Question

A triangle has corners at `(3 , 3 )`, `(1 ,2 )`, and `(5 ,4 )`. What is the radius of the triangle's inscribed circle?

Answer 1

The three points lie on same straight line and hence no distinct triangle can be formed. In other words, one could say that radius of inscribed circle is `0`.

Explanation:

If the sides of a triangle are `a`, `b` and `c`, then the area of the triangle `Delta` is given by the formula

`Delta=sqrt(s(s-a)(s-b)(s-c))`, where `s=1/2(a+b+c)`

and radius of inscribed circle is `Delta/s`

Hence let us find the sides of triangle formed by `(3,3)`, `(1,2)` and `(5,4)`. This will be surely distance between pair of points, which is

`a=sqrt((1-3)^2+(2-3)^2)=sqrt(4+1)=sqrt5=2.236`

`b=sqrt((5-1)^2+(4-2)^2)=sqrt(16+4)=sqrt20=4.472` and

`c=sqrt((5-3)^2+(4-3)^2)=sqrt(4+1)=sqrt5=2.236`

As `a+c=b`, it is apparent that the three points lie on same straight line and hence no distinct triangle can be formed.

In other words, one could say that radius of inscribed circle is `0`.