Question

What is the center of a circle with points (4,5), (-2,11), and (-4,21).

Answer 1

Coordinates for the center, I mean.

Answer 2

Not sure if my method is correct, but I find the 2 perpendicular bisectors of the points(any 2) to find the circumcenter, which should be the center.

Answer 3

You know the equation of a circle correct?

Answer 4

Yes.

Answer 5

But just in case, (x-h)^2+(y-k)^2=r^2
Where (h,k) is the center

Answer 6

The equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\) which is an equation with 3 variables and you have 3 points so if you do this:

\(( 4 - h)^2 + (5 - k)^2 = r^2\)
\((-4 - h)^2 + (21 - k)^2 = r^2\)
\((-2 - h)^2 + (11 - k)^2 = r^2\)

Now you have a system of equations in 3 variables which you can solve

Answer 7

You can solve the system like that, but doesn't it take a lot more math than finding the perpendicular bisectors of the midpoints?
I suppose I'm just trying to justify the method I used, even if I may have done it wrong.

Answer 8

Can you draw a visual representation of your approach?