A triangle has corners at $(4, 4)$ $(4 ,4 )$ , $(8, 9)$ $(8 ,9 )$ , and $(3, 1)$ $(3 ,1 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-09-05T08:55:23

Area of circumscribed circle is $584.99$ $584.99$

If the sides of a triangle are $a$ $a$ , $b$ $b$ and $c$ $c$ , 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 $(4,4)$ , $(8,9)$ and $(3,1)$ . This will be surely distance between pair of points, which is

$a=sqrt((8-4)^2+(9-4)^2)=sqrt(16+25)=sqrt41=6.4031$

$b=sqrt((3-8)^2+(1-9)^2)=sqrt(25+64)=sqrt89=9.4340$ and

$c=sqrt((3-4)^2+(1-4)^2)=sqrt(1+9)=sqrt10=3.1623$

Hence $s=1/2(6.4031+9.4340+3.1623)=1/2xx18.9994=9.4997$

and $Delta=sqrt(9.4997xx(9.4997-6.4031)xx(9.4997-9.4340)xx(9.4997-3.1623)$

= $sqrt(9.4997xx3.0966xx0.0657xx6.3374)=sqrt12.2482=3.4997$

And radius of circumscribed circle is

$(6.4031xx9.4340xx3.1623)/(4xx3.4997)=13.6458$

And area of circumscribed circle is $3.1416xx(13.6458)^2=584.99$

A triangle has corners at (4 ,4 )(4,4)(4 ,4 ), (8 ,9 )(8,9)(8 ,9 ), and (3 ,1 )(3,1)(3 ,1 ). What is the area of the triangle's circumscribed circle?