Question

A triangle has corners at `(2 , 4 )`, `(8 ,2 )`, and `(1 ,3 )`. What is the radius of the triangle's inscribed circle?

Answer 1

`0.5402`

Explanation:

Introduction: Law of Cotangents

To solve this question you will need to know about the `"Law of Cotangents"`.

Here is an article from Wikipedia about this law.

In brief, the states that the radius of a triangle's inscribed circle is

For a inscribed circle in a triangle with radius `r` and triangle with sides `a`, `b` and `c`:

`r = sqrt [ ((s - a)(s - b)(s - c)) / s ]` where `s = (a + b + c) / 2`

You can find the proof for this law in the Wikipedia article.

Step One: Find side `a`

`a = sqrt((8-2)^2+(2-4)^2) = sqrt(40)`

Step Two: Find side `b`

`a = sqrt((1-8)^2+(3-2)^2) = sqrt(50)`

Step Three: Find side `c`

`a = sqrt((2-1)^2+(4-3)^2) = sqrt(2)`

Step Four: Find the value of `s`

`s = (a + b + c) / 2 = (sqrt(40)+sqrt(50)+sqrt(2))/2 = sqrt(10) + 3sqrt(2)`

Step Five: Use the equation!

`r = sqrt [ ((s - a)(s - b)(s - c)) / s ] = 0.5402`

Assuming that you know how to use equations, I didn't put my working for using the equation.

Hope that makes sense!