\textbf{Exercise 4 — For candidates who have not followed the speciality}

5 points

On the figure given in appendix 2 to be returned with the copy:
\begin{itemize}
  \item ABCDEFGH is a rectangular parallelepiped such that $\mathrm { AB } = 12 , \mathrm { AD } = 18$ and $\mathrm { AE } = 6$
  \item EBDG is a tetrahedron.
\end{itemize}

Space is referred to an orthonormal coordinate system with origin A in which the points $\mathrm { B } , \mathrm { D }$ and E have respective coordinates $\mathrm { B } ( 12 ; 0 ; 0 ) , \mathrm { D } ( 0 ; 18 ; 0 )$ and $\mathrm { E } ( 0 ; 0 ; 6 )$.

\begin{enumerate}
  \item Prove that the plane (EBD) has the Cartesian equation $3 x + 2 y + 6 z - 36 = 0$.
  \item a. Determine a parametric representation of the line (AG).\\
b. Deduce that the line (AG) intersects the plane (EBD) at a point K with coordinates (4;6;2).
  \item Is the line (AG) orthogonal to the plane (EBD)? Justify.
  \item a. Let M be the midpoint of the segment $[ \mathrm { ED } ]$. Prove that the points B, K and M are collinear.\\
b. Then construct the point K on the figure given in appendix 2 to be returned with the copy.
  \item We denote by P the plane parallel to the plane (ADE) passing through the point K.\\
a. Prove that the plane P intersects the plane (EBD) along a line parallel to the line (ED).\\
b. Then construct on appendix 2 to be returned with the copy the intersection of the plane P and the face EBD of the tetrahedron EBDG.
\end{enumerate}