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Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Giant Holly Leaf

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).


Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why use this problem?
The problem offers the opportunity to make a conjecture about how the two perimeter lengths change and then for learners to try to prove their own conjetures.

The problem offers a non standard application of the use of the formula for arc length and an application of one of the main circle theorems.

Possible approach
Suggest learners experiment with the interactivity and make their own conjectures.

Key question
If the moving line turns through an angle, what is the connection between this angle and the changes in arc lengths on either side of the moving line?