A triangle has corners at #(5 ,7 )#, #(3 ,9 )#, and #(4 ,6 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-03-02T13:02:32

Area of the circumscribed circle is

#A_c = pi * R^2 ~~ 7.84# sq units

Steps :
1. Find the lengths of the three sides using distance formula
#d = sqrt((x2 - x1)^2 + (y2 - y1)^2)#

Find the area of the triangle using formula
#A_t = sqrt(s (s - a) (s - b) ( s - c))#

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#a = sqrt((3-4)^2 + (9-6)^2) ~~3.16#

#b = sqrt((5-4)^2 + (7-6)^2) ~~ 1.414#

#c = sqrt((5-3)^2 + (7-9)^2) ~~ 2.83#

#s = (a+b+c)/2 = (3.16+1.414+2.83)/2 ~~ 3.7#

#A_t = sqrt((3.7 (3.7-3.16) (3.7-1.414) (3.7-2.83)) ~~ 2#

Radius of the circumscribed circle is

#R = (abc) / (4 A_t) = (3.16*1.414.2.83)/ (4 * 2) ~~ 1.58#

Area of the circumscribed circle is

#A_c = pi * R^2 = pi * 1.58^2 ~~ 7.84# sq units