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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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A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

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Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

Zig Zag

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

Why do this problem :

There is an important configuration of four congruent right-angled triangles which make a square in two ways. Either the sides of the square are formed from each triangle's hypotenuse, or from putting together the two lengths adjacent to the right angle in the triangle. This problem uses two of the four matching points from each configuration. The other two points are not shown and so exploring this problem will draw students into discovering this arrangement and also provides experience in how a situation may sometimes present as only part of a more meaningful or useful 'whole'.

Four Right Angle Triangles

Possible approach :

Ask students to guess whether the four points indicated do or do not lie on a straight line, especially once the triangles are free to move as a general right-angled triangle.

There is scope for an activity to draw accurately the configuration shown in the problem, but with right-angled triangles which students form for themselves.

Alternatively use a Dynamic Geometry package, but in either case move the group into a discussion about whether these four points form one straight line or not.

The aim, as the exploration continues, is to cause some high-energy discussion shot-through with geometric reasoning, where students challenge each other to say why they are sure that this is one line. And as they do that the configuration illustrated above is discovered along the way.

Key questions :

  • What does this diagram show ?

  • Are the four points aligned or just close to it, and would that answer change if the arrangement used right-angled triangles of another proportion ?

  • How can you be sure of that ?

  • What perspective would you try to share with others to help them see and enjoy the relationships within this arrangement ?

Possible extension :

Squirty is a good companion problem if this activity is pursued using standard geometric construction with straight edge and compasses.

Possible support :

Less able students will gain much by making and then using congruent triangles to make patterns based on the configuration shown above.