Why do this problem?
This problem leads towards the theorem about the relationship
between the angle subtended by ("hanging from") an arc at the
centre of a circle and the angle subtended by the same arc at the
circumference. It's worth noting that once students know this
relationship, they will be able to deduce that angles subtended by
an arc at the circumference are all equal.
The nine-peg circle allows students to concentrate on the
geometrical structure without having to worry about the arithmetic.
A special case of this problem is
Teachers may find the article
Angle Measurement: an Opportunity for Equity
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.
Draw a triangle that encloses the centre, split it into three
isosceles triangles and find the angles.
An example may look like this:
"The angles around the centre add
up to 360 degrees and the angles at the vertices add up to 180
degrees. What else do you notice about angles at the centre and
angles at the vertices?"
Ask students to take a look at their working for
Triangles in Circles
to identify general rules that work in all
cases. Encourage students to consider all the different possible
angles at the circumference subtending a specific arc.
When the angle at the centre is a reflex angle, students may need
some help to see that the angle at the circumference is still half
the angle at the centre. For example:
allows you to create triangles in circles with
a variable number of pegs. Ask students to check that the
relationship between angles at the centre and angles at the
circumference holds in 10 peg, 12 peg, 15 peg ... circles.
What is the relationship between the angle at the centre and the
angle at the circumference?
What do we know already that might be useful here?
What are the implications of our findings? (How can other circle
theorems be deduced from this one?)
Students can work with several ready-made triangles like the
one below and asked to compare angles ACB and ADB. The sheet of nine-peg
could be used to draw these.