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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]$\overline{GH}$[/tex]?

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]



Answer :

GinnyAnswer
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]

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