Answer:
The circumcenter is the point \((4,5)\)
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
\(E(2,6), F(2,4),G(6,4)\)
Step 1
Find the slope of the side EF
we know that
The formula to calculate the slope between two points is equal to
\(m=\frac{y2-y1}{x2-x1}\)
we have
\(E(2,6), F(2,4)\)
Substitute the values
\(m=\frac{4-6}{2-2}\)
\(m=\frac{-2}{0}\) -------> 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
\(y=\frac{y1+y2}{2}\)
we have
\(E(2,6), F(2,4)\)
substitute the values
\(y=\frac{6+4}{2}=5\)
therefore
the equation of the perpendicular bisector to the side EF is equal to
\(y=5\) -------> 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
\(m=\frac{y2-y1}{x2-x1}\)
we have
\(F(2,4),G(6,4)\)
Substitute the values
\(m=\frac{4-4}{6-2}\)
\(m=\frac{0}{4}=0\)
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
\(x=\frac{x1+x2}{2}\)
we have
\(F(2,4),G(6,4)\)
substitute the values
\(x=\frac{2+6}{2}=4\)
therefore
the equation of the perpendicular bisector to the side FG is equal to
\(x=4\) -------> 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
\(y=5\) -------> equation A
\(x=4\) -------> equation B
The solution is the point \((4,5)\)
