Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the circumcenter
To find the coordinates of the circumcenter, we can use the fact that the circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle.
First, let's find the equations of the perpendicular bisectors of the sides.
The midpoint of XY is ((2+8)/2, (5+5)/2) = (5, 5). The slope of the line XY is (5-5)/(8-2) = 0.
The perpendicular bisector of XY has a slope of -1/0, which is undefined. Therefore, the equation of the perpendicular bisector of XY is x = 5.
The midpoint of XZ is ((2+8)/2, (5+1)/2) = (5, 3). The slope of the line XZ is (3-5)/(8-2) = -1/3.
The perpendicular bisector of XZ has a slope of 3/1. Using the point-slope form of a line, the equation of the perpendicular bisector of XZ is y - 3 = (3/1)(x - 5) => y = 3x - 12.
The midpoint of YZ is ((8+8)/2, (5+1)/2) = (8, 3). The slope of the line YZ is (3-5)/(8-8) = undefined.
The perpendicular bisector of YZ has a slope of 0. Therefore, the equation of the perpendicular bisector of YZ is y = 3.
To find the coordinates of the circumcenter, we need to find the intersection point of the perpendicular bisectors (x = 5 and y = 3x - 12).
Setting x = 5 in the equation y = 3x - 12, we get y = 3(5) - 12 = 3.
Therefore, the coordinates of the circumcenter are (5, 3).