What is the circumcenter? triangle ABC has vertices at A (2,3) B (-4,-3) and C(2,3)

Answer 1

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.

To find the circumcenter of triangle ABC with vertices A(2,3), B(-4,-3), and C(2,3), we first need to find the equations of the perpendicular bisectors of the sides AB, BC, and AC.

1) Perpendicular bisector of AB:
Midpoint of AB = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+(-4))/2, (3+(-3))/2) = (-1,0)
Slope of AB = (y₂-y₁)/(x₂-x₁) = (-3-3)/(-4-2) = (-6)/(-6) = 1
Slope of perpendicular bisector of AB = -1
Using the point-slope form, the equation of the perpendicular bisector of AB is:
y - 0 = -1(x - (-1))
y = -x -1

2) Perpendicular bisector of BC:
Midpoint of BC = ((x₁+x₂)/2, (y₁+y₂)/2) = ((-4+2)/2, (-3+3)/2) = (-1,0)
Slope of BC = (y₂-y₁)/(x₂-x₁) = (3-(-3))/(2-(-4)) = 6/6 = 1
Slope of perpendicular bisector of BC = -1
Using the point-slope form, the equation of the perpendicular bisector of BC is:
y - 0 = -1(x - (-1))
y = -x -1

3) Perpendicular bisector of AC:
Midpoint of AC = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+2)/2, (3+3)/2) = (2,3)
Slope of AC = (y₂-y₁)/(x₂-x₁) = (3-3)/(2-2) = 0/0 (Undefined slope)
The perpendicular bisector of AC is a vertical line passing through (2,3). Therefore, the equation of the perpendicular bisector of AC is:
x = 2

To find the circumcenter, we need to find the intersection point of the three perpendicular bisectors.
From the previous calculations, we have the equations:
y = -x - 1
y = -x - 1
x = 2

The first two equations are identical, so they represent the same line. The third equation represents a vertical line passing through (2,3).

The intersection point of these lines is (2,3). Thus, the circumcenter of triangle ABC with vertices A(2,3), B(-4,-3), and C(2,3) is at the point (2,3).