Answer
$\text{a) Distance: }
3\sqrt{55}
\text{ units}\\\text{b) Midpoint: }
\left( 2\sqrt{7}, \dfrac{7\sqrt{3}}{2} \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the Distance Formula and the Midpoint Formula to find the distance and the midpoint of the given points $\left(
-\sqrt{7},8\sqrt{3}
\right)$ and $\left(
5\sqrt{7}, -\sqrt{3}
\right).$
$\bf{\text{Solution Details:}}$
With the given points, then $x_1=
-\sqrt{7}
,$ $x_2=
5\sqrt{7}
,$ $y_1=
8\sqrt{3}
,$ and $y_2=
-\sqrt{3}
.$
Using the Distance Formula which is given by $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}
,$ then
\begin{array}{l}\require{cancel}
d=\sqrt{(-\sqrt{7}-5\sqrt{7})^2+(8\sqrt{3}-(-\sqrt{3}))^2}
\\\\
d=\sqrt{(-\sqrt{7}-5\sqrt{7})^2+(8\sqrt{3}+\sqrt{3})^2}
\\\\
d=\sqrt{(-6\sqrt{7})^2+(9\sqrt{3})^2}
\\\\
d=\sqrt{36(7)+81(3)}
\\\\
d=\sqrt{252+243}
\\\\
d=\sqrt{495}
\\\\
d=\sqrt{9\cdot55}
\\\\
d=\sqrt{(3)^2\cdot55}
\\\\
d=3\sqrt{55}
.\end{array}
Using $\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)$ or the Midpoint Formula, then the midpoint of the line segment with endpoints given above is
\begin{array}{l}\require{cancel}
\left( \dfrac{-\sqrt{7}+5\sqrt{7}}{2}, \dfrac{8\sqrt{3}+(-\sqrt{3})}{2} \right)
\\\\=
\left( \dfrac{-\sqrt{7}+5\sqrt{7}}{2}, \dfrac{8\sqrt{3}-\sqrt{3}}{2} \right)
\\\\=
\left( \dfrac{4\sqrt{7}}{2}, \dfrac{7\sqrt{3}}{2} \right)
\\\\=
\left( 2\sqrt{7}, \dfrac{7\sqrt{3}}{2} \right)
.\end{array}
Hence,
$
\text{a) Distance: }
3\sqrt{55}
\text{ units}\\\text{b) Midpoint: }
\left( 2\sqrt{7}, \dfrac{7\sqrt{3}}{2} \right)
.$