To determine 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 for any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the coordinates of the midpoint [tex]\((M)\)[/tex] are calculated as follows:
[tex]\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
Given the points [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex]:
1. Calculate the [tex]\(x\)[/tex]-coordinate of the midpoint:
[tex]\[
x = \frac{14 + 10}{2} = \frac{24}{2} = 12
\][/tex]
2. Calculate the [tex]\(y\)[/tex]-coordinate of the midpoint:
[tex]\[
y = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5
\][/tex]
Thus, the coordinates of the midpoint are:
[tex]\[
\left(12, -\frac{3}{2}\right)
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
