A circle is inscribed in a triangle. The triangle has side lengths of 60, 80, and 100.?

This triangle is a right triangle, you can use the pythagorean theorem to verify that. If you have a right triangle that is inscribed in a circle then the diameter of the circle is equal to the hypotenuse. In your triangle the hypotenuse is 100 so the diameter of the circle is also 100 and the radius is 50.

This is an ideal Pythagorean triangle:

3 (*20) // 4 (*20) // 5 (*20) <===> 3/4/5

the inscribed circle will have a radius:

r = 1/(Ha^-1 + Hb^-1 + Hc^-1)

where Hc = 2* Area∆ / Hypotenuse = 2*2400/100 = 48

therefore:

r = 1/(1/60 + 1/80 + 1/48) = 20

r = 20