Find the circumcenter of ΔABC
with A(1, 6), B(1, 4), and C(5, 4).(1 point)
Responses
(3, 5)
(3, 5)
(1, 6)
(1, 6)
(1, 4)
(1, 4)
(5, 4)
To find the circumcenter of a triangle, we can find the intersection of the perpendicular bisectors of the triangle's sides.
The midpoint of segment AB is:
((1 + 1) / 2, (6 + 4) / 2) = (1, 5)
The slope of segment AB is: (4 - 6) / (1 - 1) = undefined
The equation of the perpendicular bisector of segment AB passing through (1, 5) is x = 1.
The midpoint of segment BC is:
((1 + 5) / 2, (4 + 4) / 2) = (3, 4)
The slope of segment BC is: (4 - 4) / (5 - 1) = 0
The equation of the perpendicular bisector of segment BC passing through (3, 4) is y = 4.
The intersection of the perpendicular bisectors is the circumcenter of the triangle. In this case, the point (3, 4) satisfies both equations and is therefore the circumcenter of triangle ABC.
So, the correct response is (3, 4).