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)
O (3,5)
○ (5,3)
○ (4,2)
○ (4,4)
To find the circumcenter of triangle ABC, we need to find the intersection of the perpendicular bisectors of the sides of the triangle.
First, let's find the equation of the perpendicular bisector of side AB. The midpoint of AB can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of AB: ((2+2)/2, (5+3)/2) = (2, 4)
The slope of the line AB is (3-5)/(2-2) = undefined (since the x-coordinates are the same)
Therefore, the slope of the perpendicular bisector of AB is 0.
Using the point-slope form of a line, 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 can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint of BC: ((2+6)/2, (3+3)/2) = (4, 3)
The slope of the line BC is (3-3)/(6-2) = 0
Therefore, the slope of the perpendicular bisector of BC is undefined (since the y-coordinates are the same).
Using the point-slope form of a line, the equation of the perpendicular bisector of BC is x - 4 = 0.
Now, let's find the intersection of the equations of these two perpendicular bisectors.
Since the equation of the perpendicular bisector of AB is y - 4 = 0, we can solve for y:
y = 4
Since the equation of the perpendicular bisector of BC is x - 4 = 0, we can solve for x:
x = 4
Therefore, the intersection of these two perpendicular bisectors is (4, 4).
The circumcenter of triangle ABC is at (4, 4).
Therefore, the correct answer is option ○ (4, 4).