The coordinates of the circumcenter of the triangle are approximately **(3.17, 3.67)**.
To find the circumcenter of the triangle with vertices A(2, 2), B(2, 4), and C(8, 4), we can follow these steps:
**1. Find the midpoints of the sides:**
* Midpoint of side AB: M1 = ((2 + 2)/2, (2 + 4)/2) = (2, 3)
* Midpoint of side BC: M2 = ((8 + 2)/2, (4 + 4)/2) = (5, 4)
**2. Find the slopes of the perpendicular bisectors:**
* Slope of perpendicular bisector of AB: m1 = -1/((4 - 2)/(2 - 2)) = -1/2
* Slope of perpendicular bisector of BC: m2 = (4 - 2)/(8 - 5) = 1/3
**3. Convert slopes to equations of the perpendicular bisectors:**
* Equation of perpendicular bisector of AB: y - 3 = -1/2 (x - 2)
* Equation of perpendicular bisector of BC: y - 4 = 1/3 (x - 5)
**4. Solve the system of equations:**
Solve the system of equations formed by the equations of the perpendicular bisectors to find the x and y coordinates of the point where they intersect. This point will be the circumcenter.
By solving the system, we find:
* x ≈ 3.17
* y ≈ 3.67
Therefore, the coordinates of the circumcenter of the triangle are approximately **(3.17, 3.67)**.