In a right triangle, one leg measures 6 units, and the other leg is 8 units. A circle is inscribed within the triangle, touching all three sides. Determine the radius of the inscribed circle (inradius) of this triangle.

Recall that the formula to find the inradius $r$ of a triangle is given by:

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

where $A$ is the area of the triangle and $s$ is the semi-perimeter. The semi-perimeter $s$ can be calculated using:

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

Here, $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. Find the length of the hypotenuse using the Pythagorean theorem.