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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?


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem :

This problem could work well as a 'poster' - a visual challenge placed where students will see it. Or presented at the end of a lesson as something to try to solve.

Possible approach :

This printable worksheet may be useful: Semi-detached.

This problem can worked well as something short and closed, but there is also an opportunity to invite questions which open up beyond the initial challenge. Finding the area of a square in a quadrant or squares fitted in the space between either of the two squares in the main problem and the circle.

Key questions :

  • What is the challenge and how might you start ?
  • How did you do it ? Can you explain ?
  • How might you extend this problem ?
  • Can you calculate the area of squares fitted into other places within this diagram ?

Possible extension :

The approach suggested above indicates one route to extension within this context, or for another challenge fitting squares into shapes try Squirty. The problem Semi-square offers another opportunity to work out areas of squares inside circles.

Possible support :

Tilted Squares could be an excellent and accessible challenge for slightly less experienced students.