Final answer:
The circumcenter of right triangle XYZ, with vertices X(2,5), Y(8,5), and Z(8,1), is at the midpoint of the hypotenuse XZ, which is point (5,3).
Step-by-step explanation:
The student has asked how to find the coordinates of the circumcenter of triangle XYZ with vertices X(2,5), Y(8,5), and Z(8,1).
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle meet. In a right-angled triangle, such as XYZ where angle Y is a right angle (since sides XY and YZ are perpendicular), the circumcenter is located at the midpoint of the hypotenuse.
Here, the hypotenuse XZ has endpoints X(2,5) and Z(8,1). The midpoint of XZ can be found by averaging the x-coordinates (2+8)/2 and the y-coordinates (5+1)/2, which yields the coordinates (5,3). Therefore, the coordinates of the circumcenter are (5,3).