The coordinates of the circumcenter are (0, -3/2).
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It's also the center of the circle that passes through all three vertices of the triangle.
For side TU:
The midpoint of TU is M = ((-6 + 0)/2, (-5 - 1)/2) = (-3, -3).
The slope of TU is -4/6 = -2/3, so the perpendicular bisector has a slope of 3/2.
Using the point-slope form of the equation and point M, we can find the perpendicular bisector equation for TU.
For side UV:
The midpoint of UV is N = ((0 + 0)/2, (-1 - 5)/2) = (0, -3).
UV is a horizontal line, so its perpendicular bisector is a vertical line passing through N.
Solve the system of equations formed by the perpendicular bisector equations for TU and UV. This will give you the coordinates of the circumcenter, O.
Following the steps above, the perpendicular bisector equations and their intersection point are:
Perpendicular bisector of TU: y = (3/2)x - 3/2
Perpendicular bisector of UV: x = 0
Intersection point (circumcenter O): O = (0, -3/2)
Therefore, the circumcenter of the triangle with vertices T, U, and V is O(0, -3/2).