A triangle has corners at $(4 ,6 )$ , $(2 ,9 )$ , and $(8 ,4 )$ . What is the area of the triangle's circumscribed circle?

Question

1332 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-15T09:49:32

Area of circumscribed circle is $194.5068$

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 $(4,6)$ , $(2,9)$ and $(8,4)$ . This will be surely distance between pair of points, which is

$a=sqrt((2-4)^2+(9-6)^2)=sqrt(4+9)=sqrt13=3.6056$

$b=sqrt((8-2)^2+(4-9)^2)=sqrt(36+25)=sqrt61=7.8102$ and

$c=sqrt((8-4)^2+(4-6)^2)=sqrt(16+4)=sqrt20=4.4721$

Hence $s=1/2(3.6056+7.8102+4.4721)=1/2xx15.8879=7.944$

and $Delta=sqrt(7.944xx(7.944-3.6056)xx(7.944-7.8102)xx(7.944-4.4721)$

= $sqrt(7.944xx4.3384xx0.1338xx3.4719)=sqrt16.01=4.0013$

And radius of circumscribed circle is

$(3.6056xx7.8102xx4.4721)/(4xx4.0013)=7.8685$

And area of circumscribed circle is $3.1416xx(7.8685)^2=194.5068$

A triangle has corners at (4 ,6 )(4 ,6 ), (2 ,9 )(2 ,9 ), and (8 ,4 )(8 ,4 ). What is the area of the triangle's circumscribed circle?