Question

Find the circumcenter of triangle EFG with E(2, 6) F(2,4) and G)6,4

Answer 1

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 $(4,5)$

Answer 2

(4,5) you draw lines perpendicularly through each side and the point where they meet is the answer