To determine the midpoint of a line segment, we use the midpoint formula. The midpoint [tex]\((M_x, M_y)\)[/tex] of a line segment with endpoints [tex]\(G(x_1, y_1)\)[/tex] and [tex]\(H(x_2, y_2)\)[/tex] is given by:
[tex]\[
M_x = \frac{x_1 + x_2}{2}
\][/tex]
[tex]\[
M_y = \frac{y_1 + y_2}{2}
\][/tex]
Given the endpoints [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex], we can substitute the coordinates into the midpoint formula.
First, calculate the x-coordinate of the midpoint:
[tex]\[
M_x = \frac{14 + 10}{2} = \frac{24}{2} = 12
\][/tex]
Next, calculate the y-coordinate of the midpoint:
[tex]\[
M_y = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -\frac{3}{2}
\][/tex]
So, the midpoint of [tex]\(\overline{GH}\)[/tex] is:
[tex]\[
\left(12, -\frac{3}{2}\right)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\left(12, -\frac{3}{2}\right)}
\][/tex]
