A triangle has corners at $(2 ,5 )$ , $(3 ,1 )$ , and $(4 ,2 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-07-10T15:51:58

$color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"$

![http://mathibayon.blogspot.com/2015/01/ derivation-of-formula-for-radius-of-circumcircle.html

$"Area of Triangle " = A_T = (a b c) / (4 R)$

$A (2,5), B (3,1), C(4,2)$

$a = sqrt((2-3)^2 + (5-1)^2) = sqrt 17 = 4.1231$

$b = sqrt((3-4)^2 + (1-2)^2) = sqrt 2 = 1.4142$

$c = sqrt((4-2)^2 + (2-5)^2) = sqrt 13 = 3.6056$

$"Semi perimeter of the triangle " s = (a + b + c ) / 2 = 4.5715$

$A_T = sqrt (s (s-a) (s-b) (s - c))$

$A_T = sqrt(4.5715 (4.5715 - 4.1231) (4.5715 - 1.4142) (4.5715 - 3.6056)) = 2.5$

$R = (a b c ) / (4 * A_T) = (sqrt 17 * sqrt 2* sqrt 13) / (4 * 2.5) = 2.1024$

$color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"$

A triangle has corners at (2 ,5 )(2 ,5 ), (3 ,1 )(3 ,1 ), and (4 ,2 )(4 ,2 ). What is the area of the triangle's circumscribed circle?