how do you find the circumcenter of A(-2,-2), B(-1,3) & C(3,2)?

Answer 1

The circumcenter is the point which is equidistant from the three given points. You can do this by finding the perpendicular bisectors of 2 pairs of the points and finding their point of intersection.
(-1, 3) & (3,2):
mdpt: ((-1 + 3)/2, (3 + 2)/2) = (1, 5/2)
slope: (y2 - y1)/(x2 - x1) = (2 - 3)/(3 - (-1)) = -1/5
So the slope of the perpendicular bisector is +5
Use the point-slope form:
m = (y - y1)/(x - x1)
5 = (y - 5/2)/(x - 1)
5x - 5 = y - 5/2
(1.) y = 5x - 5/2

Do the same for (-2, -2) & (-1, 3):
mdpt: ((-2 -1)/2, (-2 +3)/2) = (-3/2, 1/2)
slope: (y2 - y1)/(x2 - x1) = (3 - (-2))/(-1 - (-2)) = 5/1 = 5
So the slope of the perpendicular bisector is -1/5
Use piont-slope form:
m = (y - y1)/(x - x1)
-1/5 = (y - 1/2)/(x - (-3/2))
-1/5(x + 3/2) = y - 1/2
(-1/5)x - 3/10 = y - 1/2
(-1/5)x + 2/10 = y
(2.) y = (-1/5)x + 1/5

Find the intersection of (1.) & (2.):
5x = 5/2 = (-1/5)x + 1/5
50x - 25 = -2x + 2
52x = 27
x = 27/52
y = 5(27/52) - 5/2
y = 135/52 - 5/2
y = 5/52
So the circumcenter is (27/52, 5/52)

Answer 2

Given : A (-2, -2) , B (-1, 3) , C (3, 2)

The circumcentre lies at the junction of the three
perpendicular side bisectors. It is only necessary
to find the equation to two of them.

Let O be the circumcentre point.
Let D be the midpoint of AB.
Let E be the midpoint of BC.

Find point D, which is [(-2 + -1)/2, (-2 + 3)/2] = (-3/2, 1/2).
Find point E, which is [(-1 + 3)/2, (3 + 2)/2] = (1, 5/2)

Find slope of AB, which is (-2 - 3)/(-2 - -1) = -5/-1 = 5.
DO is perpendicular to AB, so slope of DO = -1/5.
Now that we have point D and the slope of DO, the
equation to DO can be found.
Let the equation be : y = mx + b, so, 1/2 = (-1/5)(-3/2) + b
Therefore, b = 1/5.
Thus, the equation to DO is : y = (-1/5)x + 1/5

Find slope of BC, which is (3 - 2)/(-1 - 3) = -1/4.
EO is perpendicular to BC, so slope of EO = 4.
We have point E and slope of EO, so let the equation
to EO be y = mx + b, so, 5/2 = 4(1) + b.
Therefore, b = -3/2.
Thus, the equation to EO is : y = 4x - 3/2.

Now find the intersection of DO and EO.
(-1/5)x + 1/5 = 4x - 3/2
From this, x = 17/42.
Substituting this into, say, EO, gives :
y = 4(17/42) - 3/2, or, y = 5/42.

Thus, circumcentre is at (17/42, 5/42).