Find the circumcenter of ABC with A(1,6), B(1,4), and C (5,4)
To find the circumcenter of a triangle, we need to find the intersection point of the perpendicular bisectors of its sides.
First, let's find the midpoint of each side:
The midpoint of AB is ((1+1)/2, (6+4)/2) = (1, 5).
The midpoint of BC is ((1+5)/2, (4+4)/2) = (3, 4).
The midpoint of AC is ((1+5)/2, (6+4)/2) = (3, 5).
Next, let's find the slopes of the perpendicular bisectors of the sides AB, BC, and AC:
The slope of the perpendicular bisector of AB is the negative reciprocal of the slope of AB, which is (6-4)/(1-1) = undefined.
The slope of the perpendicular bisector of BC is the negative reciprocal of the slope of BC, which is (4-4)/(5-1) = 0.
The slope of the perpendicular bisector of AC is the negative reciprocal of the slope of AC, which is (6-4)/(1-5) = -1/2.
Finally, for each perpendicular bisector, let's use point-slope form to find its equation:
The perpendicular bisector of AB passes through the midpoint (1, 5), so its equation is y - 5 = undefined * (x - 1).
Since the slope is undefined, the equation simplifies to x = 1.
The perpendicular bisector of BC passes through the midpoint (3, 4), so its equation is y - 4 = 0 * (x - 3).
Since the slope is 0, the equation simplifies to y = 4.
The perpendicular bisector of AC passes through the midpoint (3, 5), so its equation is y - 5 = -1/2 * (x - 3).
Multiplying through the equation gives you 2y - 10 = -x + 3.
Rearranging the equation gives you x + 2y = 13.
To find the circumcenter, we need to find the intersection point of these three equations.
Solving the system of equations x = 1, y = 4, and x + 2y = 13 gives you x = 1, y = 4.
Therefore, the circumcenter of triangle ABC is at the point (1, 4).