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)

Question

Pick one

Answer 1

The circumcenter of triangle ABC can be found by finding the intersection point of the perpendicular bisectors of the sides AB, BC, and AC.

First, let's find the equation of the perpendicular bisector of side AB:
The midpoint of AB is ((2+2)/2, (5+3)/2) = (2, 4).
The slope of AB is (3-5)/(2-2) = undefined.
Thus, the slope of the perpendicular bisector of AB is 0.
Using the point-slope form, the equation of the perpendicular bisector of AB is y - 4 = 0.

Next, let's find the equation of the perpendicular bisector of side BC:
The midpoint of BC is ((2+6)/2, (3+3)/2) = (4, 3).
The slope of BC is (3-3)/(6-2) = 0.
Since the slope of BC is 0, the slope of the perpendicular bisector is undefined (perpendicular to the slope of BC).
Thus, the equation of the perpendicular bisector of BC is x - 4 = 0.

Now, let's find the equation of the perpendicular bisector of side AC:
The midpoint of AC is ((2+6)/2, (5+3)/2) = (4, 4).
The slope of AC is (3-5)/(6-2) = -1/2.
Thus, the slope of the perpendicular bisector of AC is 2 (negative reciprocal of -1/2).
Using the point-slope form, the equation of the perpendicular bisector of AC is y - 4 = 2(x - 4), which simplifies to y - 4 = 2x - 8.

To find the circumcenter, we need to find the intersection point of the three perpendicular bisectors:
Solving the system of equations formed by the three perpendicular bisectors:
1) y - 4 = 0
2) x - 4 = 0
3) y - 4 = 2x - 8

From equations 1) and 2), we get (x, y) = (4, 4).
Substituting x = 4 into equation 3), we get y - 4 = 2(4) - 8, which simplifies to y = 4.

Therefore, the circumcenter of triangle ABC is located at (4, 4).

The correct response is (4, 4).