To find the midpoint of a line segment with given endpoints, you can use the midpoint formula. The midpoint formula is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the endpoints. For the endpoints [tex]\(G(14,3)\)[/tex] and [tex]\(H(10,-6)\)[/tex], we have:
1. Identify the [tex]\(x\)[/tex]-coordinates of [tex]\(G\)[/tex] and [tex]\(H\)[/tex]:
- [tex]\(x_1 = 14\)[/tex]
- [tex]\(x_2 = 10\)[/tex]
2. Identify the [tex]\(y\)[/tex]-coordinates of [tex]\(G\)[/tex] and [tex]\(H\)[/tex]:
- [tex]\(y_1 = 3\)[/tex]
- [tex]\(y_2 = -6\)[/tex]
3. Apply the coordinates to the midpoint formula:
- Midpoint [tex]\(x\)[/tex]-coordinate: [tex]\(\frac{x_1 + x_2}{2} = \frac{14 + 10}{2} = \frac{24}{2} = 12\)[/tex]
- Midpoint [tex]\(y\)[/tex]-coordinate: [tex]\(\frac{y_1 + y_2}{2} = \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5\)[/tex]
Therefore, the midpoint of [tex]\(\overline{GH}\)[/tex] is:
[tex]\[
\left( 12, -\frac{3}{2} \right)
\][/tex]
Given the answer choices:
A. [tex]\((6, -15)\)[/tex]
B. [tex]\(\left( -2, -\frac{9}{2} \right)\)[/tex]
C. [tex]\(\left( 12, -\frac{3}{2} \right)\)[/tex]
D. [tex]\((24, -3)\)[/tex]
E. [tex]\((18, 12)\)[/tex]
The correct answer is:
C. [tex]\(\left( 12, -\frac{3}{2} \right)\)[/tex]
