Find the circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3).(1 point) Responses (4,2) (4,2) (3,5) (3,5) (4,4) (4,4) (5,3)
The circumcenter of triangle ABC can be found by finding the intersection point of the perpendicular bisectors of each side of the triangle.
To find the equation of the perpendicular bisector of side AB, we first find the midpoint of AB:
Midpoint of AB = ((2+2)/2, (5+3)/2) = (2, 4)
The slope of AB can be found as:
m_AB = (5-3)/(2-2) = 2/0 (undefined slope)
The perpendicular slope to AB is negative reciprocal of its slope, so the slope of the perpendicular bisector is 0.
Therefore, the equation of the perpendicular bisector of AB passing through the midpoint is x = 2.
To find the equation of the perpendicular bisector of side BC, we first find the midpoint of BC:
Midpoint of BC = ((2+6)/2, (3+3)/2) = (4, 3)
The slope of BC can be found as:
m_BC = (3-3)/(6-2) = 0/4 = 0
The perpendicular slope to BC is undefined.
Therefore, the equation of the perpendicular bisector of BC passing through the midpoint is y = 3.
The point of intersection of the perpendicular bisectors can be found by solving the system of equations:
x = 2
y = 3
The solution is (2, 3).
Therefore, the circumcenter of triangle ABC is at (2, 3).
The correct response is (2, 3).