Question

Find the circumference of the triangle whose varieties are (1,3) (0,2) (-3,1)

Answer 1

Let the O = (a, b) is the co-ordinate of circumcircle of given triangle.

Let A = (-3,1) , B = (0,-2) and C = (1,3)

then, AO = BO = CO = radius of circumcircle.

so, AO² = BO² = CO²

use distance formula,

(a + 3)² + (b - 1)² = a² + (b + 2)² = (a - 1)² + (b - 3)²

now, (a + 3)² + (b - 1)² = a² + (b + 2)²

a² + 6a + 9 + b² - 2b + 1 = a² + b² + 4b + 4

6a - 6b + 6 = 0

a - b + 1 = 0 ------(1)

Similarly, a² + (b + 2)² = (a - 1)² + (b - 3)²

a² + b² + 4b + 4 = a² - 2a + 1 + b² - 6b + 9

2a + 10b - 6 = 0

a + 5b - 3 = 0 ------(2)

from equations (1) and (2),

5a + a +5 - 3 = 0

6a + 2 = 0 , a = -1/3

and b = -1/3 + 1 = 2/3

Hence, centre of circumcircle = (-1/3, 2/3)