Answer:
Center of the circle:
[tex]\displaystyle \left(\frac{7}{2},\, 6\right)[/tex].
Radius of the circle:
[tex]\displaystyle \frac{1}{2}\, \sqrt{137}[/tex].
Step-by-step explanation:
The center of a circle is at the midpoint of its diameters. The midpoint of a segment between [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] is:
[tex]\displaystyle \left(\frac{x_{0} + x_{1}}{2},\, \frac{y_{0} + y_{1}}{2}\right)[/tex].
The midpoint of the diameter in this question would be:
[tex]\displaystyle \left(\frac{(-2) + 9}{2},\, \frac{4 + 8}{2}\right)[/tex].
Simplify to obtain:
[tex]\displaystyle \left(\frac{7}{2},\, 6\right)[/tex].
The radius of a circle is equal to [tex](1/2)[/tex] of the diameter. The length of a segment between [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] is:
[tex]\displaystyle \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}}[/tex].
The diameter of the circle in this question is equal to the length of the segment between [tex](-2,\, 4)[/tex] and [tex](9,\, 8)[/tex]:
[tex]\begin{aligned} & \sqrt{(9 - (-2))^{2} + (8 - 4)^{2}} \\ =\; & \sqrt{121 + 16}\\ =\; & \sqrt{137}\end{aligned}[/tex].
The radius of this circle would be:
[tex]\displaystyle \frac{1}{2}\, \sqrt{137}[/tex].
