To find the midpoint of a line segment given its endpoints, you can use the midpoint formula. 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 the formula:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
In this particular problem, the endpoints are [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex].
1. Identify the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates of the endpoints:
- [tex]\(G_x = 14\)[/tex]
- [tex]\(G_y = 3\)[/tex]
- [tex]\(H_x = 10\)[/tex]
- [tex]\(H_y = -6\)[/tex]
2. Calculate the [tex]\(x\)[/tex]-coordinate of the midpoint:
[tex]\[
x_{\text{mid}} = \frac{G_x + H_x}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12
\][/tex]
3. Calculate the [tex]\(y\)[/tex]-coordinate of the midpoint:
[tex]\[
y_{\text{mid}} = \frac{G_y + H_y}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5
\][/tex]
Therefore, the coordinates of the midpoint [tex]\(M\)[/tex] are [tex]\((12, -1.5)\)[/tex].
From the given options, the correct answer is:
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
