Consider a cube ABCDEFGH with side length 1. The point I is the midpoint of segment [BD]. We define the point L such that $\overrightarrow { \mathrm { IL } } = \frac { 3 } { 4 } \overrightarrow { \mathrm { IG } }$. We use the orthonormal coordinate system ( $A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
a. Specify the coordinates of points $\mathrm { D } , \mathrm { B } , \mathrm { I }$ and G. No justification is required. b. Show that point L has coordinates $\left( \frac { 7 } { 8 } ; \frac { 7 } { 8 } ; \frac { 3 } { 4 } \right)$.
Verify that a Cartesian equation of plane (BDG) is $x + y - z - 1 = 0$.
Consider the line $\Delta$ perpendicular to plane (BDG) passing through L. a. Justify that a parametric representation of line $\Delta$ is: $$\left\{ \begin{aligned}
x & = \frac { 7 } { 8 } + t \\
y & = \frac { 7 } { 8 } + t \text { where } t \in \mathbb { R } . \\
z & = \frac { 3 } { 4 } - t
\end{aligned} \right.$$ b. Show that lines $\Delta$ and (AE) intersect at point K with coordinates $\left( 0 ; 0 ; \frac { 13 } { 8 } \right)$. c. What does point L represent for point K? Justify your answer.
a. Calculate the distance KL. b. We admit that triangle DBG is equilateral. Show that its area equals $\frac { \sqrt { 3 } } { 2 }$. c. Deduce the volume of tetrahedron KDBG. We recall that:
the volume of a pyramid is given by the formula $V = \frac { 1 } { 3 } \times \mathscr { B } \times h$ where $\mathscr { B }$ is the area of a base and $h$ is the length of the height relative to this base;
a tetrahedron is a pyramid with a triangular base.
We denote by $a$ a real number belonging to the interval $] 0 ; + \infty \left[ \right.$ and we note $K _ { a }$ the point with coordinates ( $0 ; 0 ; a$ ). a. Express the volume $V _ { a }$ of pyramid $\mathrm { ABCD } K _ { a }$ as a function of $a$. b. We denote $\Delta _ { a }$ the line with parametric representation $$\left\{ \begin{aligned}
x & = t ^ { \prime } \\
y & = t ^ { \prime } \\
z & = - t ^ { \prime } + a
\end{aligned} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right.$$ We call $L _ { a }$ the point of intersection of line $\Delta _ { a }$ with plane (BDG). Show that the coordinates of point $L _ { a }$ are $\left( \frac { a + 1 } { 3 } ; \frac { a + 1 } { 3 } ; \frac { 2 a - 1 } { 3 } \right)$. c. Determine, if it exists, a strictly positive real number $a$ such that tetrahedron $\mathrm { GDB } L _ { a }$ and pyramid $\mathrm { ABCD } K _ { a }$ have the same volume.
Consider a cube ABCDEFGH with side length 1.
The point I is the midpoint of segment [BD]. We define the point L such that $\overrightarrow { \mathrm { IL } } = \frac { 3 } { 4 } \overrightarrow { \mathrm { IG } }$.\\
We use the orthonormal coordinate system ( $A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
\begin{enumerate}
\item a. Specify the coordinates of points $\mathrm { D } , \mathrm { B } , \mathrm { I }$ and G.
No justification is required.\\
b. Show that point L has coordinates $\left( \frac { 7 } { 8 } ; \frac { 7 } { 8 } ; \frac { 3 } { 4 } \right)$.\\
\item Verify that a Cartesian equation of plane (BDG) is $x + y - z - 1 = 0$.\\
\item Consider the line $\Delta$ perpendicular to plane (BDG) passing through L.\\
a. Justify that a parametric representation of line $\Delta$ is:
$$\left\{ \begin{aligned}
x & = \frac { 7 } { 8 } + t \\
y & = \frac { 7 } { 8 } + t \text { where } t \in \mathbb { R } . \\
z & = \frac { 3 } { 4 } - t
\end{aligned} \right.$$
b. Show that lines $\Delta$ and (AE) intersect at point K with coordinates $\left( 0 ; 0 ; \frac { 13 } { 8 } \right)$.\\
c. What does point L represent for point K? Justify your answer.\\
\item a. Calculate the distance KL.\\
b. We admit that triangle DBG is equilateral. Show that its area equals $\frac { \sqrt { 3 } } { 2 }$.\\
c. Deduce the volume of tetrahedron KDBG.
We recall that:
\begin{itemize}
\item the volume of a pyramid is given by the formula $V = \frac { 1 } { 3 } \times \mathscr { B } \times h$ where $\mathscr { B }$ is the area of a base and $h$ is the length of the height relative to this base;
\item a tetrahedron is a pyramid with a triangular base.
\end{itemize}
\item We denote by $a$ a real number belonging to the interval $] 0 ; + \infty \left[ \right.$ and we note $K _ { a }$ the point with coordinates ( $0 ; 0 ; a$ ).\\
a. Express the volume $V _ { a }$ of pyramid $\mathrm { ABCD } K _ { a }$ as a function of $a$.\\
b. We denote $\Delta _ { a }$ the line with parametric representation
$$\left\{ \begin{aligned}
x & = t ^ { \prime } \\
y & = t ^ { \prime } \\
z & = - t ^ { \prime } + a
\end{aligned} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right.$$
We call $L _ { a }$ the point of intersection of line $\Delta _ { a }$ with plane (BDG). Show that the coordinates of point $L _ { a }$ are $\left( \frac { a + 1 } { 3 } ; \frac { a + 1 } { 3 } ; \frac { 2 a - 1 } { 3 } \right)$.\\
c. Determine, if it exists, a strictly positive real number $a$ such that tetrahedron $\mathrm { GDB } L _ { a }$ and pyramid $\mathrm { ABCD } K _ { a }$ have the same volume.
\end{enumerate}