To determine the center of the circle given that the line segment [tex]\(\overline{MN}\)[/tex] is the diameter, we need to find the midpoint of [tex]\(M\)[/tex] and [tex]\(N\)[/tex]. The midpoint formula, which calculates the coordinates of the midpoint [tex]\((x_m, y_m)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], is:
[tex]\[
(x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the coordinates of [tex]\(M\)[/tex] and [tex]\(N\)[/tex]:
- [tex]\(M = (-20, 39)\)[/tex]
- [tex]\(N = (20, -31)\)[/tex]
We substitute [tex]\(M\)[/tex] and [tex]\(N\)[/tex] into the midpoint formula:
[tex]\[
(x_m, y_m) = \left( \frac{-20 + 20}{2}, \frac{39 + (-31)}{2} \right)
\][/tex]
Now, perform the arithmetic inside the parentheses:
1. For the [tex]\(x\)[/tex]-coordinate:
[tex]\[
x_m = \frac{-20 + 20}{2} = \frac{0}{2} = 0.0
\][/tex]
2. For the [tex]\(y\)[/tex]-coordinate:
[tex]\[
y_m = \frac{39 + (-31)}{2} = \frac{39 - 31}{2} = \frac{8}{2} = 4.0
\][/tex]
Therefore, the center of the circle is at:
[tex]\[
(x_m, y_m) = (0.0, 4.0)
\][/tex]
In the context of the given multiple-choice answers, this corresponds to the point:
[tex]\[
(0, 4)
\][/tex]
Thus, the center of the circle is [tex]\((0, 4)\)[/tex].
