A circle lies inside a quarter-circle, as shown below. The circle is tangent to side AO and arc AB. Find the radius of the circle.

Call the center of the  small circle  M

Let N be the point of tangency between the small circle and OA

Let r be the radius of the small circle

Triangle  OMN  is  right

OM = 3 - r

MN = r

ON = 2 

OM^2 = MN^2 + ON^2

(3 -r)^2  = r^2 + 2^2

r^2 - 6r + 9  = r^2 + 4

6r  = 5

r = 5/6