Find the center of the circle that you can circumscribe about a triangle with vertices e(4, 4), f(4, 2), and g(8, 2)

The perpendicular bisectors of the 3 sides will meet at the centre of the circle.

midpoint of EF will be (4+4)2 , (4+2)/2  = (4,3)

equation of EF is  x = 4 so the equation of the line perpendicular to EF will be 
y = 3

midpoint of FG is (6,2)  so line perpendicualr to it is x = 6

so center of the circle is ( 6 , 3)