Question

A triangle has corners at `(1 ,1 )`, `(7 ,9 )`, and `(4 ,2 )`. What is the area of the triangle's circumscribed circle?

Answer 1

Area of triangle's circumscribed circle is `143.0`

Explanation:

Vertices of triangle are `A(1,1), B(7,9) , (4,2)`
Side `AB=a=sqrt((1-7)^2+(1-9)^2)= 10.0`
Side `BC=b=sqrt((7-4)^2+(9-2)^2)= 7.6`
Side `CA=c=sqrt((4-1)^2+(2-1)^2)= 3.16`

Semi perimeter of triangle `S=(10.0+7.6+3.16)/2=10.38`

Area of the triangle `A_t=sqrt(s(s-a)(s-b)(s-c))`
`=sqrt(10.38(10.38-10.0)(10.38-7.6)(10.38-3.16)) = sqrt79.17=8.9`

Circumscribed circle radius is `R=(a*b*c)/(4.A_t)`
`R=(10.0*7.6*3.16)/(4*8.9) =6.74`

Area of circumscribed circle is `A_c =pi*R^2=pi*6.74^2=143.0`