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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?

### Flexi Quad Tan

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.

# Pythagoras on a Sphere

##### Stage: 5 Challenge Level:

 All angles are in radians. (1) Without loss of generality take coordinate axes so that $A$ is the point$(0,0,1)$, the xz-plane contains the point $C$ and the yz-plane contains the point $B$. (2) Thinking of $A$ as the North Pole then $C$ has latitude $u$ and longitude 0 and $B$ has latitude $v$ and longitude $\pi/2$. (3) Find the 3D coordinates of $B$ and $C$. Where the origin O is the centre of the sphere ${\bf OA, OB}$ and ${\bf OC}$ are vectors of unit length. (4) Use scalar products and vectors ${\bf OA, OB}$ and ${\bf OC}$ to find the lengths of the arcs $AB, BC$ and $CA$ in terms of $u$ and $v$. The required result follows.