Partly Circles

Age 14 to 16

Challenge Level Yellow star

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

This printable worksheet may be useful: Partly Circles.
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 images in Getting Started might help stimulate discussion.

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?

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 support

Use some, or all, of the images in the Getting Started section 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.

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.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ