Final answer:
To find the coordinates of the circumcenter, we need to find the intersection point of the perpendicular bisectors of the triangle's sides. The midpoint of XY is (5,5) and the midpoint of YZ is (8,3). The intersection point of the perpendicular bisectors is (8,5).
Step-by-step explanation:
To find the coordinates of the circumcenter, we need to find the intersection point of the perpendicular bisectors of the triangle's sides. The perpendicular bisector of a side passes through the midpoint of that side and is perpendicular to it.
Let's find the midpoint of side XY first. The x-coordinate of the midpoint is (2+8)/2 = 5, and the y-coordinate is (5+5)/2 = 5. So the midpoint of XY is M(5,5).
The slope of XY is (5-5)/(8-2) = 0. The slope of the perpendicular bisector is the negative reciprocal of the slope of XY, which is undefined. This means that the perpendicular bisector of XY is a vertical line passing through the midpoint M(5,5).
Now, let's find the midpoint of side YZ. The x-coordinate of the midpoint is (8+8)/2 = 8, and the y-coordinate is (5+1)/2 = 3. So the midpoint of YZ is N(8,3).
The slope of YZ is (1-5)/(8-8) = undefined. The slope of the perpendicular bisector is 0. This means that the perpendicular bisector of YZ is a horizontal line passing through the midpoint N(8,3).
The circumcenter of the triangle is the intersection point of the perpendicular bisectros. Since one of the bisectors is vertical and the other is horizontal, their intersection point is (8,5).