ANSWER
The circumcenter of triangle DEF has coordinates [tex](4.5,-1)[/tex]
EXPLANATION
The circumcenter of triangle DEF is the point where all the three perpendicular bisectors of the three sides of the triangle intersect.
Let us find the perpendicular bisector of any two sides, because the third one will also meet these two perpendicular bisectors at the same point.
The perpendicular bisector of ED passes through the midpoint of the line connecting D(1,3) and E(8,3).
We must therefore find the midpoint to obtain,
[tex]N=(\frac{1+8}{2},\frac{3+3}{2})=(4.5,3)[/tex]
The slope of ED is
[tex]n=\frac{3-3}{8-1}=0[/tex].
The slope of the perpendicular bisector is the negative reciprocal of the slope of ED
[tex]slope=-\frac{1}{0}[/tex] which undefined.
The equation of a line that has an undefined slope is given by
[tex]x=x_1[/tex].
Therefore the equation of the perpendicular bisector of ED is [tex]x=4.5--(1)[/tex]
We must also find the equation of the perpendicular bisector of DF.
The midpoint of D(1,3) and F(1,-5) is [tex]M=(\frac{1+1}{2}, \frac{3+-5}{2})=(1,-1)[/tex]
The slope of DF is
[tex]m=\frac{-5-3}{1-1} =\frac{-8}{0}[/tex] this is an undefined slope but let us keep it like this for now.
The perpendicular bisector will have a slope that is the negative reciprocal of this undefined slope
[tex]slope=-\frac{1}{\frac{-8}{0}}= \frac{0}{8}=0[/tex]. This means that the slope is parallel to the x-axis.
The equation of lines that are parallel to the x-axis is given by the formula,
[tex]y=y_1[/tex].
Therefore the equation of the perpendicular bisector is [tex]y=-1--(2)[/tex].
The perpendicular bisectors will intersect at, [tex](4.5,-1)[/tex]
Hence the circumcenter is [tex](4.5,-1)[/tex]
