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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:

 You only need elementary trigonometry and scalar products Given any right-angled triangle $\Delta ABC$ on a sphere of unit radius, right angled at $A$, and with lengths of sides $a, b$ and $c$, then Pythagoras' Theorem in Spherical Geometry is $$\cos a = \cos b \cos c.$$ Prove this result. Find a triangle containing three right angles on the surface of a sphere of unit radius. What are the lengths of the sides of your triangle? Use the Pythagoras' Theorem result above to prove that all spherical triangles with three right angles on the unit sphere are congruent to the one you found.
To find out more about Spherical Geometry read the article 'When the Angles of a Triangle Don't Add Up to 180 degrees.