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## 'Partly Circles' printed from http://nrich.maths.org/

### Why do this problem?

This problem brings together some key ideas including the value of
similar triangles, employing useful lines, some algebra and that
problems can be solved even when information seems limited. The
problem can be used to make sense of other people's mathematical
arguments. The aim of bringing the three problems together is to
encourage discussion about their links and connectedness and
therefore investigate relationships between problems and
problem-solving experiences.

### Possible approach

The first problem is designed to draw attention to similar
triangles and to see these learners need to add lines.

Present the problem and give time for reflection and then
discussion. If ideas are not forthcoming suggest the addition of
two lines (you might add them to the diagram without discussion)
and then use of angles in the same segment. If necessary spend some
time adding and removing pairs of lines talking about what is known
about the triangles created:

- They are right-angled
- They have equal angles
- They are similar

Consider other ways of writing the result or substitute integers
for three of the letters asking what the fourth must be (if a=$3$,
b=$5$ and c=$6$ what must d be?)

The next two problems can be tackled in any order and the
accompanying

Word 2003 file has
images that might stimulate additional discussion if
needed.

Once solutions for all three problems are available spend time
discussing connections learners can make and those that are worth
drawing particular attention to.

### Key questions

- What lines can you draw that highlight additional
properties
- Can you identify similar triangles?
- Why don't you need to worry about $ \pi$?
- What do the first and scond problems have in common?
- How about the second and last problems?
- Can you think of other problems where you have used similar
ideas?

### Possible extension

Allow learners time to work on the problems with limited
support and focus on what they see as "key moments" that helped
them to solving the problems.

### Possible support

Use some, or all, of the

guidance sheets to
help learners in structuring their own solutions using the images
and statements on offer. Then focus on what is the same and what is
different about each of the three problems.

Or

Bull's
Eye