#### Answer

(a) The midpoint of the line segment $~\overline{P_1P_2}~$ is: $(0, 1, -1)$
(b) $d = 13.3$

#### Work Step by Step

$P_1 = (-2,3,5)$
$P_2 = (2,-1,-7)$
(a) We can find the midpoint between these two points:
$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2} ,\frac{z_1+z_2}{2})$
$= (\frac{-2+2}{2},\frac{3+(-1)}{2} ,\frac{5+(-7)}{2})$
$ = (0, 1, -1)$
The midpoint of the line segment $~\overline{P_1P_2}~$ is: $(0, 1, -1)$
(b) We can find the length $d$ of $\overline{P_1P_2}$:
$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
$d = \sqrt{(2-(-2))^2+(-1-3)^2+(-7-5)^2}$
$d = \sqrt{(4)^2+(-4)^2+(-12)^2}$
$d = \sqrt{16+16+144}$
$d = \sqrt{176}$
$d = 13.3$