### Why do this problem ?

This
problem challenges students to apply what they know about
angles in triangles. The nine-peg circle allows students to
concentrate on the geometrical structure without having to worry
about the arithmetic. It offers a good preparation for the problems

Subtended angles and

Right angles which lead towards the circle theorems.

Teachers may find the article

Angle Measurement: an Opportunity for Equity of interest.

### Possible approach

Demonstrate how the geoboard works - clicking on a coloured rubber
band, dragging it onto a peg and then "stretching" it out onto two
more pegs to make a triangle.

"How many different triangles can
you make which consist of the centre point and two of the points on
the edge? "

"Can you find all the angles of
the triangles?"
At some stage students will want to know what makes triangles
"different". For the purposes of this exercise, triangles that are
congruent are considered to be the same.

After a short time bring the class together to discuss findings.
Issues that you could touch upon:

- caclulating angles at the centre for different numbers of
pegs
- rules for finding the base angles of the isosceles
triangles

Draw a triangle on the geoboard with three vertices on the
edge of the circle which encloses the centre point.

"Can you find all the angles
of this triangle?"

How this builds on their previous work may not be immediately
obvious to most students. After a few minutes, if no progress is
being made that can be shared, add one isosceles triangle inside
the original triangle (see the

solution ). A further isosceles triangle can be added if
needed.

Discuss efficient methods.

"Now draw all the possible
triangles with three vertices on the edge of the circle and find
their angles."

At some stage discuss how to cope with triangles that do not
enclose the centre.

This

handout
presents the whole problem on one sheet.

### Key questions

What do we know already that might be useful here?

### Possible extension

The

virtual geoboard environment allows you to create triangles in
circles with a variable number of pegs. The virtual geoboard
resource also has links to sheets with printed circles with
different numbers of pegs, a full set of these sheets can also be
found

here.

Ask students to work out the angles of triangles in 10-peg,
12-peg, 15-peg ... circles.

Lens Angle
provides an interesting challenge that requires students to apply
the properties of triangles in circles.

### Possible support

Before moving on to the main activity, students find the angles of
triangles with a vertex at the centre in circles with different
number of pegs on the edge.

Students can start by finding the angles of triangles in 9-peg and
12-peg circles which contain the centre of the circle. They can
then move on to triangles that do not contain the centre of the
circles.

Printable sheets with all the possible triangles in 9-peg and
12-peg circles can be found

here and

here .

Sethe extension section of these notes for other useful
links.