Question

A triangle has corners at `(6 , 3 )`, `(3 ,5 )`, and `(2 ,9 )`. What is the radius of the triangle's inscribed circle?

Answer 1

`color(cyan)("Radius of incircle " r = A_t / s = 5 / 7.47 = 0.67 " units"`

Explanation:

`"Incircle radius " r = A_t / s`

`A(6,3), B(3,5), C(2,9)`

`a = sqrt((3-2)^2 + (5-9)^2) = sqrt17`

`b = sqrt((6-2)^2 + (3-9)^2) = sqrt52`

`c = sqrt((6-3)^2 + ( 3-5)^2) = sqrt13`

`"Semi-perimeter " s = (a + b + c) / 2 = (sqrt17 + sqrt52 + sqrt13) / 2 = 7.47`

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

`A_t = sqrt(7.47 (7.47-sqrt17) (7.47 - sqrt52) (7.47 - sqrt13)) = 5`

`color(cyan)("Radius of incircle " r = A_t / s = 5 / 7.47 = 0.67 " units"`