Final answer:
The circumcenter of triangle PQR is the center of the circle that circumscribes the triangle. To find the circumcenter, find the point where the perpendicular bisectors of the triangle's sides intersect.
Step-by-step explanation:
The circumcenter of triangle PQR is the center of the circle that circumscribes the triangle. To find the circumcenter, we need to find the point where the perpendicular bisectors of the triangle's sides intersect.
1. Find the midpoint of each side of the triangle:
PQ midpoint = (P.x + Q.x) / 2 , (P.y + Q.y) / 2
QR midpoint = (Q.x + R.x) / 2 , (Q.y + R.y) / 2
RP midpoint = (R.x + P.x) / 2 , (R.y + P.y) / 2
2. Find the slopes of the lines perpendicular to each side:
PQ slope = -1 / ((Q.y - P.y) / (Q.x - P.x))
QR slope = -1 / ((R.y - Q.y) / (R.x - Q.x))
RP slope = -1 / ((P.y - R.y) / (P.x - R.x))
3. Use the midpoint and slope to find the equations of the perpendicular bisectors:
PQ bisector: y - PQ midpoint.y = PQ slope * (x - PQ midpoint.x)
QR bisector: y - QR midpoint.y = QR slope * (x - QR midpoint.x)
RP bisector: y - RP midpoint.y = RP slope * (x - RP midpoint.x)
4. Solve the system of equations formed by the three bisectors to find the point of intersection, which is the circumcenter of triangle PQR.