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
(5,3)
(3,5)
(4,4)
(4,2)
To find the circumcenter of triangle ABC, we can use the properties of perpendicular bisectors.
First, let's find the midpoint of each side of the triangle.
The midpoint of side AB is ((2+2)/2, (5+3)/2) = (2, 4).
The midpoint of side BC is ((2+6)/2, (3+3)/2) = (4, 3).
The midpoint of side AC is ((2+6)/2, (5+3)/2) = (4, 4).
Now, let's find the slope of the line passing through AB. The slope of AB is (5-3)/(2-2) = undefined (since the denominator is 0). Therefore, the perpendicular bisector of AB is a vertical line passing through the midpoint of AB, which is (2, 4).
Next, let's find the slope of the line passing through BC. The slope of BC is (3-3)/(6-2) = 0. Therefore, the perpendicular bisector of BC is a horizontal line passing through the midpoint of BC, which is (4, 3).
Since the perpendicular bisectors of AB and BC intersect at the midpoint of BC, which is (4, 3), this point is the circumcenter of triangle ABC.
The answer is (4, 3).