To find the midpoint of the line segment [tex]$\overline{GH}$[/tex] with endpoints [tex]$G (14, 3)$[/tex] and [tex]$H (10, -6)$[/tex], we use the midpoint formula. The midpoint formula states that the midpoint [tex]$M$[/tex] of a line segment with endpoints [tex]$(x_1, y_1)$[/tex] and [tex]$(x_2, y_2)$[/tex] is 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]$G$[/tex] are [tex]$(14, 3)$[/tex] and the coordinates of point [tex]$H$[/tex] are [tex]$(10, -6)$[/tex]. Plugging these coordinates into the midpoint formula, we get:
[tex]\[
M = \left( \frac{14 + 10}{2}, \frac{3 + (-6)}{2} \right)
\][/tex]
Simplifying the expressions inside the parentheses, we have:
[tex]\[
M = \left( \frac{24}{2}, \frac{-3}{2} \right)
\][/tex]
Calculating these values results in:
[tex]\[
M = \left( 12, -1.5 \right)
\][/tex]
Therefore, the correct answer is:
C. [tex]$\left( 12, -\frac{3}{2} \right)$[/tex]
