Question

A triangle has corners at `(9 ,4 )`, `(3 ,2 )`, and `(5 ,8 )`. What is the area of the triangle's circumscribed circle?

Answer 1

`color(orange)("Area of Circum circle " A_R = pi R^2 = 39.27 " sq units"`

Explanation:

`"Area of Triangle " = A_T = (a b c) / (4 R)`

`A (9,4), B (3,2), C(5,8)`

`a = sqrt((5-3)^2 + (8-2)^2) = sqrt 40 = 6.3246`

`b = sqrt((5-9)^2 + (8-4)^2) = sqrt 32 = 5.6569`

`c = sqrt((9-3)^2 + (2-2)^2) = sqrt 40 = 6.3246`

`"Semi perimeter of the triangle " s = (a + b + c ) / 2 = 9.153`

`A_T = sqrt s (s-a) (s-b) (s - c))`

`A_T = sqrt(9.153 (9.153 - 6.3246) (9.153 - 5.6569) (9.153 - (6.3246)) = 16`

`R = (a b c ) / (4 * A_T) = (sqrt40 * sqrt32* sqrt40) / (4 * 16) = 3.5355`

`color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 3.5355^2 = 39.27 " sq units"`