Question

A triangle has sides with lengths of 5, 9, and 4. What is the radius of the triangles inscribed circle?

Answer 1

Notice that `5+4=9`, which means this is not a real triangle, but a segment with length `9` divided in two parts, `5` and `4`.
So, we cannot talk about inscribed circle.

Explanation:

However, it would be educational to know how to solve this problem in general for real triangles.

Assume, we have a triangle with sides `a`, `b` and `c`. If the radius of an inscribed circle is `r`, the area of this triangle is, obviously,
`S = 1/2(a+b+c)*r`

On the other hand, this same area, according to Heron's formula, is equal to
`S = sqrt(p(p-a)(p-b)(p-c))`,
where `p=(a+b+c)/2`.

From this we can derive an equation
`1/2(a+b+c)*r = sqrt(p(p-a)(p-b)(p-c))`

Solving the above for `r`, we obtain the radius of an inscribed circle:
`r = 2sqrt(p(p-a)(p-b)(p-c))/(a+b+c)`