Select the correct answer.

The endpoints of [tex]\(\overline{GH}\)[/tex] are [tex]\(G (10,1)\)[/tex] and [tex]\(H(3,5)\)[/tex]. What is the midpoint of [tex]\(\overline{GH}\)[/tex]?

A. [tex]\((-4,9)\)[/tex]

B. [tex]\(\left(\frac{7}{2}, 2\right)\)[/tex]

C. [tex]\(\left(\frac{13}{2}, 3\right)\)[/tex]

D. [tex]\((13,6)\)[/tex]



Answer :

To find the midpoint of the line segment with endpoints [tex]\( G (10, 1) \)[/tex] and [tex]\( H (3, 5) \)[/tex], we use the midpoint formula. The midpoint formula for a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

[tex]\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]

Starting with the coordinates of point [tex]\( G \)[/tex], which are [tex]\( (10, 1) \)[/tex], and the coordinates of point [tex]\( H \)[/tex], which are [tex]\( (3, 5) \)[/tex]:

1. Calculate the [tex]\( x \)[/tex]-coordinate of the midpoint.
[tex]\[ \text{Midpoint}_x = \frac{10 + 3}{2} = \frac{13}{2} = 6.5 \][/tex]

2. Calculate the [tex]\( y \)[/tex]-coordinate of the midpoint.
[tex]\[ \text{Midpoint}_y = \frac{1 + 5}{2} = \frac{6}{2} = 3 \][/tex]

Thus, the coordinates of the midpoint are:

[tex]\[ (6.5, 3.0) \][/tex]

Therefore, the closest answer in the given options is:

C. [tex]\(\left(\frac{13}{2}, 3\right)\)[/tex]