Find the circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3).
(4,4)
(4,2)
(5,3)
(3,5)
To find the circumcenter of triangle ABC, we need to find the midpoint of each side of the triangle and then find the intersection point of the perpendicular bisectors of these sides.
1) Midpoint of AB:
The x-coordinate of the midpoint of AB is (2+2)/2 = 2 and the y-coordinate is (5+3)/2 = 4. So, the midpoint of AB is (2,4).
2) Midpoint of BC:
The x-coordinate of the midpoint of BC is (2+6)/2 = 4 and the y-coordinate is (3+3)/2 = 3. So, the midpoint of BC is (4,3).
3) Midpoint of AC:
The x-coordinate of the midpoint of AC is (2+6)/2 = 4 and the y-coordinate is (5+3)/2 = 4. So, the midpoint of AC is (4,4).
Next, we need to find the slopes of the perpendicular bisectors of each side of the triangle.
1) Slope of the perpendicular bisector of AB:
The slope of the line AB is (3-5)/(2-2) = undefined (vertical line). So, the slope of the perpendicular bisector of AB is 0.
2) Slope of the perpendicular bisector of BC:
The slope of the line BC is (3-3)/(6-2) = 0 (horizontal line). So, the slope of the perpendicular bisector of BC is undefined.
3) Slope of the perpendicular bisector of AC:
The slope of the line AC is (3-5)/(6-2) = -1/2. So, the slope of the perpendicular bisector of AC is 2.
Now, we can find the equations of the perpendicular bisectors and then find their intersection point:
1) Perpendicular bisector of AB:
Since the midpoint of AB is (2,4), the equation of the perpendicular bisector passing through (2,4) is: y-4 = 0*(x-2) -> y = 4.
2) Perpendicular bisector of BC:
Since the midpoint of BC is (4,3), the equation of the perpendicular bisector passing through (4,3) is: x-4 = undefined*(y-3) -> x = 4.
3) Perpendicular bisector of AC:
Since the midpoint of AC is (4,4) and the slope is 2, the equation of the perpendicular bisector passing through (4,4) is: y-4 = 2*(x-4) -> y = 2x-4.
Now, we can find the intersection point of these perpendicular bisectors by solving the equations:
y = 4 (equation of perpendicular bisector of AB)
x = 4 (equation of perpendicular bisector of BC)
y = 2x-4 (equation of perpendicular bisector of AC)
From the equations, we can see that the intersection point is (4,4).
Therefore, the circumcenter of triangle ABC is (4,4).
So, the correct answer is (4,4).