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A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

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As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

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Coordinated Crystals

Age 16 to 18 Challenge Level:

Why do this problem?

This problem is a useful exercise in visualisation and 3D vectors. It provides a natural context for the mathematics and has many extension possibilities. It is an example of a problem where a clear geometric image really facilitates the work with vectors and where the use of the scalar product really facilitates the calculation of the angles.

Possible approach

Initially focus on trying to understand the atomic structure. Encourage discussion and the drawing of diagrams? Share these. Which ways of thinking about the atomic structure are the simplest and clearest?

To understand how close the various atoms are to each other requires clear thinking. It will be easiest to think in terms of each atom $A$ surrounded by a 'box' of $B$ atoms, in which case it will be easier to see which distances, and therefore angles, are possible.

The extension concerning the other crystal configurations is mathematically very interesting.

You could consider structures well known from chemistry or encourage students to research the idea following the link from the problem.

Key questions

What sort of atom lies at the origin?
What is the configuration of all of the $A$ atoms or all of the $B$ atoms?
What angle is formed between the atom at the origin and its two closest neighbours?

Possible extension

Consider creating a version of the problem with face-centred cubic packing, where the first challenge is to determine an algebraic form of the location of the different atoms.

Possible support

Focus on the central atom and its nearest neighbours only.