To find the midpoint of a line segment given its endpoints, we can use the midpoint formula:
[tex]\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Here, the endpoints of the segment [tex]\(\overline{GH}\)[/tex] are [tex]\(G (14,3)\)[/tex] and [tex]\(H(10,-6)\)[/tex]. Let's label the coordinates of [tex]\(G\)[/tex] and [tex]\(H\)[/tex] as follows:
- [tex]\(G_x = 14\)[/tex]
- [tex]\(G_y = 3\)[/tex]
- [tex]\(H_x = 10\)[/tex]
- [tex]\(H_y = -6\)[/tex]
Now, we will apply the midpoint formula to these coordinates:
1. Calculate the [tex]\(x\)[/tex]-coordinate of the midpoint:
[tex]\[ \frac{G_x + H_x}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]
2. Calculate the [tex]\(y\)[/tex]-coordinate of the midpoint:
[tex]\[ \frac{G_y + H_y}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5 \][/tex]
Therefore, the coordinates of the midpoint of [tex]\(\overline{GH}\)[/tex] are [tex]\( (12, -1.5) \)[/tex].
Thus, the correct answer is:
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
