A triangle has corners at $(5 ,7 )$ , $(2 ,1 )$ , and $(3 ,4 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-03-01T06:33:56

Area of the circumcircle

$color(green)(A_c = pi R^2 = pi * 12.7032^2 = 506.9629$ sq units

$A (5,7), B (2,1), C(3,4)$ . To find the radius of the Incircle.

Using distance formula

$a = sqrt((2-3)^2 + (1-4)^2) ~~ 3.16$

$b = sqrt((5-3)^2 + (7-4)^2) ~~ 3.6$

$c = sqrt ((5-2)^2 + (7-1)^2) = 6.7$

Semi perimeter of the triangle ABC

$s = (a + b + c) / 2 = (3.16 + 3.6 + 6.7) / 2 = 6.73$

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Formula for Area of triangle, knowing three sides is

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

$=> sqrt((6.73 * (6.73 - 3.16) * (6.73 - 3.6) * (6.73 - 6.7)) ~~ 1.5$

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Radius of the circum circle

$R = (abc) / (4 A_t) = (3.16 * 3.6 * 6.7) / (4 * 1.5) = color(green)(12.7032$

Area of circumcircle

$A_c = pi R^2 = pi * (12.7032)^2 ~~ 506.9629$

A triangle has corners at (5 ,7 )(5 ,7 ), (2 ,1 )(2 ,1 ), and (3 ,4 )(3 ,4 ). What is the area of the triangle's circumscribed circle?