The endpoints of [tex]\overline{PQ}[/tex] are [tex]\( P(4,1) \)[/tex] and [tex]\( Q(4,8) \)[/tex]. Find the midpoint of [tex]\overline{PQ}[/tex].

A. [tex]\( (4, 4.5) \)[/tex]
B. [tex]\( (0, -3.5) \)[/tex]
C. [tex]\( (4.5, 4) \)[/tex]
D. [tex]\( (6, 3.5) \)[/tex]

Please select the best answer from the choices provided.



Answer :

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]