Radius of Inscribed Circle in Right Triangle

Consider a right triangle where one leg measures 6 units and the other leg measures 8 units. A circle is inscribed within this triangle. What is the radius of the inscribed circle?

To find the radius of the inscribed circle (also known as the inradius), you can use the formula:

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

where:

- $ A $ is the area of the triangle,

- $ s $ is the semi-perimeter of the triangle, given by $ s = \f\frac{a + b + c}{2} $ with $ a $ and $ b $ being the lengths of the legs and $ c $ being the length of the hypotenuse.

First, compute the hypotenuse using the Pythagorean theorem:

$$ c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 $$

Next, find the area of the triangle:

$$ A = \f\frac{1}{2} \times base \times height = \f\frac{1}{2} \times 6 \times 8 $$