A triangle has corners at #(5 ,7 )#, #(2 ,1 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-01-09T03:56:26

The area of the circumscribed circle is #325/18pi#.

To find the area of the triangle's circumscribed circle, we need to find its radius.

[Step1] Find the equation of the circle.
The equation of a circle is the form #x^2+y^2+ax+by+c=0#.
Substitute the coordinate of the three vertices.

#5^2+7^2+5a+7b+c=0#
⇒ #5a+7b+c=-74# [1]

#2^2+1^2+2a+b+c=0#
⇒ #2a+b+c=-5# [2]

#3^2+6^2+3a+6b+c=0#
⇒ #3a+6b+c=-45# [3]

The solution for [1],[2] and [3] is
#(a,b,c)=(-35/3, -17/3, 24)#.

Then, the equation of the circle is
#x^2+y^2-35/3x-17/3y+24=0#
#(x-35/6)^2+(y-17/6)^2=-24+(35/6)^2+(17/6)^2#
#(x-35/6)^2+(y-17/6)^2=325/18#.

[Step2] Find the area of the circle.
The equation tells us that the radius of the circle is #r=sqrt(325/18)#, and its area is #pir^2=325/18pi#.