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Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

Hexy-metry

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

Why do this problem?

This problem requires the solver to reason geometrically and make use of symmetry. By re-presenting the information in a different way, for example by adding additional lines (a useful technique in geometrical problems) more structure can be revealed. It is an interesting idea that adding something, and therefore apparently making it more complex, can sometimes make a problem more accessible. Then of course there is an opportunity to use the cosine rule in a non-standard context.

Possible approach

Use the images in this document to make cards.

Display the problem and ask learners to work in groups rearranging the cards in ways which help to make connections for them.

Share ideas and relationships that groups notice before going on to solve the problem. Encourage careful reasoning and convincing arguments. For example:

  • "Why can you say those two angels are equal?

There are several ways of solving the problem so share different approaches and discuss what helped to move thinking forward.

Two observations may be worth drawing attention to :

  • the reflection symmetry between $a$ followed by $b$ and $b$ followed by $a$,
  • the angle at the circumference as half the angle subtended at the centre onto the same arc (in this case an arc greater than a semicircle) .

Key questions

  • How can you use the symmetry of the figure to determine some of the angles ?
  • What things in the images helped to make connections for you?
  • What do we need to find, and can that be found directly ?

Possible extension

For a similar level challenge try Bendy Quad

And for something much harder try Biggest Bendy

Possible support

For less able students concentrate on the interesting property that all angles in the hexagon are equal. This might be approached practically : drawing a circle, split into thirds to emphasise the symmetry, then splitting each third into the same two unequal proportions before measuring all six angles.

Ask the students if they can explain why these should all be equal .