In order to find the circumcenter by using the given vertices of the triangle, take into account the equation of a circle with center (h,k):
(x - h)² + (y - k)² = r²
replace the given points to obtain a system of equations:
(4 , 4):
(4 - h)² + (4 - k)² = r² (1)
(4 , 2):
(4 - h)² + (2 - k)² = r² (2)
(8 , 2):
(8 - h)² + (2 - k)² = r² (3)
subtract the equation (2) to the equation (1):
(4 - h)² + (4 - k)² - (4 - h)² - (2 - k)² = r² - r² simplify like terms both sides
(4 - k)² - (2 - k)² = 0 expand the factors
16 - 8k + k² - 4 + 4k - k ² = 0 simplify like terms
-4k + 12 = 0 subtract 12 both sides
4k = 12 divide by 4 both sides
k = 12/4
k = 3
Next, to find h, subtract equation (3) to equation (2):
(8 - h)² + (2 - k)² - (4 - h)² - (2 - k)² = r² - r²
(8 - h)² - (4 - h)² = 0
64 - 16h + h² - 16 + 8h - h² = 0
48 - 8h = 0
8h = 48
h = 48/8
h = 6
Hence, the center of the circumference is (3,6)
