Question

A triangle has corners at `(2 , 2 )`, `(1 ,3 )`, and `(6 ,4 )`. What is the radius of the triangle's inscribed circle?

Answer 1

The radius is approximately 0.54.

Explanation:

There is a formula to calculate the radius of inscribed circle based on the perimeter and area of the triangle.
If the area is `A` and the perimeter is `P` the radius of the circle is `r=2A/P`.
So our goal is to find area and perimeter.

First the perimeter, it is given by the distance of the points in pairs.
Lets call the three points `p_1=(2,2)`, `p_2=(1,3)`, `p_3=(6,4)` the length of the three sides are

side between `p_1` and `p_2` is
`s_1 = d(p_1, p_2)=sqrt((2-1)^2+(2-3)^2)=sqrt(2)\approx1.41`

side between `p_2` and `p_3` is
`s_2 = d(p_3, p_3)=sqrt((1-6)^2+(3-4)^2)=sqrt(26)\approx5.1`

side between `p_3` and `p_1` is
`s_3 = d(p_3, p_1)=sqrt((6-2)^2+(4-2)^2)=sqrt(20)\approx4.47`

The perimeter is then `P=s_1 + s_2 + s_3 = 1.41+5.1+4.47=10.98`
The Area can be calculated with the formula
`A=sqrt(P/2(P/2-s_1)(P/2-s_2)(P/2-s_3))`
`=sqrt(5.49(5.49-1.41)(5.49-5.1)(5.49-4.47))`
`=sqrt(5.49*4.08*0.39*1.02)`
`=sqrt(8.91)`
`A\approx2.98`.

Now it is time for the initial formula

`r=2A/P=2*2.98/10.98\approx0.54`.