Question

A triangle has corners at `(9 ,3 )`, `(4 ,9 )`, and `(2 ,8 )`. What is the area of the triangle's circumscribed circle?

Answer 1

61.34 approx

Explanation:

Radius R of the circumcircle of a triangle is given by the formula

`R= (abc)/(4Delta)` where a,b,c are the side lengths and `Delta` is the area of the triangle. The three sides of the triangle in the present case would be

`sqrt((9-4)^2 +(3-9)^2) = sqrt 61`

`sqrt((9-2)^2 +(3-8)^2)=sqrt74`

`sqrt((4-2)^2 +(9-8)^2)= sqrt5`

Area of triangle can be had using the formula
`1/2 [x_1 (y_2-y_3)+x_2(y_3-y_1)+x_2(y_1 -y_2]`

=`1/2[9(9-8) +4(8-3) +2(3-9)]`= 8.5

Radius R =`sqrt(61*74*5) /(4(8.5))`

Area of the circumcircle would be `pi (61*74*5)/34^2`= 61.34 aprox