A triangle has corners at $(6 ,8 )$ , $(1 ,2 )$ , and $(3 ,9 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-05-27T08:44:40

Area of the triangle's circumscribed circle is $48.005$ .

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 $(6,8)$ , $(1,2)$ and $(3,9)$ . This will be surely distance between pair of points, which is

$a=sqrt((1-6)^2+(2-8)^2)=sqrt(25+36)=sqrt61=7.810$

$b=sqrt((3-1)^2+(9-2)^2)=sqrt(4+49)=sqrt53=7.280$ and

$c=sqrt((3-6)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.162$

Hence $s=1/2(7.810+7.280+3.162)=1/2xx18.252=9.126$

and $Delta=sqrt(9.126xx(9.126-7.810)xx(9.126-7.280)xx(9.126-3.162)$

= $sqrt(9.126xx1.316xx1.846xx5.964)=sqrt132.2226=11.499$

And radius of circumscribed circle is

$(7.810xx7.280xx3.162)/(4xx11.499)=3.909$

And area of circumscribed circle is $3.1416xx(3.909)^2=48.005$

A triangle has corners at (6 ,8 )(6 ,8 ), (1 ,2 )(1 ,2 ), and (3 ,9 )(3 ,9 ). What is the area of the triangle's circumscribed circle?