A circle passes through the points (-2,0), (2,0), and (3,2). Find the center of the circle. Enter your answer as an ordered pair.
The center is the point where the perpendicular bisectors of two chords intersect.
Plan: to find the equations of the two perpendicular bisectors and then find what point they have in common.
The first chord has endpoints (-2,0) and (2,0). This chord is horizontal and has a midpoint of (0,0).
The perpendicular bisector will be the y-axis and the y-axis has an equation of x = 0.
The second chord has endpoints (2,0) and (3,2). This chord has its midpoint at (2.5,1).
The slope of this chord is: (2 - 0)/(3 -2 ) = 2. Therefore, the slope of the perpendicular bisector is -1/2.
The equation of the perpendicular bisector is: y - 1 = -0.5(x - 2.5)
Simplifying this equation gives: 2x + 4y = 9.
The point where x = 0 and 2x + 3y = 9 intersect is: (0, 2.25)