Question

A triangle has sides with lengths of 5, 8, and 3. What is the radius of the triangles inscribed circle?

Answer 1

There exists no triangle with the given measurements because the given measurements form a straight line.

If you still want to calculate the are then I have explained below.

If `a, b and c` are the three sides of a triangle then the radius of its inscribed circle is given by

`R=Delta/s`

Where `R` is the radius `Delta` is the are of the triangle and `s` is the semi perimeter of the triangle.

The area `Delta` of a triangle is given by

`Delta=sqrt(s(s-a)(s-b)(s-c)`

And the semi perimeter `s` of a triangle is given by
`s=(a+b+c)/2`

Here let `a=5, b=8 and c=3`

`implies s=(5+8+3)/2=16/2=8`

`implies s=8`

`implies s-a=8-5=3, s-b=8-8=0 and s-c=8-3=5`

`implies s-a=3, s-b=0 and s-c=5`

`implies Delta=sqrt(8*3*0*5)=sqrt0=0`

`implies R=0/8=0`

Mathematically, since the area comes out to be zero therefore it proves that it is a straight line it is not a triangle, and since there is no triangle so there is no in-circle and thus no in-radius.