A triangle has corners at $(4 ,7 )$ , $(3 ,4 )$ , and $(6 ,9 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-06-04T07:50:27

Area of circumscribed circle is $133.47$

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 $(4,7)$ , $(3,4)$ and $(6,9)$ . This will be surely distance between pair of points, which is

$a=sqrt((3-4)^2+(4-7)^2)=sqrt(1+9)=sqrt10=3.1623$

$b=sqrt((6-3)^2+(9-4)^2)=sqrt(9+25)=sqrt34=5.831$ and

$c=sqrt((6-4)^2+(9-7)^2)=sqrt(4+4)=sqrt8=2.8284$

Hence $s=1/2(3.1623+5.831+2.8284)=1/2xx11.8217=5.9109$

and $Delta=sqrt(5.9109xx(5.9109-3.1623)xx(5.9109-5.831)xx(5.9109-2.8284))$

= $sqrt(5.9109xx2.7486xx0.0799xx3.0825)=sqrt4.0014=2.0004$

And radius of circumscribed circle is

$(3.1623xx5.831xx2.8284)/(4xx2.0004)=6.518$

And area of circumscribed circle is $3.1416xx(6.518)^2=133.47$

A triangle has corners at (4 ,7 )(4 ,7 ), (3 ,4 )(3 ,4 ), and (6 ,9 )(6 ,9 ). What is the area of the triangle's circumscribed circle?