Question

A triangle has corners at `(3 , 3 )`, `(1 ,2 )`, and `(5 ,9 )`. What is the radius of the triangle's inscribed circle?

Answer 1

`"radius of incircle" ~~ 0.602`

Explanation:

Denote the lengths of the sides of the triangle as:
`color(white)("XXX")a`: between `(3,3)` and `(1,2)`
`color(white)("XXX")b`: between `(1,2)` and `(5,9)`
`color(white)("XXX")c`: between `(5,9)` and `(3,3)`
and also
`color(white)("XXX")s` as the semi-perimeter `=(a+b+c)/2`

Then the radius of the triangle's inscribed circle is
`color(white)("XXX")r=("Area"_triangle)/s`
or using Heron's formula
`color(white)("XXX")r=(sqrt(s(s-a)(s-b)(s-c)))/s`

Using the Pythagorean Theorem (and a calculator)
`color(white)("XXX")a=sqrt((3-1)^2+(3-2)^2)~~2.236`
`color(white)("XXX")b=sqrt((1-5)^2+(2-9)^2)~~8.062`
`color(white)("XXX")c=sqrt((5-3)^2+(9-3)^2)~~6.325`

`color(white)("XXX")s~~8.311`

`"Area"_triangle=sqrt(s(s-a)(s-b)(s-c))=5`

`"radius of incircle" = 5/8.311 ~~0.602`