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Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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Quads

The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?

Right Angles

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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 are going to be working at individual computers, 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.

If students will be working on paper ask them to draw right-angled triangles on their 9-peg and 12-peg circles.

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.