Find the coordinates of the circumcenter for ∆def with coordinates d(1,1) e (7,1) and f(1,5). show your work

Draw an accurate picture of the points in the coordinate axis, as shown in the figure.

D, F have equal x coordinate, so DF is perpendicular to the x-axis.
D, E have equal x coordinate, so DE is perpendicular to the y-axis.

this means DF and DE are perpendicular, so DEF is a right triangle, with m(D)=90°.

Let A be the midpoint of DF. The x coordinate of A is 1 and the y coordinate can be found as follows:

subtract the y coordinate of D from the y coordinate of F, that is 5-1=4.
Divide by 2 and add to the y coordinate of D, that is 4/2+1=2+1=3 is the y coordinate of A.

Let B be the midpoint of DE, follow the previous procedure to find the coordinates of B.

A(1, 3) and B(4, 1).

draw the perpendicular through A and the perpendicular through B, thet meet at point C(4, 3), which is the circumcenter.

C(4, 3)