### Why do this problem?

This problem builds on

Triangles in Circles and

Subtended Angles . It leads to the theorem about right-angles in circles.

Teachers may find the article

Angle Measurement: An Opportunity For Equity to be of interest.

### Possible approach

If students will be working on paper ask them to draw right-angled triangles on their

9-peg and

12-peg circles. Alternatively, they could work on the interactive pegboard.

How many different triangles can they find?

How do they know they are right-angled?

What is special about the right-angled triangles?

Draw together conjectures which might mention the number of dots on the circle and the need to be able to join two points to form a diameter. Challenge students to justify these conjectures with convincing arguments. Eventually link this to the work on

Subtended Angles .

### Key questions

What do we know already that might be useful here?

What is the relationship between the angle at the centre and the angle at the circumference?

What are the implications of our findings for circles in general (without dots)?

### Possible extension

Ask students to prove the general case.

### Possible support

Students may need to spend more time convincing themselves when it is possible to make right-angled triangles.

Here is a sheet with 10-peg, 12-peg and 16-peg circles to support them in investigating other cases.