The endpoints of [tex]\(\overline{GH}\)[/tex] are [tex]\(G(14, 3)\)[/tex] and [tex]\(H(10, -6)\)[/tex]. What is the midpoint of [tex]\(GH\)[/tex]?

A. [tex]\((24, -3)\)[/tex]
B. [tex]\((6, -15)\)[/tex]
C. [tex]\(\left(12, -\frac{3}{2}\right)\)[/tex]
D. [tex]\(\left(-2, -\frac{9}{2}\right)\)[/tex]



Answer :

To find the midpoint of a line segment with endpoints [tex]\( G(14, 3) \)[/tex] and [tex]\( H(10, -6) \)[/tex], we 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 two endpoints.

Here, the coordinates of [tex]\( G \)[/tex] are [tex]\( (14, 3) \)[/tex] and the coordinates of [tex]\( H \)[/tex] are [tex]\( (10, -6) \)[/tex].

Let's calculate the midpoint:

1. Calculate the [tex]\(x\)[/tex]-coordinate of the midpoint:
[tex]\[ \frac{14 + 10}{2} = \frac{24}{2} = 12 \][/tex]

2. Calculate the [tex]\(y\)[/tex]-coordinate of the midpoint:
[tex]\[ \frac{3 + (-6)}{2} = \frac{3 - 6}{2} = \frac{-3}{2} = -1.5 \][/tex]

So, the coordinates of the midpoint are:
[tex]\[ \left(12, -1.5\right) \][/tex]

Therefore, the correct answer is:
[tex]\[ \text{C.} \left(12, -\frac{3}{2}\right) \][/tex]