Find the radius of the circle and write the standard form of the circle's equation

Given two points on a circle which are connected by a line segment through the center of the circle, we know that this line segment is the diameter of the circle. Since we have the coordinates of the two points, we will use the midpoint formula [tex]M = ( \dfrac{x_1+x_2}{2}), ( \dfrac{y_1+y_2}{2}) = ( \frac{12}{2}), ( \frac{14}{2}) = (6,7)[/tex]. 

The center [tex](h, k)[/tex] of the circle is the point (6, 7). We can use the distance formula for the center and one of the given points to find the radius. [tex]d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} = \sqrt{(6-2)^2+(7-5)^2} = \sqrt{16+4} \\ d= \sqrt{20} = 2 \sqrt{5} [/tex].

The standard formula for a circle with a center  [tex](h,k)[/tex] and radius [tex]r[/tex] is [tex](x-h)^2+(y-k)^2=r^2[/tex].

Substituting our known values, we get [tex](x-6)^2+(y-7)^2=20[/tex].