Find circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3).
To find the circumcenter of triangle ABC, we need to find the intersection point of the three perpendicular bisectors of the sides.
1. First, let's find the midpoint of each side:
Midpoint of AB = ((2 + 2)/2, (5 + 3)/2) = (2, 4)
Midpoint of BC = ((2 + 6)/2, (3 + 3)/2) = (4, 3)
Midpoint of AC = ((2 + 6)/2, (5 + 3)/2) = (4, 4)
2. Next, let's find the slopes of the lines passing through each side:
Slope of AB = (5 - 3)/(2 - 2) = undefined (vertical line)
Slope of BC = (3 - 3)/(6 - 2) = 0 (horizontal line)
Slope of AC = (5 - 3)/(2 - 6) = 1/(-2) = -1/2
3. Then, let's find the equations of the perpendicular bisectors:
Equation of the perpendicular bisector of AB passing through (2, 4):
Since this line is vertical, its equation can be written as x = 2.
Equation of the perpendicular bisector of BC passing through (4, 3):
Since this line is horizontal, its equation can be written as y = 3.
Equation of the perpendicular bisector of AC passing through (4, 4):
Since we know the midpoint (4, 4) and the slope (-1/2), we can use the point-slope form:
y - 4 = (-1/2)(x - 4)
2(y - 4) = - (x - 4)
2y - 8 = -x + 4
x + 2y = 12
4. Finally, we find the intersection point of the three perpendicular bisectors:
Solving the system of equations:
x = 2 (from the equation of the bisector of AB)
y = 3 (from the equation of the bisector of BC)
x + 2y = 12 (from the equation of the bisector of AC)
We can see that the intersection point is (2, 3).
Therefore, the circumcenter of triangle ABC is located at (2, 3).