First, let's graph the trapezoid.
The midsegment refers to a segment that goes from the midpoint of BC and the midpoint of AD.
Let's find the midpoints using the following formula
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex][tex]\begin{gathered} M_{BC}=(\frac{8+12}{2},\frac{-4+2}{2})=(\frac{20}{2},-\frac{2}{2})=(10,-1) \\ M_{AD}=(\frac{2+0}{2},\frac{0+10}{2})=(\frac{2}{2},\frac{10}{2})=(1,5) \end{gathered}[/tex]Now, we use the distance formula to find the length of the midsegment
[tex]\begin{gathered} d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2} \\ d=\sqrt[]{(-1-5)^2+(10-1)^2}=\sqrt[]{(-6)^2+(9)^2} \\ d=\sqrt[]{36+81}=\sqrt[]{117} \\ d\approx10.8 \end{gathered}[/tex]