HELP! How do I find the coordiantes of the center of this circle and describe what method I used?

Question

Circle R passed through points A (6,8) B (16, -4) and C (10, 8). Find the coordinates of the center of the circle.

Answer 1

The centre is at the point (x, y) = (8, -1/2). How?:

Points A and C both have y = 8. The centre P must lie on the perpendicular to the chord AC, therefore its x coordinate is
(6 + 10) / 2 = 8.

Let the equation for the circle be (x - 8)^2 + (y - c)^2 = r^2.

Then A ==> 4 + (8 - c)^2 = r^2 ......(A)
Also B ==> 64 + (4+ c)^2 = r^2 ......(B)

Subtracting (B) from (A), (8 - c)^2 = 60 + (4 + c)^2. The c^2 terms cancel (thank goodness), leading to 24 c = - 12 or c = - 1/2.

So the centre is at (x, y) = (8, - 1/2).

Live long and prosper.

Answer 2

In this case ,looking at the points you see that A and C have the same ordinate . So AC is paralel to ox and the center is localed on the perpendicular to AC at its midpoint x=(6+10)/2=8
the slope of AB is (12/-10= -6/5 and the midpoint is (11,2)
Take the perpendicular to AB through this `point
y-2=5/6(x-11) and intercept it with x=8
y= 2-15/6 = -3/6=-1/2.The center is (8,-1/2 )