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 for a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the coordinates [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex], we can substitute these values into the formula. Let's calculate each component step by step.
1. Calculate the x-coordinate of the midpoint:
[tex]\[
\text{x-coordinate} = \frac{14 + 10}{2}
\][/tex]
Add the x-coordinates:
[tex]\[
14 + 10 = 24
\][/tex]
Then divide by 2:
[tex]\[
\frac{24}{2} = 12
\][/tex]
2. Calculate the y-coordinate of the midpoint:
[tex]\[
\text{y-coordinate} = \frac{3 + (-6)}{2}
\][/tex]
Add the y-coordinates:
[tex]\[
3 + (-6) = -3
\][/tex]
Then divide by 2:
[tex]\[
\frac{-3}{2} = -1.5
\][/tex]
So, the coordinates of the midpoint are:
[tex]\[
(12, -1.5)
\][/tex]
Given the options provided, the correct answer is:
[tex]\[
\text{C.} \quad \left(12, -\frac{3}{2}\right)
\][/tex]
This is equivalent to [tex]\((12, -1.5)\)[/tex].
