\item The line $(LK)$ intersects the axes at two points $E(x_E,0)$ and $F(0,y_F)$. Compute the values of $x_E$ and $y_F$.
\end{enumerate}
}





\end{document}