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

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\partie{Homework \#7}


In the figure below, the angles $\widehat{ABC}$ and $\widehat{BCD}$ are right and, in units, $AB=BE=EF=FG=GC=1$, $BC=4$
and $CD=2$. The segments linking the vertices $A$ and $D$ to every point on the segment $[BC]$ have been drawn. 

\begin{center}
\psset{unit=1.5cm}
\pspicture(-0.5,-0.5)(4.5,2.5)
\pspolygon(0,0)(4,0)(4,2)(0,1)
\uput[dl](0,0){$B$}
\uput[dr](4,0){$C$}
\uput[ul](4,2){$D$}
\uput[ur](0,1){$A$}
\uput[dl](1,0){$E$}
\uput[dl](2,0){$F$}
\uput[dl](3,0){$G$}
\psline(0,1)(1,0)
\psline(0,1)(2,0)
\psline(0,1)(3,0)
\psline(0,1)(4,0)
\psline(4,2)(1,0)
\psline(4,2)(2,0)
\psline(4,2)(3,0)
\psline(4,2)(0,0)
\endpspicture
\end{center}

\probpart{-- Static experiments with Geogebra}

\begin{enumerate}
\item Reproduce the picture using Geogebra.
\item \begin{enumerate}
  \item Use Geogebra to compute the lengths $AE$, $AF$, $AG$ and $AC$. Give in each case an
  approximation to 3DP.
  \item Do the same for the lengths $DB$, $DE$, $DF$ and $DG$.
  \item Two of the eight lengths you've computed are equal. Explain why.
\end{enumerate}
\item \begin{enumerate}
    \item Use Geogebra to compute the sums $AB+BD$, $AE+ED$, $AF+FD$, $AG+GD$ and $AC+CD$ to 3DP.
    \item Which of these 5 sums is the lowest ? Do you find this surprising ?
\end{enumerate}
\end{enumerate}

\probpart{-- Dynamic experiments with Geogebra}

\begin{enumerate}
\item Do a new figure with Geogebra, with only the points $A$, $B$, $C$, $D$ and the segments $[AB]$, $[BC]$ and $[CD]$
\item Define the lengths $m=AM$ and $n=MD$ and then their sum
$d=m+n$.
\item Ask Geogebra to show the values of the numbers to 5 DP, and move $M$ along $[BC]$ to find the position where
$d$ is minimal. Zoom in if you need more precision.
\item Write down the value of $BM$ such that $d$ is minimal. This
value is close to a simple fraction, which one ?
\end{enumerate}

\probpart{-- Using functions}

In this part, we name $x$ the distance $BM$ and $d(x)$ the sum of
distances $AM+MD$.

\begin{enumerate}
\item Write the lengths $MC$, $AM$, and $MD$ as functions of
$x$.
\item Give the expression of $d(x)$ as a function of $x$.
\item Draw the graph of this function with Geogebra or with your
calculator. Print it or, if you can't, draw it on your paper.
\item Find graphically, and with $5$ DP, the minimum of the function
$d$ and the values of $x$ for which this minimum is reached.
\item Explain how the result of the previous question confirms the
conclusion of part A.
\end{enumerate}

\probpart{-- Some properties of the minimal point}

Let $E$ be the point on the segment $[BC]$ such that
$\vect{BE}=\frac13\vect{BC}$.

\begin{enumerate}
\item Do a new figure with Geogebra and place precisely the point
$E$. Print it or copy it on your paper at the end of the exercise.
\item What can you say about the directions of the vectors $\vect{BE}$ and
$\vect{CE}$ ? Compute their norms (lengths) as fractions.
\item What can you deduce about the sum $2\vect{BE}+\vect{CE}$ ?
\item Use the definition of $E$ and Chasles' relation to prove again the equality
you found in the previous question.
\item Use Geogebra to build the point $A'$, symmetric of $A$ around
the point $B$.
\item Draw the line $(A'D)$. What do you notice ?
\item What is the position of a point $M$ on the segment $[BC]$
such that the sum $A'M+MD$ is minimal ? Deduce an explanation of the
property you noticed in the previous question.
\end{enumerate}




\end{document}
