A triangle has corners at $(7 ,3 )$ , $(5 ,8 )$ , and $(4 ,6 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-05-24T10:10:12

Area of triangle's circumscribed circle is $25.16$

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 $(7,3)$ , $(5,8)$ and $(4,6)$ . This will be surely distance between pair of points, which is

$a=sqrt((7-5)^2+(3-8)^2)=sqrt(4+25)=sqrt29=5.385$

$b=sqrt((4-5)^2+(6-8)^2)=sqrt(1+4)=sqrt5=2.236$ and

$c=sqrt((7-4)^2+(3-6)^2)=sqrt(9+9)=sqrt18=4.243$

Hence $s=1/2(5.385+2.236+4.243)=1/2xx11.864=5.932$

and $Delta=sqrt(5.932xx(5.932-5.385)xx(5.932-2.236)xx(5.932-4.243)$

= $sqrt(5.932xx0.547xx3.696xx1.689)=sqrt20.3693=4.513$

And radius of circumscribed circle is

$(5.385xx2.236xx4.243)/(4xx4.513)=2.83$

And area of circumscribed circle is $3.1416xx(2.83)^2=25.16$

A triangle has corners at (7 ,3 )(7 ,3 ), (5 ,8 )(5 ,8 ), and (4 ,6 )(4 ,6 ). What is the area of the triangle's circumscribed circle?