We consider the cube ABCDEFGH with edge length 1 represented opposite.\\
We denote K the midpoint of segment [HG].\\
We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AE } } )$.

\begin{enumerate}
  \item Justify that the points $\mathrm { C } , \mathrm { F }$ and K define a plane.
  \item a. Give, without justification, the lengths KG, GF and GC.\\
b. Calculate the area of triangle FGC.\\
c. Calculate the volume of tetrahedron FGCK.\\
\end{enumerate}

We recall that the volume $V$ of a tetrahedron is given by:
$$V = \frac { 1 } { 3 } \mathscr { B } \times h ,$$
where $\mathscr { B }$ is the area of a base and $h$ the corresponding height.\\
3. a. We denote $\vec { n }$ the vector with coordinates $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)$.

Prove that $\vec { n }$ is normal to the plane (CFK).\\
b. Deduce that a Cartesian equation of the plane (CFK) is:
$$x + 2 y + z - 3 = 0 .$$

\begin{enumerate}
  \setcounter{enumi}{3}
  \item We denote $\Delta$ the line passing through point G and perpendicular to the plane (CFK). Prove that a parametric representation of the line $\Delta$ is:
\end{enumerate}

$$\left\{ \begin{aligned}
x & = 1 + t \\
y & = 1 + 2 t \\
z & = 1 + t
\end{aligned} \quad ( t \in \mathbb { R } ) \right)$$

\begin{enumerate}
  \setcounter{enumi}{4}
  \item Let L be the point of intersection between the line $\Delta$ and the plane (CFK).\\
a. Determine the coordinates of point L .\\
b. Deduce that $\mathrm { LG } = \frac { \sqrt { 6 } } { 6 }$.
  \item Using question 2., determine the exact value of the area of triangle CFK.
\end{enumerate}