A triangle has sides with lengths of 5, 9, and 8. What is the radius of the triangles inscribed circle?

The radius ($r$) of the inscribed circle in a triangle can be found using the formula:

$r = \frac{A}{s}$

where $A$ is the area of the triangle and $s$ is the semi-perimeter of the triangle, given by:

$s = \frac{a + b + c}{2}$

where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

To find the area of the triangle, we can use Heron's formula:

$A = \sqrt{s(s - a)(s - b)(s - c)}$

Given that the lengths of the sides of the triangle are $a = 5$, $b = 9$, and $c = 8$, we can calculate $s$ and $A$:

$s = \frac{5 + 9 + 8}{2} = 11$

$A = \sqrt{11(11 - 5)(11 - 9)(11 - 8)}$ $= \sqrt{11(6)(2)(3)}$ $= \sqrt{396}$

Now, we can find the radius $r$:

$r = \frac{\sqrt{396}}{11}$

$r \approx \frac{19.9}{11}$

$r \approx 1.81$

So, the radius of the inscribed circle in the triangle is approximately $1.81$.