\begin{exercice}
On considère un parallélépipède rectangle $ABCDEFGH$ tel que $AB = 3$, $AD = 5$ et $AE = 2$. On note $O$ son centre.
\begin{center}
 	\def \dx {-0.8}
 	\def \r {0.3}
	\begin{tikzpicture}[general, scale=1]
	%points
	\coordinate (A) at (0,0); \draw (A) node[above left] {$A$};
	\coordinate (B) at (\dx,-0.5); \draw (B) node[below left] {$B$};
	\coordinate (C) at (5+\dx,-0.5); \draw (C) node[below] {$C$};
	\coordinate (D) at (5,0); \draw (D) node[right] {$D$};
	\coordinate (E) at (0,2); \draw (E) node[above left] {$E$};
	\coordinate (F) at (\dx,1.5); \draw (F) node[left] {$F$};
	\coordinate (G) at (5+\dx,1.5); \draw (G) node[below right] {$G$};
	\coordinate (H) at (5,2); \draw (H) node[above right] {$H$};
	\coordinate (O) at ({(5+\dx)/2},0.75); \draw (O) node[above] {$O$};
	%arêtes
	\draw (F)--(B)--(C)--(D)--(H)--(G)--(C); \draw (G)--(F)--(E)--(H);
	\draw[dashed] (A)--(B); \draw[dashed] (A)--(D); \draw[dashed] (A)--(E);
	%diagonales
	\draw[thin, dashed, color=C1] (E)--(C); \draw[thin, dashed, color=C1] (B)--(H);
	%angle
	\fill[color=A1,opacity=0.3,domain=204:330] (O)
						--plot ({(5+\dx)/2+\r*cos(\x)}, {0.75+\r*sin(\x)})
						--cycle;
	\end{tikzpicture}
\end{center}

\vspace{-\baselineskip}
En se plaçant dans le repère orthonormé $\left(A\,; \dfrac{1}{3}\vv{AB}, \dfrac{1}{5}\vv{AD}, \dfrac{1}{2}\vv{AE} \right)$, déterminer une mesure de l'angle $\widehat{BOC}$ au centième de degré près.
\end{exercice}
