Please help me with this math question?

There are different ways to look at this problem.

The formula for a circle with a given center point (h, k) is
(x - h)^2 + (y -k)^2 = r^2

where r = the radius
and (x, y) is any point on the circle.

We know 3 points on the circle, and are trying to find the center point (h,k).
This gives us 3 different equations for the same circle with the same radius.

Point A (0,0) is (0 - h)^2 + (0 - k)^2 = r^2
Point B (-3, 0) is (-3 -h)^2 + (0 - k)^2 = r^2
Point C (1,2) is (1 - h)^2 + (2 - k)^2 = r^2

Take 2 of these equations and expand the terms and set them equal to each other
(since they both equal r^2)

(0 - h)^2 + (0 - k)^2 = (1 - h)^2 + (2 - k)^2
h^2 + k^2 = 1 - 2h + h^2 + 4 - 4k + k^2
0 = 1 - 2h + 4 - 4k
2h + 4k = 5
h + 2k = 5/2
h = 5/2 - 2k

Do the same with the remaining equation for B
(0 - h)^2 + (0 - k)^2 = (-3 -h)^2 + (0 - k)^2
h^2 + k^2 = 9 + 6h + h^2 + k^2
0 = 9 + 6h
-9 = 6h
-3/2 = h

Now we know what h is (the x coordinate of the center point)
so plug that into the other solution to solve for k
h = 5/2 - 2k
-3/2 = 5/2 - 2k
8/2 = 2k
4 = 2k
2 = k

so the center point (h,k) = (-3/2, 2)
Plug that into the standard equation for the circle to plot your circle and check the other points on the circle.
(x + 3/2)^2 + (y - 2)^2 = r^2

Plug h and k into the A point equation to get r^2
(-3/2)^2 + 2^2 = r^2
9/4 + 4 = r^2
25/4 = r^2

(x + 3/2)^2 + (y - 2)^2 = 25/4

The perpendicular bisector of a chord of a circle passes through the center of the circle. Chose two sets of points. Find the equations of the lines that are their perpendicular bisectors. Their intersection is the center of the circle.

How to find the perpendicular bisector of a line segment? It passes through the midpoint of the segment. Its slope is the negative reciprocal of the slope of the line segment. (If it exists.) To find the equation of the line use the point slope formula. (If it exists).

How to find the midpoint of a line segment? Average the x and y coordinates.

updated:
AB: M(-3/2,0) m=0 m'=inf -> x=-3/2
AC: M(1/2,1) m=2 m'=-1/2
(y-1)=-1/2(x-1/2)
y-1=-1/2 x + 1/4
y = -1/2 x + 5/4

y = -1/2 (-3/2) + 5/4
y = 3/4 + 5/4 = 8/4 = 2

x = -3/2, y=2
O = (-3/2,2)