To find the midpoint of a line segment given its endpoints, we use the midpoint formula. The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( G(x_1, y_1) \)[/tex] and [tex]\( H(x_2, y_2) \)[/tex] is calculated as follows:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the endpoints [tex]\( G(14, 3) \)[/tex] and [tex]\( H(10, -6) \)[/tex], let's apply the formula step-by-step:
1. Identify the coordinates of the endpoints:
- [tex]\( G(14, 3) \)[/tex] where [tex]\( x_1 = 14 \)[/tex] and [tex]\( y_1 = 3 \)[/tex]
- [tex]\( H(10, -6) \)[/tex] where [tex]\( x_2 = 10 \)[/tex] and [tex]\( y_2 = -6 \)[/tex]
2. Calculate the x-coordinate of the midpoint:
[tex]\[
\frac{x_1 + x_2}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12
\][/tex]
3. Calculate the y-coordinate of the midpoint:
[tex]\[
\frac{y_1 + y_2}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -\frac{3}{2}
\][/tex]
4. Combine the results to get the midpoint:
[tex]\[
M = \left( 12, -\frac{3}{2} \right)
\][/tex]
Therefore, the midpoint of [tex]\( \overline{GH} \)[/tex] is [tex]\( \left( 12, -\frac{3}{2} \right) \)[/tex].
So the correct answer is:
C. [tex]\( \left( 12, -\frac{3}{2} \right) \)[/tex]
