Question

A triangle has corners at `(3 ,7 )`, `(2 ,5 )`, and `(5 ,4 )`. What is the area of the triangle's circumscribed circle?

Answer 1

`color(maroon)("Area of circumcircle " = A_c = pi * R^2 = 10.52`

Explanation:


`A(3,7), B(2,5), C(5,4)`

`a = sqrt ((x_b-x_c)^2 + (y_b - y_c)^2)`

`a = sqrt((2-5)^2 + (5-4)^2) = sqrt10`

Similarly, `b = sqrt((5-3)^2 + (4-7)^2) = sqrt13`

`c = sqrt((3-2)^2 + (7-5)^2) = sqrt5`

`"Semiperimeter " = (a + b + c) / 2 = (sqrt10 + sqrt13 + sqrt5) /2 = 4.5`

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

`A_t = sqrt(4.5 (4.5 - sqrt10) (4.5 - sqrt13 ) (4.5 - sqrt5)) = 3,49`

`"circum-radius " = R = (abc) / (4 A_t)`

`R = (sqrt10 * sqrt13 * sqrt5) / (4 * 3.49) = 1.83`

`"Area of circumcircle " = A_c = pi * R^2 = pi * 1.83^2 = 10.52`