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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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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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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.

Efficient Packing

Age 14 to 16 Challenge Level:

Why do this problem ?

This problem will allow students to engage with calculations of areas of disks, often used as an approximation to real objects in science. They will also practice visualisation skills.
To find the best answer students will need to resort to trigonometry, although this is not necessary to make a satisfactory attempt at the question.

Possible approach

Whilst students might be familiar with the concept of the area of a disk they might be unable immediately to see how to apply this to the question in hand. Some discussion might facilitate this. Note that there are two 'obvious' different ways to pack disks. Which will work best? Discuss this. When they do the packing calculation, be sure to note that students must be clear as to exactly how may rows and columns of disks will fit into the grid. They can do this exactly (using trigonometry) or by drawing an accurate diagram and taking a measurement. Or, perhaps, they will produce some other valid method.

Key questions

  • Describe your method of packing the disks clearly in words.
  • How many different sensible packing methods might you try?
  • Can you be sure that each disk is completely contained within the confines of the square?
  • What order of magnitude checks could you make to test that your answer is sensible?

Possible extension

Can students repeat the question by filling an equilateral triangle with side 1m?

Possible support

Provide coins or counters for the students physically to work with.

Students might struggle with the 'open' nature of the questions. To begin, they might like to read the Student Guide to Getting Started with Rich Tasks