Answer:
The circumcenter is the point [tex](4,5)[/tex]
Step-by-step explanation:
we know that
The circumcenter is the point where the perpendicular bisectors of a triangle intersect
so
In this problem we have
The coordinates of triangle EFG are
[tex]E(2,6), F(2,4),G(6,4)[/tex]
Step 1
Find the slope of the side EF
we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
[tex]E(2,6), F(2,4)[/tex]
Substitute the values
[tex]m=\frac{4-6}{2-2}[/tex]
[tex]m=\frac{-2}{0}[/tex] -------> the slope is undefined
The side EF is parallel to the y-axis
therefore
The segment perpendicular to the side EF will be parallel to the x-axis
and the equation of the perpendicular bisector to the side EF is equal to the y-coordinate of the midpoint EF
Step 2
Find the y-coordinate of the Midpoint EF
The formula to calculate the y-coordinate of the midpoint between two points is equal to
[tex]y=\frac{y1+y2}{2}[/tex]
we have
[tex]E(2,6), F(2,4)[/tex]
substitute the values
[tex]y=\frac{6+4}{2}=5[/tex]
therefore
the equation of the perpendicular bisector to the side EF is equal to
[tex]y=5[/tex] -------> equation A
Step 3
Find the slope of the side FG
we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
[tex]F(2,4),G(6,4)[/tex]
Substitute the values
[tex]m=\frac{4-4}{6-2}[/tex]
[tex]m=\frac{0}{4}=0[/tex]
The side FG is parallel to the x-axis
therefore
The segment perpendicular to the side FG will be parallel to the y-axis
and the equation of the perpendicular bisector to the side FG is equal to the x-coordinate of the midpoint FG
Step 4
Find the x-coordinate of the Midpoint FG
The formula to calculate the x-coordinate of the midpoint between two points is equal to
[tex]x=\frac{x1+x2}{2}[/tex]
we have
[tex]F(2,4),G(6,4)[/tex]
substitute the values
[tex]x=\frac{2+6}{2}=4[/tex]
therefore
the equation of the perpendicular bisector to the side FG is equal to
[tex]x=4[/tex] -------> equation B
Step 5
Find the intersection point equation A and equation B
we know that
the intersection point of the perpendicular bisector to the side EF and the perpendicular bisector to the side FG is called the circumcenter
[tex]y=5[/tex] -------> equation A
[tex]x=4[/tex] -------> equation B
The solution is the point [tex](4,5)[/tex]
