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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

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Can you work systematically to prove this?

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Now try changing the number of points round the edge.
Can you do it now?

Can you show by calculation that the angle is a right angle?
What do you notice about the side of the triangle opposite the right angle?
Try this with other numbers of points round the edge.
When is it possible to make a right-angled triangle?
In this interactivity, the points are equally spaced around a circle. Imagine that they are not.
Can you explain the conditions which will give a right-angled triangle?
Can you prove this?

For printable sets of circle templates for use with this activity, please see Printable Resources page.

Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board.
The boards, moulded in crystal clear ABS that can be used on an OHP (185 cm in diameter), together with a teacher's guide, are available from Geoff at Education Initiatives