Answer
$\text{a) Distance: }
8\sqrt{2}
\\\text{b) Midpoint: }
\left( -9, -3 \right)$
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 $\left(
-5,-7
\right)$ and $\left(
-13,1
\right).$
$\bf{\text{Solution Details:}}$
With the given points, then $x_1=
-5
,$ $x_2=
-13
,$ $y_1=
-7
,$ and $y_2=
1
.$
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{(-5-(-13))^2+(-7-1)^2}
\\\\
d=\sqrt{(-5+13)^2+(-7-1)^2}
\\\\
d=\sqrt{(8)^2+(-8)^2}
\\\\
d=\sqrt{64+64}
\\\\
d=\sqrt{128}
\\\\
d=\sqrt{64\cdot2}
\\\\
d=\sqrt{(8)^2\cdot2}
\\\\
d=8\sqrt{2}
.\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{-5+(-13)}{2}, \dfrac{-7+1}{2} \right)
\\\\=
\left( \dfrac{-5-13}{2}, \dfrac{-7+1}{2} \right)
\\\\=
\left( \dfrac{-18}{2}, \dfrac{-6}{2} \right)
\\\\=
\left( -9, -3 \right)
.\end{array}
Hence,
$
\text{a) Distance: }
8\sqrt{2}
\\\text{b) Midpoint: }
\left( -9, -3 \right)
.$