### Why do this problem :

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.

### Possible approach :

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.

- What possible values can x and y have and does the relationship
hold in these cases?
- Do they notice anything about the relationship between x and
y?
- How many possible values of x and y can there be, and does the
relationship hold for any of them?

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.

### Key questions :

- Look at your diagram, what's there that might help ?
- What's there anyway ?
- Paying attention to any triangles in your diagram, what do you
think about the relationship between them. What might be true
?
- Is it ? Why ?
- How might that help ?
- What is it that this problem asks us to find out ?

### Possible extension :

The
Line and Its Strange Pair
### Possible support :

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.