The line [tex]$\overline{MN}$[/tex] is the diameter of a circle where the coordinates of points [tex]$M$[/tex] and [tex]$N$[/tex] are [tex]$M(-20,39)$[/tex] and [tex]$N(20,-31)$[/tex]. What is the center of the circle?

A. [tex]$(6,3)$[/tex]
B. [tex]$(-1,13)$[/tex]
C. [tex]$(0,4)$[/tex]
D. [tex]$(-9,-2)$[/tex]



Answer :

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].