A triangle has corners at $(9 ,4 )$ , $(7 ,1 )$ , and $(3 ,9 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-09-05T06:56:15

Area of circumscribed circle is $63.5496$

If the sides of a triangle are $a$ , $b$ and $c$ , then the area of the triangle $Delta$ is given by the formula

$Delta=sqrt(s(s-a)(s-b)(s-c))$ , where $s=1/2(a+b+c)$

and radius of circumscribed circle is $(abc)/(4Delta)$

Hence let us find the sides of triangle formed by $(9,4)$ , $(7,1)$ and $(3,9)$ . This will be surely distance between pair of points, which is

$a=sqrt((7-9)^2+(1-4)^2)=sqrt(4+9)=sqrt13=3.6056$

$b=sqrt((3-7)^2+(9-1)^2)=sqrt(16+64)=sqrt80=8.9443$ and

$c=sqrt((3-9)^2+(9-4)^2)=sqrt(36+25)=sqrt61=7.8102$

Hence $s=1/2(3.6056+8.9443+7.8102)=1/2xx20.3601=10.1801$

and $Delta=sqrt(10.1801xx(10.1801-3.6056)xx(10.1801-8.9443)xx(10.1801-7.8102)$

= $sqrt(10.1801xx6.5745xx1.2358xx2.3699)=sqrt16.01=14.0006$

And radius of circumscribed circle is

$(3.6056xx8.9443xx7.8102)/(4xx14.0006)=4.4976$

And area of circumscribed circle is $3.1416xx(4.4976)^2=63.5496$

A triangle has corners at (9 ,4 )(9 ,4 ), (7 ,1 )(7 ,1 ), and (3 ,9 )(3 ,9 ). What is the area of the triangle's circumscribed circle?