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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?

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.

### Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

# Coordinated Crystals

##### Stage: 5 Challenge Level:

A certain crystal, $X$, is formed from two types of atom, $A$ and $B$. The atoms of $A$ are found at all the points, and only the points, with coordinates $(l, m, n)$ for any whole numbers $l, m, n$; the atoms of $B$ are found at all the points, and only the points, with coordinates $(l+0.5, m+0.5, n+0.5)$.

Think about the geometry of this crystal. Can you visualise its structure? Can you devise a clear pictorial representation? How simply can you describe its structure in words?

How close are the various $A$ and $B$ atoms to each other? What bond angles are formed?

What crystal structure does this represent?

Can you represent any other crystal structures in a similar way?

NOTES AND BACKGROUND
Developing an understanding of the symmetry properties of crystals leads to insights into many of the chemical and physical properties of chemicals. Due to the mathematical constraints of three-dimensional geometry, there are a limited number of possibilities for the symmetry structures. You can read more about this topic at http://en.wikipedia.org/wiki/Unit_cell#Unit_cell