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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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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?

Running Race

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Imagine a circle of diameter $1$ unit, and another circle of a similar diameter, and another.

What is the total perimeter of these three circles (you can leave your answer with $\pi$ in it)?

What is the perimeter of a circle of diameter $3$ units?


Now imagine two circles - one of diameter $2$ units and one of diameter $3$ units.

What is the total perimeter of these two circles?

What is the perimeter of a circle of diameter 5units?


What do you notice?

Is this generally the case with combinations of circles whose total diameters are equal?

Can you explain why?

This problem is taken from the UKMT Mathematical Challenges.