Final answer:
The circumcenter of a triangle with vertices A(5, -8), B(7, -4), and C(5, -4) is calculated by finding the intersection of the perpendicular bisectors of the sides, resulting in the circumcenter at (6, -6).
Step-by-step explanation:
To find the circumcenter of a triangle with vertices A(5, -8), B(7, -4), and C(5, -4), you need to find the intersection point of the perpendicular bisectors of the sides of the triangle. In this case, two of the vertices, B and C, have the same y-coordinate, which means that the perpendicular bisector of side BC is a horizontal line that passes through the midpoint of A and BC. The midpoint of BC is at (6, -4), and the midpoint of AC is at (5, -6).
Since vertices A and C have the same x-coordinate, the perpendicular bisector of AC is a vertical line passing through the midpoint. Therefore, the intersection of the horizontal line through the midpoint of BC and the vertical line through the midpoint of AC will give us the circumcenter. In this case, the circumcenter is at (6, -6), which is the midpoint of A and BC.