\textbf{Exercise 4 Geometry in space}

In the figure below, ABCDEFGH is a rectangular parallelepiped such that $\mathrm{AB} = 5$, $\mathrm{AD} = 3$ and $\mathrm{AE} = 2$.\\
The space is equipped with an orthonormal coordinate system with origin A in which the points B, D and E have coordinates respectively $(5; 0; 0)$, $(0; 3; 0)$ and $(0; 0; 2)$.

\begin{enumerate}
  \item a. Give, in the coordinate system considered, the coordinates of points H and G.\\
b. Give a parametric representation of the line (GH).
  \item Let M be a point of the segment $[\mathrm{GH}]$ such that $\overrightarrow{\mathrm{HM}} = k\overrightarrow{\mathrm{HG}}$ with $k$ a real number in the interval $[0; 1]$.\\
a. Justify that the coordinates of M are $(5k; 3; 2)$.\\
b. Deduce from this that $\overrightarrow{\mathrm{AM}} \cdot \overrightarrow{\mathrm{CM}} = 25k^{2} - 25k + 4$.\\
c. Determine the values of $k$ for which AMC is a triangle right-angled at M.
\end{enumerate}

For the rest of the exercise, we consider that point M has coordinates $(1; 3; 2)$.\\
We admit that triangle AMC is right-angled at M.\\
We recall that the volume of a tetrahedron is given by the formula $\frac{1}{3} \times$ Area of the base $\times h$ where $h$ is the height relative to the base.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item We consider the point K with coordinates $(1; 3; 0)$.\\
a. Determine a Cartesian equation of the plane (ACD).\\
b. Justify that point K is the orthogonal projection of point M onto the plane (ACD).\\
c. Deduce from this the volume of the tetrahedron MACD.
  \item We denote P the orthogonal projection of point D onto the plane (AMC). Calculate the distance DP; give a value rounded to $10^{-1}$.
\end{enumerate}