Question

A triangle has corners at `(7 , 9 )`, `(3 ,7 )`, and `(4 ,8 )`. What is the radius of the triangle's inscribed circle?

Answer 1

Incenter radius `r = A_t / s = 1 / 4.5243 = color(green)(0.221)`

Explanation:

Given the coordinates of the three vertices of a triangle ABC,
the coordinates of the incenter O are

Given A(7,9), B(3,7), C(4,8)

Using distance formula we can calculate a, b, c.

`a = sqrt((4-3)^2 + (8-7)^2) = 1.4142`

`b = sqrt((9-8)^2 + (7-4)^2) = 3.1623`

`c = sqrt((7-9)^2 + (3-7)^2) = 4.4721`

Semi perimeter `s = (a + b + c)/2 = (1.4142 + 3.1623 + 4.4721)/2 = 4.5243`

Area of triangle `A_t = sqrt(s (s-a) (s-b) (s-c))`

`A_t = sqrt(4.5243(4.5243-1.4142)(4.5243-3.1623)(4.5243-4.4721)) = 1`

Incenter radius `r = A_t / s = 1 / 4.5243 = color(green)(0.221)`