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

1 Answer
Feb 23, 2017

#12.5pi~~39.3#

Explanation:

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Given #A(5,1), B(2,7) and C(7,2)#

By the distance formula, we get :
#AB^2=(2-5)^2+(7-1)^2=45#
#AC^2=(7-5)^2+(2-1)^2=5#
#BC^2=(7-2)^2+(2-7)^2=50#
#=> BC^2=AB^2+AC^2#
This means that #DeltaABC# is a right triangle, with #BC# as the hypotenuse.
In a right triangle, the hypotenuse is the circum-diameter,
#=># the circumradius #r# of #DeltaABC =(BC)/2#

Therefore, the area of the circumscribed circle of #DeltaABC#
#=pir^2=pi(BC)^2/4=pixx50/4=12.5pi#