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 $pir^2=325/18pi$ .

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