I is true. Note that the slopes of $\ell$ and $L$ are the same. Hence, $\Gamma$ is a straight line parallel to $\ell$ and $L$.
II is not true. Note that the coordinates of $A$ and $B$ are $\left( \dfrac{37}{9}, 0\right)$ and $\left(0, \dfrac{-85}{16} \right)$ respectively.
Note also that the slope of $\Gamma$ is $\dfrac{-3}{4}$.
Hence, we have
$\begin{array}{cl}
& m_{AB} \times m_\Gamma \\
= & \dfrac{0-\frac{-85}{16}}{\frac{37}{9}-0} \times \dfrac{-3}{4} \\
= & \dfrac{-2295}{2368} \\
\neq & -1
\end{array}$
Therefore, $\Gamma$ is not perpendicular to $AB$.
III is true. Note that the slopes of $\ell$ and $L$ are the same. Hence, $\Gamma$ is a straight line passing through the mid-point of the $y$-intercepts of $\ell$ and $L$.
Note that the $y$-intercepts of $\ell$ and $L$ are $\dfrac{37}{12}$ and $\dfrac{-85}{16}$ respectively. Then the coordinates of the mid-point of the $y$-intercepts
$\begin{array}{cl}
= & \left(0, \dfrac{1}{2}\left(\dfrac{37}{12}+\dfrac{-85}{16}\right)\right) \\
= & \dfrac{-107}{96}
\end{array}$
The equation of $\Gamma$ is
$\begin{array}{rcl}
\dfrac{y-\frac{-107}{96}}{x-0} & = & \dfrac{-3}{4} \\
4y +\dfrac{107}{24} & = & -3x \\
72x+96y+107 & = & 0
\end{array}$
The mid-point of $AB$
$\begin{array}{cl}
= & \left(\dfrac{1}{2}\times \dfrac{37}{9},\dfrac{1}{2} \times \dfrac{-85}{16}\right) \\
= & \left(\dfrac{37}{18},\dfrac{-85}{32}\right)
\end{array}$
Sub. the mid-point of $AB$ into the left side of the equation of $\Gamma$, we have
$\begin{array}{cl}
& 72 \times \dfrac{37}{18}+96\times\dfrac{-85}{32}+107 \\
= & 0
\end{array}$
Therefore, $\Gamma$ passes through the mid-point of $AB$.
