To find the midpoint of the line segment [tex]\(\overline{AB}\)[/tex] with endpoints [tex]\(A(-3, 8)\)[/tex] and [tex]\(B(-7, -6)\)[/tex], we use the midpoint formula. The midpoint formula states that the coordinates of the midpoint [tex]\( M \)[/tex] of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are given by:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Here, the coordinates of point [tex]\(A\)[/tex] are [tex]\(A(-3, 8)\)[/tex], so [tex]\(x_1 = -3\)[/tex] and [tex]\(y_1 = 8\)[/tex]. The coordinates of point [tex]\(B\)[/tex] are [tex]\(B(-7, -6)\)[/tex], so [tex]\(x_2 = -7\)[/tex] and [tex]\(y_2 = -6\)[/tex].
Substitute these values into the midpoint formula:
[tex]\[
M_x = \frac{-3 + (-7)}{2} = \frac{-3 - 7}{2} = \frac{-10}{2} = -5
\][/tex]
[tex]\[
M_y = \frac{8 + (-6)}{2} = \frac{8 - 6}{2} = \frac{2}{2} = 1
\][/tex]
Therefore, the coordinates of the midpoint [tex]\( M \)[/tex] are [tex]\( (-5, 1) \)[/tex].
So, the midpoint of [tex]\(\overline{AB}\)[/tex] is [tex]\((-5, 1)\)[/tex].
