To find the midpoint of a line segment defined by two endpoints, you use the midpoint formula. The formula for the midpoint [tex]\((M_x, M_y)\)[/tex] of a segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}
\][/tex]
Here, the endpoints of the line segment [tex]\(\overline{PQ}\)[/tex] are given as [tex]\(P(4,1)\)[/tex] and [tex]\(Q(4,8)\)[/tex].
Let's apply the midpoint formula step-by-step:
1. Identify the coordinates of the endpoints:
[tex]\[
x_1 = 4, \quad y_1 = 1, \quad x_2 = 4, \quad y_2 = 8
\][/tex]
2. Calculate the [tex]\(x\)[/tex]-coordinate of the midpoint:
[tex]\[
M_x = \frac{x_1 + x_2}{2} = \frac{4 + 4}{2} = \frac{8}{2} = 4
\][/tex]
3. Calculate the [tex]\(y\)[/tex]-coordinate of the midpoint:
[tex]\[
M_y = \frac{y_1 + y_2}{2} = \frac{1 + 8}{2} = \frac{9}{2} = 4.5
\][/tex]
Therefore, the coordinates of the midpoint are [tex]\((4, 4.5)\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{(4, 4.5)}
\][/tex]
