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.

Thanks! ;D

Answer 1

The center of the circle is equidistant from all three points.

Using the first two points we have

(x - 6)² + (y - 8)² = (x - 16)² + (y + 4)²
x² - 12x + 36 + y² - 16y + 64 = x² - 32x + 256 + y² + 8y + 16
20x - 24y = 172
5x - 6y = 43

Using the first and last points we have

(x - 6)² + (y - 8)² = (x - 10)² + (y - 8)²
x² - 12x + 36 + (y - 8)² = x² - 20x + 100 + (y - 8)²
8x = 64
x = 8

Plugging in to the first equation we have

5x - 6y = 43
5*8 - 6y = 43
40 - 6y = 43
-6y = 3
y = -1/2

The center of the circle is (8,-1/2).

The radius r is

r = √{(8 - 6)² + (8 - (-1/2))²} = √{2² + (17/2)²} = √{(16 + 289)/16}
r = √305 / 4

The equation of the circle is

(x - 8)² + (y + 1/2)² = 305/16