What are the coordinates of the circumcenter of triangle ABC with vertices A (1,4), B (1,2), and C (6,2)
To find the circumcenter of triangle ABC, we need to find the intersection of the perpendicular bisectors of any two sides of the triangle.
First, let's find the midpoint of each side of the triangle:
Midpoint of AB:
x-coordinate: (1 + 1) / 2 = 1
y-coordinate: (4 + 2) / 2 = 3
Midpoint of BC:
x-coordinate: (1 + 6) / 2 = 7/2 = 3.5
y-coordinate: (2 + 2) / 2 = 4 / 2 = 2
Now, let's find the slope of each side of the triangle:
Slope of AB:
(2 - 4) / (1 - 1) = -2 / 0 (undefined)
Slope of BC:
(2 - 2) / (6 - 1) = 0 / 5 = 0
Since the slope of AB is undefined, the perpendicular bisector of AB is a vertical line passing through the midpoint of AB at x = 1.
Since the slope of BC is 0, the perpendicular bisector of BC is a horizontal line passing through the midpoint of BC at y = 2.
The intersection of the lines x = 1 and y = 2 is the point (1, 2), which is also the circumcenter of triangle ABC.
Therefore, the coordinates of the circumcenter of triangle ABC are (1, 2).