Solution:
To find the midpoint of a line segment with endpoints [tex]\( G \)[/tex] and [tex]\( H \)[/tex], we use the midpoint formula. The midpoint formula for the endpoints [tex]\( G (x_1, y_1) \)[/tex] and [tex]\( H (x_2, y_2) \)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the endpoints [tex]\( G (10, 1) \)[/tex] and [tex]\( H (3, 5) \)[/tex]:
1. Identify the coordinates of [tex]\( G \)[/tex] and [tex]\( H \)[/tex]:
- [tex]\( G (10, 1) \)[/tex]: Where [tex]\( x_1 = 10 \)[/tex] and [tex]\( y_1 = 1 \)[/tex]
- [tex]\( H (3, 5) \)[/tex]: Where [tex]\( x_2 = 3 \)[/tex] and [tex]\( y_2 = 5 \)[/tex]
2. Apply the midpoint formula:
[tex]\[
\left( \frac{10 + 3}{2}, \frac{1 + 5}{2} \right)
\][/tex]
3. Perform the calculations:
- For the x-coordinate:
[tex]\[
\frac{10 + 3}{2} = \frac{13}{2} = 6.5
\][/tex]
- For the y-coordinate:
[tex]\[
\frac{1 + 5}{2} = \frac{6}{2} = 3.0
\][/tex]
4. Combine the results to get the midpoint:
[tex]\[
\left( 6.5, 3.0 \right)
\][/tex]
So, the midpoint of [tex]\( \overline{ GH } \)[/tex] is [tex]\( \left( 6.5, 3.0 \right) \)[/tex].
Among the given choices, the correct answer matches option C:
[tex]\[
\boxed{\left( \frac{13}{2}, 3 \right)}
\][/tex]
