This problem offers students a valuable context in which to visualise the effect of constraints (the fact that the centre of the circle is on the hypotenuse). They can be encouraged to establish the relationships within the context (for example by utilising properties of similar triangles), and then find a way to use those to make a route to a solution, which might include working backwards as well as algebraic manipulation.
This printable worksheet may be useful: Sitting Pretty
Invite the group to visualise the circle, perhaps rolling into place. Draw attention to its size and position and identify the constraints under which this circle exists.
You might wish to invite learners to try fixing the radius of the circle (at 3cm say) and constructing triangles around it.
Ask students to create their own diagram. The questions below at the right moment may help to steer the thought process with a light touch. Adding lines that represent the constraints of the situation will be helpful, so students must feel free to redraw their diagrams until they are happy that they have arrived at a good representation.
Some students might not immediately be ready for this because they are not sufficiently familiar with the similar triangle context, but it is more likely that the algebra demand is too great. It may be that this problem works well as a 'guided example' where the teacher still asks questions, and shares the 'blocks', which at each point need to be identified and a way ahead sought, but takes the group on a pre-determined journey to the solution. Perhaps asking learners to work with some real values to explore possibilities and verify the relationship for these cases.
Partly Circles is another challenging problem where students can apply their understanding of circle geometry.