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#